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(p ∨ (q ⇒ r)) ∧ (p⇒ ¬q) ∧ (r ⇒ q) ∧ (¬q ⇒ r).

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(¬p∧ q ⇒ p)⇔ p∨ q g6]N p = 02q = 1F[I

((¬0∧ 1⇒ 0)⇔ 0∨ 1) = ((1∧ 1⇒ 0)⇔ 1) = ((1⇒ 0)⇔ 1) = (0⇔ 1) = 0

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p q ¬p ¬p ∧ q ¬p ∧ q ⇒ p p ∨ q (¬p ∧ q ⇒ p)⇔ p ∨ qø ø Ö ø Ö ø øø Ö Ö Ö ø Ö øÖ ø ø ø Ö Ö ÖÖ Ö ø ø Ö Ö Ö

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p ∧ q ⇒ prH¢IWAN\Y7Y^>QgGK|IWôN,:=DJIW:=IWuN\^2 b Y7

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p q p ∧ q p ∧ q ⇒ pø ø ø Öø Ö ø ÖÖ ø ø ÖÖ Ö Ö Ö

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23 = 8KM257YB=:=<?>Wr©TN\YguNo<Y^257Y464uNKM25mYXY798

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¤Er313257`KMD6B=IW:;FfrJB=<?>ED6S6=f^>WbJY7<YguN\46257p ∧ q ⇒ p

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p ∧ q = 12p = 0

r¢s57AWIW4625S646A\X98=N8;7Y:;FD6B[NfKMg6<Y2KRNGb ZWgG>QIWuN8;798x:=A6PhN\g64625AC2`:[eD6B[NfKMg6<Y2K|7\bMXY<?>E52

p = q = 1rM1sN\:=<Y7QD6B=<?>ED6S6:=<YXY<f7Y46257

g6IWD6B=ILK#N\g6<Y2hPiI4uN\:x<fNaF[7Y^g6I:=D6B=<Y7YXY<Y46IW=Xf2$/p = 0

2p = 1

1b|XY<?>E52@46257ÎWZEPiId>ED6B[NfKMg6<Y2K|7\r FOe\g,<YguN\46257

p ∧ q ⇒ p8;7Y:;FMFON\SGF[IW5IWZW25eGr

k :=ACB=VCXY257T4uN\:=<Y7B=IW<YS6ÎLK#N\46257`ÎWY4uN<fN\D625:[N\@FON\ABb

p ∧ q ⇒ pøÖ

p = 0p = 1 q = 1

D6B=<Y7YXY<Y46IW=Wr

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kl4625IW:=A\IK#N\46257 D6B=<Y7?D6B[ILK#N\g6<YIW467|g6B=S6ZW25^:=DJIW:=IWJ7Y^8;7Y:;FHD6B=<?>EA6PhN\g67Y^J! ,Xc %RV+W&oX[Q * rCHIW57YZdNÌW4o4uNTFU>E^,bdY7s<fN\A6PhN\guN\^>WbCY73g6ILK#ICg6<YIW4uNTF[7Y<fN46257M<fN\XO_6IEg6<Y2

ñpø þ ³O³´ ?µ ®,¶¯w³!·

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28;7Y:;F346257?KR>E^257YB=4uNGr

NCPiVWY^>,46257KMD6B=IW:;FfbY7525XY<YuN √24625738;7Y:;F`46257?KR>E^257YB=4uNGbwXY<?>E52 bJY7s8;7Y:;FT525XY<Yue

KR>E^257YB=4ueGr¢U:=F[46257U8=e<fNaF[7Y^ 525XY<Yd>lXfNCPiA\IKM2F[7a2bFON\AC257\bHY7 √

2 = ab

IWB[N\<a2

b:[exKM<YZW5mYg646257D6257YB;KM:=<Y7TûY/hXY<?>E52SJPhN\^7YA a

b

8;7Y:;F46257Y:=ACB[N\XfN\54d>Ë1r¡¢I,DcIEXY2]eCZdNGbwY72 = (

√2)2 = a2

b2

bdXY<?>E522b2 = a2 rWkl4625IW:=ACSa8;7Y^>:;FOe\gbCY73525XY<?uN a2 8;7Y:;F|DuN\B=<?>E:;FONGrk0F[7YgG>8;7Yg64uN\A525XY<YuNaB=VLKM46257Y8;7Y:;FxDuN\B=<?>E:;FON2325:;F[4625798;7p525X?<YuNQXfNCPiAWILKM2FON

cb

FON\AN@Y7a = 2c

rakVLK|XY<fN\:2b2 = (2c)2 b\XY<?>E52 b2 = 2c2

rW¢IEg6IW646257 8=N\ADJIWD6B=<Y7Yg64625IKM4625IW:=ACSa8;7Y^>WbLY7 525XY<YuN

b8;7Y:;F DuN\B=<?>E:;FONGr¡ I8;7Yg64uN\A@:;F[IW2\K:=D6B=<Y7YXY<Y46IW[XY2C<y N\AdF[7Y^,b

Y7 525XY<Yd>a2b:[eMKM<YZ\5mYg646257 D6257YB;KM:=<Y7\r\NaF[7Y^ 525XY<YuN √

246257 ÎWY7 C>E KR>E^257YB=4uNGr

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p⇒ (q ∨ (p⇒ ¬r)) 8;7Y:;FMFON\SGF[IW5IWZW25eGr¢IEg6IW646257|8=N\AvDcIWD6B=<Y7Yg64625IÎWY7Y^>v<YB=IW625@F[I4uNgGK#N:=DJIW:=IWd>WrÖ rM¡¢N\J7Y5AaNGr6¡ K#I\B[<>E^>xFON\c7Y5A\m`K#N\B;F[IW=XY2 5IWZW2XY<Y4C>EXO_4uN\:=<Y7YZWI<YguN\462]NGr

p q r . . . p⇒ (q ∨ (p⇒ ¬r))ø ø ø. . .

Öø ø Ö

. . .Ö

ø Ö ø. . .

Öø Ö Ö

. . .Ö

Ö ø ø. . .

ÖÖ ø Ö

. . .ø

Ö Ö ø. . .

ÖÖ Ö Ö

. . .Ö

k25guN\abY7g6]Np = r = 1

2q = 0

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p ⇒ (q ∨ (p ⇒ ¬r))øD Ö ø

q = 0ø

p = 1ø

r = 1

k25guN\\bRY7g6IW:=<Y525=^>g6Ip = r = 1

2q = 0

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• B[NfK|IKR>uPhe\XY<YIW467YZWIo=B=ICg6AaNGr

p ∨ ¬p

ñdù þ ³O³´ ?µ ®,¶¯w³!·

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a2bbaFON\AC257|Y7

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¨MIW<?K#N\Y^>525XY<YJm(√

2)√

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a = b =√

2r6z\7Y=52J<fN\

8;7Y:;F346257?KR>E^257YB=4uNGbEF[I(

(√

2)√

2)√

2

= (√

2)√

2·√

2 = (√

2)2 = 2

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√2)

√2 2 b =

√2r

• B[NfK|I:=D6B=<Y7YXY<Y46IW=Xf2 r¬(p ∧ ¬p)

• B[NfK#N25g67Y^DcI\F[7Y4CF[46IW=XY2!N\F[7YB=4uNaF9>CKR>x2A\IW4625S646AWX98;2 r

p ∨ p⇔ p p ∧ p⇔ p

• B[NfK|IDJICgGK|V\8;46798R467YZdN\X98;2 r

p⇔ ¬(¬p)

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(p ∨ q) ∨ r ⇔ p ∨ (q ∨ r), (p ∧ q) ∧ r ⇔ p ∧ (q ∧ r)

• B[NfK#ND6B=<Y7Y^257Y4646IW=XY2 N\F[7YB=4uNaFU>CKR>x2!A\IW4625S646A\X98;2 r

p ∨ q ⇔ q ∨ p, p ∧ q ⇔ q ∧ p

• B[NfK|IB=IW<Yg6<Y257Y546IW=XY2AWIW4625S646A\X98;2KM<YZW5mYg67Y^N\F[7YB=4uNaFU>CKR>Wr

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• B[NfK|ID6B=<Y7YXO_6IEg64625IW=XY2 25^D625AN\X98;2¢/h<?K#N\467TF[7YTD6B[NfK#7Y^%:;>E5IWZW25<Y^SB1r

(p⇒ q) ∧ (q ⇒ r)⇒ (p⇒ r)

• B[NfK|I7Y525^24uN\X98;2w25^D62AaN\X98;2 r

(p⇒ q)⇔ ¬p ∨ q

• B[NfK|I7Y525^24uN\X98;2wB=VLKM46ILKRN\Y46IW=XY2 r

(p⇔ q)⇔ (p⇒ q) ∧ (q ⇒ p)

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• B[NfK#Ng67n,IWB=ZdN\4uNGr

¬(p ∨ q)⇔ ¬p ∧ ¬q, ¬(p ∧ q)⇔ ¬p ∨ ¬q

• B[NfK|I467YZdN\X98;2!25^D652AN\X98;2 r

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ñ%6 þ ³O³´ ?µ ®,¶¯w³!·

• p ∨ 0 = p, p ∨ 1 = 1, p ∧ 0 = 0, p ∧ 1 = p

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R = (p ∨ (q ⇒ r)) ∧ (p⇒ ¬q) ∧ (r ⇒ q) ∧ (¬q ⇒ r).

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R⇔ (p ∨ ¬q ∨ r) ∧ (¬p ∨ ¬q) ∧ (¬r ∨ q) ∧ (q ∨ r)⇔⇔

(

((p ∨ r) ∧ ¬p) ∨ ¬q)

∧ ((¬r ∧ r) ∨ q)⇔⇔

(

((p ∧ ¬p) ∨ (r ∧ ¬p)) ∨ ¬q)

∧ (0 ∨ q)⇔⇔

(

(0 ∨ (r ∧ ¬p)) ∨ ¬q)

∧ q ⇔⇔ (r ∧ ¬p ∧ q) ∨ (¬q ∧ q)⇔ (r ∧ ¬p ∧ q) ∨ 0⇔⇔ ¬p ∧ q ∧ r

NaF[7Y^B[N\DJIWB;F!FO7?4M8;7Y:;FHD6B[NLKMg6<Y2K#>/ NRKM257YB=<?>E^>N\:=D625B[N\4dF[IKM2m1JFU>E5A\IMKRF[7YgG>WbZWgG>

p = 02q = r = 1

rL|<?>E52d;KM2]N\g67YAs46257d>uPE<fN\:;F[B[N\:=<YIW4d>Wba¦37Y46B;>@DJIWDJ7Pi462hP:[N\ÎWcV\8;:;FUK|I6buNF[7Y:;FON\^7Y4dF3ICg64uN\57Y<Y25IW46I6r

¤Ersz\7Y=52JXO_6IEg6<Y2wIN\4uN\525<Ym3DJIW57?^25AD6B=<Y7YgGKR>EJIWB=XY<?>GXO_!bGF[IDcIWAaN\Y7Y^>WbGY7@D6B=<?>O-4uN8;^4625798gGK#Na8DuN\46ILKM257^>E5252#:=25mKICXY7Y46257,:;K|IW25X_A\IW4dF[B=AaN\46gG>EguNaF[VLKTrnpN\^>vg6IB=IW<YDuNaF[B=<Y7Y462]ND6B[NfKMg6<Y2K|7<YguN\4627

P = (p⇒ ¬q) ∧ (q ⇒ ¬r) ∧ (r ⇒ ¬p). F[IW:=Sa8=e\X`7Y525^24uN\X98;m@25^D652AaN\X98;2wg6IW:;FON8;7Y^>

P ⇔ (¬p ∨ ¬q) ∧ (¬q ∨ ¬r) ∧ (¬r ∨ ¬p).NCPiVWY^>T46257 KMD6B=IW:;FfbW?7RD6B=<?>E4uN8;^4625798¢gGK#N@:=DJIW=B=VCg<YguN\

p, q2r:[esD6B[NLK2-

g6<Y2K|7\r¢¥@<Y4uN\XY<fNxF[I6b!Y7D6B=<?>E4uNa8;^4625798@gGK#N:=DJIW=B=VEgQ<YguN\ ¬p,¬q 2 ¬r :[ey NCPi:=<?>CK|7\rCs57RKRF[7YgG>@8;7Yg64uN<MDJILKR>EY:=<?>GXO_NaF[7YB=4uNaF9>CK^S6:=2ud>EMy NCPi:=<?>CKRNGb<fNaF[7Y^%XfNCPi7`<YguN\46257

PF[7Y\rJ¥sF[B=<?>EôN\4uN:=D6B=<Y7YXY<Y46IW=AWI\6XY<?>,g6ILK|VEgr

ñCò6587 ³d¯w³ µ ª5³ ñ91sN<fN\A\IW6XY<Y7Y46257T<YB=IW625^>`8;7Y:=<YXY<Y7R8;7Yg64ueSdK#N\ZWm\rc|<YmY:;F[IK ôNaF[7YôNaFU>EXY7@:=DJI,-

FU>EAaN\^>:=25ms<MF9KM257YB=g6<Y7Y462]N\^2uDJIW:;FON\XY2p⇒ q

bCZWg6<Y257\bED6B=<?>EDcIW^462 8;^>\bpF[I<fNCPiIWY7=-

462]NGbNqF[IýF[7Y<fNGr':«X¢+W&[w*S.&RV+W&"¹ %X¢[Q ,(RV'," g6IF[7YZWIFUKM257YB=g6<Y7Y462]N4uN\<?>CK#N\^>

FUKM257YB=g6<Y7Y46257q ⇒ p

rcN\SCK#N\Y^>oFO7?BONa<\buY7<YguNa46257

(p⇒ q)⇔ (q ⇒ p)

RV+W&H8;7Y:;FFON\SGF[IW5IWZW25eGr\n,25ÎsF[7YZWI`:;F[S6g67Y4dF[IW^Æ<YguN\B=<fN@:=25m#I@F9>E^<fN\DJIW^254uN\|2G^>E52<fNCPiIWY7Y46257<F[7Y<feGbwNoFUKM257YB=g6<Y7Y46257<FUKM257YB=g6<Y7Y46257Y^g6Iv46257YZWIoIEgGKMB=I\F[4d>E^,r1sN\57Y?>IF9>E^%DuN\^25m?FON\@2!<Yg67YX>Eg6IK#N\46257:=25m`F[7YZWIKR>E:;F[B=<Y7YZdN\\r

ØÍÙ; < Íàx'YÖ r ã ]NxDJIW4625Y:=<?>EXO_<YguN\Q:=D6B[NLKMg6<Y2\b XY<?>Q254GyhIWB=ôNaX98=N

p = 08;7Y:;FKR>E:;FON\B=XY<fN.-

8=e\XfNg6IKR>E525XY<Y7Y462]NQK#N\B;F[IW=XY2@5IWZW25XY<Y46798<YguN\462]Nl<PiIWYIW467YZWI6rMz\7Y=523FON\Acb F[IKR>E<Y4uN\XY<?>ERF[m|KRN\B;F[IW=\b8;7Y=52G4627\baF[I@DJIWAaN\<fN\abWY7#IW6257|K#N\B;F[IW=XY2G:[e@ÎWY52K|7\r/ NO1

(p⇒ q)⇒ rk

/hB1p ∧ (q ⇒ r)

k/ X1 ¬(p ∨ q)⇔ ¬p ∧ ¬q k/hgB1

p ∧ q ⇒ p ∨ r r¤Er3©@F[VWB=7<4uNfKM2]N\:=VLKKDcIW4625Y:=<?>EXO_ýKR>EB[N\Y7Y462]N\XO_ÎWY4uNoIWD6S6=XY246257<Y^257=-462]N8=e\X`:=7Y46:=SB3/ NO1

(p ∧ (q ∨ (¬r)))⇒ ((¬s)⇔ (r ∨ q)) k/hB1(p⇒ r)⇒ (p ∧ q) k/ X1((p ∨ q) ∨ r) ∧ (¬s) ∧ ¬(r ∨ s) r

ÚGr D6B[NfKMg6<Y25\bEY7MKR>E^257Y4625IW467Mg6IF[798 DJIWB;>D6B[NLK#NB[N\XO_CS646ACSx<YguNav25:;F[I\F[46257M:[eFON\SGF[IW5IWZW25N\^2 r

ñ r D6B[NfKMg6<Y25\buY7TDJIW4625Y:=<Y7TD6B[NLK#NB[N\XO_CS646ACSp<YguN\,:[eFON\SGF[IW5IWZ\2]N\^2 r/ NO1

(p⇒ q)⇔ (¬q ⇒ ¬p) /hD6B[NfK#IF[B[N\46:=DJIW<?>EX98;2>=1=k/hB1p ∧ (p⇒ q)⇒ q

/ D6B[NfK|IICg6B;>CKRN\462]Np1k/ X1

p ∧ (p ∨ q)⇔ p, p ∨ (p ∧ q)⇔ p/hD6B[NfK#NDcIEXO_JPhN\462]N\462]NO1=k

/hgB1 ¬p⇒ (p⇒ q)/hD6B[NLK|I ã S646:[N <YA\I\FONO1r

?@%@=£.ÈWz¢§.©iwz| ª £W©i=§px.¥m¡my_

ñp² þ ³O³´ ?µ ®,¶¯w³!·

Ý rswuN\guN\\b6XY<?>DcIW4625Y:=<Y7<YguN\462]N:[eFON\SGF[IW5IWZW25N\^2 r/ NO1

(p⇒ q)⇔ (q ⇒ p)k

/hB1p ∨ q ⇒ p ∧ q k/hX*1(p⇒ q) ∧ (p⇒ r)⇒ (p⇒ q ∧ r) k/hgB1((p⇒ q)⇒ r)⇔ (p⇒ (q ⇒ r))

k/h7*1

((p⇒ q)⇒ r)⇒ (p⇒ (q ⇒ r))k

/iy;1(p⇒ q)⇒ (p ∧ r ⇒ q)

k/hZ!1

(p⇒ q ∨ r)⇒ (p⇒ q) ∨ (r ⇒ q)k

/h_B1(p ∧ q ⇒ r) ∧ (p ∧ q ⇒ ¬r)⇒ (¬p ∧ ¬q ∧ ¬r) r

åGrRk ^25798;:=XY7M<Y4uN\ACSKM:;FONfKM253<Y^257Y464ueT<YguN\4625ILK#e

p, q5S6

rFON\Acbdd>I\F[B=<?>O-

ôN\467`<YguN\46257`<PiIWYIW467Td>uPiIFON\SGF[IW5IWZW25eGr/ NO1

p⇒ (¬q ⇒ )k

/hB1(p⇒ q) ∧ r ⇒ p ∨

rì rRk÷^25798;:=XY7@<Y4uN\ACS

KM:;FONLKM25@:;>E^cIW:=DJVa8;4625AaN5IWZW2XY<Y467YZWIFON\AcbGC>vIaFOB=<?>O-

ôN\467<YguN\46257<PiIWYIW467od>uPiIpFON\SGF[IW5IWZW25eGrwk D6B=<?>EDuN\g6ACSKM25mYXY7984625`8;7Yg646798ÎWY2K|IW=XY2DJICguN\`KM:=<?>E:;F[AE257\r/ NO1

p (p ∨ q) k/hB1((p⇒ q)¬q)⇒ ¬p r

×Gr313257YXO_p, q, r

IW<Y4uN\X?<fNa8=eTDJ7?KM4673<YguN\462]NGbCNp′, q′, r′

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p, q, r, p′, q′, r′1r

/ NO1(q ⇒ r ∧ p) ∨ ¬r k/hB1 ¬(p ∨ r)⇒ (q ⇒ p)

k/hX*1

p⇒ (q ⇒ r)k

/hgB1(p ∧ q)⇔ r

ró rsN\A6PhN\guN\^>WbGY7IDJ7?KM46798R525XY<Y6257@4uNaF[S6B[N\546798

nKM257Y^>Wb6Y7

• n 8;7Y:;F3DJICg6<Y257Y54uND6B=<Y7Y< ñ 2• 8;7Y=52 n 8;7Y:;F3DJIEg6<Y257Y4uND6B=<Y7Y<¤EbGF[I n 8;7Y:;F3DJICg6<Y257Y54uND6B=<Y7Y<ÚGr

|<?>x:;FOe\gKR>E4625AaNGbuY7T525XY<YuNn8;7Y:;F3DJICg6<Y257Y54uND6B=<Y7Y< Ö ¤p3

ñCò6587 ³d¯w³ µ ª5³ ñ Å

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/hB1zW7?[2R:;F[IWDuNýD6B=IEXY7Y4dF[ILKRN46257xS657YZW46257v<Y^2]N\46257\b F[IýKM<YB=IW:=4ueK#>EguNaF[AC2B[<Ye\g6IK|75S6ÊDJIa8=NLKM2:=25mJ7?<YB=IWJIEXY257\rz\7Y=52KR>EguNaF[AC2oB=<fe\g6IK|746257KM<YB=IW:=4ueGb F[IDJIEguNaF[AC2<YIW:;FON\4ue,IW64625YIW467\r¢z\7Y=52DJICguNaF[AC2 <YIW:;FON\4ueIWC-4625YIW4672 :;F[IWDuN,D6B=ICXY7Y4CF[ILK#Np46257S657YZW46257<Y^2]N\4627\bwF[IpJ7Y<?B[I\JIEXY257:=25m46257TDcI\8=NfKM2 rcNaF[7Y^æKR>EguNaF[AC2 B=<fe\g6ILK|7TKM<YB=IW:=4ueGr

/ X1,zW7?[2x + 3 =

√3− x b|F[I x2 + 6x + 9 = 3 − x

b KM25mYXx = −65S6

x = −1r|NaF[7Y^ 525XY<Yd> −6

2 −1:[eQB=IW<?KM2]e\<fN\462]N\^2|B=VLKM4uN\462]N

x+ 3 =√

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Ö ¤Er3©@F[VWB=7:=DJIW=B=VCgl<YguN\@bp ⇒ p

b(p ⇒ p) ⇒ p

b((p ⇒ p) ⇒ p) ⇒ p

brrr :[eFON\SGF[IW5IWZW25N\^2½3

Ö ÚGr3©3IWB=<?>E:;FONa8=e\X3<3DJICguN\4C>EXO_vD6B[NfK0B[N\X_CS646ACSv<YguN\oDJIWAaN\<fN\abEY7s:=DcV\8;4625AC2c5IWZW2l-XY<Y467 ∧,⇒,⇔ :[eog67?êu4625ILKRNa5467<fNoDJIWÎCXfe:=DcV\8;4625A\VLK ∨ 2 ¬ /hXY<?>E52 DJIWAaN.-<fN\\bGY7@<YguN\462]N

p∧ q b p⇒ q2p⇔ q

:[eB=VLKM46ILKRN\Y467`<?guN\4625IW^,bCK AdF[VWB;>EXO_KR>E:;F[mYD6Sa8=eFU>E5AWI:=DcV\8;4625AC2 ∨ 2 ¬ 1r

Ö ñ r33<fN\:[N\g64625\bY7|467YZdN\X98=N346257!8;7Y:;FHg67?êu4625ILKRN\54uNMD6B=<Y7Y<#N\F[7YB=4uNaFU>CK|m|2CA\IW4625S646Ap-X98;m\r

ÖLÝ r33<fN\:[N\g64625\b!Y725^D62AaN\X98=Nv46257s8;7Y:;Fg67?êu4625ILK#N\54uNxD6B=<Y7Y<oN\F[7YB=4uNaF9>CK|m2A\I,-4625S646A\X98;m\r

Ö åGr ã 7?êu4625Sa8;7Y^>x:=DJVa8;4625A | <?K#N\4d>A),[Q&;;)paBAËeV&DCo&[w#4uN\:;F[mYD6Sa8=e\XYIËbp|q ⇔ ¬p ∨ ¬q.

1sN\D625:[N\sFON\c7Y5AWmsF[7YZWI:=DcV\8;4625ANGr63<fN\:[N\g64625\bGY7`:=DJVa8;4625AC2 ¬,∨,∧,⇒,⇔ :[eg67?êu4625ILK#N\5467T<YNDJIWÎCXfeF[7YZWI:=DcV\8;4625ANGr

ÖLì r ã 7?êu4625Sa8;7Y^>x:=DJVa8;4625A ⊥ <?K#N\4d>AVd P RV+_),+W&"FEc+W&[QU;&G #4uN\:;F[mYD6Sa8=e\XYIËbp⊥q ⇔ ¬p ∧ ¬q.

ñp× þ ³O³´ ?µ ®,¶¯w³!·

1sN\D625:[N\3FON\J7Y5A\msF[7YZWI:=DJVa8;4625AaNGr63<fN\:[N\g64625\bGY7`:=DJVa8;4625AC2 ¬,∨,∧,⇒,⇔ :[eg67?êu4625ILK#N\5467`<fNDJIW^IEXfeF[7YZWI:=DJVa8;4625AaNGr

Ö ×Grsn,VLKM25^>WbY7gGK#Ný:=DJVa8;4625AC2 2 :[eB=VLKM46ILK#N\Y467\b8;7Y=52|FON\SGF[IW5IWZW25e8;7Y:;F<YguN\46257p q ⇔ p q r3<fN\:[N\g64625\bY7s8;7Y:;F Ö åvDuN\B[N\^2¢46257YB=VKM46ILK#N\Y4d>EX_:=DcV\8;4625A\VLKgGKMSuN\B=ZWS6^7Y4dF[ILKR>EX_r

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¤ ø r ã 7?êu4625Sa8;7Y^>x:=DcV\8;4625AO<?K#N\4d> #%^l&[=R#%(',XZaX¢'*ºa!UQS*# P a!UQaLM4uN\:;F[mYD6Sa8=e\XYIËb

pOq ⇔ (¬p ∧ q) ∨ (p ∧ ¬q).1sN\D625:[N\sFON\c7Y5A\m@F[7YZWI:=DJVa8;4625AaNGr6¢IWAaN\<fN\\b6Y7`DJI\4625Y:=<Y7T<?guN\462]N:[eFON\SGF[IW5I,-ZW2]N\^2 r/ NO1 ¬(pOp)

k/hB1

pOq ⇔ qOpk

/hX*1(pOq)Or ⇔ pO(qOr)

k/hgB1

pOq ⇔ ¬(p⇔ q)r

MNw¤( ypOq

*W2|2~;y |ïPOm= %p,= %

q Q ¦¥xz y!xw¥ .*v*xzypm ª ¡mwv*£,2xÈ *¥ Ô|2z y T¤W¥yl±

Ê++!SR

TVUµÊ)x

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A1

<fN\D625:=Sa8;7Y^>a ∈ A.

¸¹ºWá!ÓÈÇ Àu¼2 ∈ N, −5 ∈ Z, 1

4∈ Q,

√3 ∈ R r

ã ]N<fN\<Y4uN\XY<Y7Y462]NGb Y7aRV+W& P &;E&^_&"&RV&".<Y625IWB=S

A/aRV+W&ÒR#%^_&Q¸T'g6I

A1

D625:=<Y7Y^>a 6∈ A. g

¸¹ºWá!ÓÈÇ Àu¼ −1 6∈ N, 276∈ Z, π 6∈ Q r

k FU>E^^25798;:=XYS,4uN\57Y?>S6XY<?>E4625uN\B=g6<YIo25:;F[I\F[4ueSdK#N\ZWm\rw¢IW46257?K#N\TK :=<YA\IW574uN3IWZWVEPGôN\^>`g6IsXY<?>E46257Y462]NM<Y7|<Y625IWB[N\^2W525XY<YJILKR>E^2d5S646Dr<Y625IWB[N\^2WD6S646AdF[VKsWÈ¥xzy ¤W¥y

a 6∈ A¡m (ª ©W~=Wz| ¬(a ∈ A)

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(0, 0), (1, 1)2(9, 10)

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)

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ÚGr313257YXO_PIW<Y4uN\XY<fNý<Y625VWB525XY<Y4uNaF[S6B[N\54d>EX_DuN\B=<?>E:;FU>EXO_b N

N<Y625VWB525XY<Y

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P = n ∈ N : 2|n, N = n ∈ N : ¬2|n.

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(a,+∞) = x ∈ R : x > a k

ùO6 74X ª5±E¬õ

• Vd*º¨[Q *i#! ,"Ó),RË+W# (−∞, b] = x ∈ R : x ≤ b[a,+∞) = x ∈ R : x ≥ a r

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Φ(x)ÎWY7Y^>vIWACB=7Y=525`<Y625VWB

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a ∈ Φ(x) : x ∈ A ⇔ a = Φ(x)g6]NDJ7?KM467YZWI

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B ⊆ Cr FOe\g¼/hDJI\46257?KRN\

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³

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A ⊆ B ⇔ A \B = ∅.ãx¾E¿ ¼ wZWIEg646257x<,D6B[NfK#7Y^ 7Y525^24uN\X98;2#B=VLKM46IK#N\Y46IW=XY2MKR>E:;FON\B=XY<?>DcIWAaN\<fN\gGK#NKR>E4625AN\462]NGr(⇒)

npN\^>DJIWAN\<fN\\bJY7A \ B = ∅ rwNCPiVWY^>x46257TKMD6B=IW:;FfbcY7 A \ B 6= ∅ bcXY<?>E52Y725:;F[4625798;7

x ∈ A \ B rJk0F[7YgG>p<g67?êu4625X98;2¢B=VWY4625X?>p<Y625IWB=VLK KR>E4625AaNGbwY7 x ∈ A 2x 6∈ B rus57T<<fNCPiIWY7Y462]NKM27Y^>WbuY7 A ⊆ B

buXY<?>E5268;7Y=52x ∈ A b6F[I x ∈ B rwNaF[7Y^KR>CKM4625IW:=A\ILK#N\525=^>WbCY7

x ∈ B 2 x 6∈ B r6¥sF[B=<?>EôN\4uN:=D6B=<Y7YXY<Y46IW[À\IW6XY<?>og6IK|VEgD6257YB;KM:=<Y7YZWIKR>E4625AN\462]NGr(⇐)

¡ >E^ B[N\<Y7Y^ ôN\^>S6<fN\:[N\g64625\bdY7A ⊆ B

rENCPiVWY^>46257RKRD6B[I\:=FfbdY7A 6⊆ B

bXY<?>E52 Y725:;F[4625798;7

xbcFON\AC257Y7

x ∈ A 2 x 6∈ B rcs57TF[IoIW<Y4uN\XY<fNGbJY7 x ∈ A \ B bJXYI8;7Y:;F3:=D6B=<Y7YXY<Y467<T<fNCPiIWY7Y4627Y^,b6Y7A \B = ∅ r g q

zWN\ApPhNaFUK|Io:=DcIW:;F[B=<Y7YX\bwg6<Y2]NCPhN\46257`B=VWY4625X?>x46257#8;7Y:;Fs:;>E^7?F[B;>EXY<Y467WbcF[<Y4rc4uNIWZWVEPA \B 6= B \ A ru¡¢7U8#5K#N\gG>dT46257`ôN4uN\:;F[mYD6467\bcuN\B=g6<YI25:;F[I\F[467Tg6<Y2]NdPhN\46257\rs ½Æ;~.v¯Wz£À|2z£,oxz©WI.yg ª ©W~*¥(¡½Æxzy u ª yVi=*W þy y (p ⇔ q) ⇔ (¬p ⇔ ¬q)

i=©i¥xz§% ª =xwIg=,z A 6⊆ B ⇔ A \ B 6= ∅OýCDxw¤«|2z£,xz©WI.yg¢§%.v*I.£.y T¡ ª I§.©Wxzwv iQy £T|v*Q*v*xzy .= "pzx ª £.y w¥xz£.T¤W¥y ª ©Wxz i=£.y ïx¢§*©iQ@Zz y |2y £,¥ ¡my!©W~;£.Q@=z£.¤W¥y_

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ãx»dè Ò Á Ðî?À¾£Ô+W&;U;eA+BÊ=QOa ! ,Xc ,^lRV',"Ð+ÈS*ÊT+W ,[w#%"Ð+(Î~±©(y-/+Èì`p*í¿z®|0Oæj&Kp®r/+¢í

S*ÊT+W ,[Qd,XA+BR#*ST',XZ#%"Ò'ÀSÊT+Wd%[

A4B = x : (x ∈ A ∧ x 6∈ B) ∨ (x ∈ B ∧ x 6∈ A) = (A \B) ∪ (B \ A).

N\SdK#N\Y^>\baY7x8;7Y:;F7Y57Y^7Y4dF[7Y^B=VWY4625X?>T:;>E^7?F[B;>EXY<Y46798

A4B KRF[7YgG>2dF9>E5A\IKRF[7YgG>WbuZ\gG>8;7Y:;FM7Y57Y^7Y4dF[7Y^æg6IWA6PhN\g646257 8;7Yg6467YZWI<Y7`<Y625IWB=VLKA2Br6B[NLKMgue38;7Y:;F

B=VLKM46257Y\buY7x ∈ A4B ⇔ x ∈ A O x ∈ B,ZWg6<Y257

O8;7Y:;F:=DcV\8;4625AC257Y^ 5IWZW25XY<Y4d>E^ <Yg67?êu4625ILK#N\4C>E^ KjIW:;FONaF[4625^ <fN\guN\4625S-DcI,-

D6B=<Y7Yg646257YZWIvB=IW<Yg6<Y2]NCPiSrc¡HN<fN\57YY46IW=uN\B=g6<YIv4uN\^:=25mD6B=<?>EguNvD6B=<?>,g6ILK#ICg6<Y7Y4625SK@PhN\:=46IW=XY2 B=VWY4625X?>v:=>E^7?F[B=>EX?<Y467;8?rwN\RIEg6DJILKM257Yg6462!g62]N\ZWB[N\^Fµ 7Y464uNF[I

³ ´

³ 4 ´ã <Y2]NCPhN\462]N:=S6^>o2JD6B=<Y7YACB=I\8;Sx<Y625IWB=VKICg6DJILKM2]N\guNa8=e:=DJVa8;4625AWIW^æN\F[7YB=4uNaF9>CKR>

23A\IW4625S646A\X98;2 r|1sNaF[S6B[N\5467KR>EguN8;7:=25m,<fNaF[7Y^ Dd>CFON\46257pIlg6<Y2]NCPhN\46257<?KM2]e\<fN\467p<Y7:=DJVa8;4625AC257Y^ 467YZdN\X98;2 r z\7Yg64uN\AB=IW<YDuNaF[B;>CK#N\46257<Y625IWB=SýKM:=<?>E:;F[AE25XO_l7Y57Y^7Y4dF[VLKÊ462574uN\57Yfe\X?>EXO_g6IS6:;FON\5IW467YZWI<Y625IWB=S

A46257 8;7Y:;FMÎWY52K|74D6B=ILKRN\g6<?2F[I:=<?>E6A\Ig6I

:=D6B=<Y7YXY<Y46IW=Xf2 b\8=N\AWIY7T46257`25:;F[4625798;7`<Y625VWB#KM:=<?>E:;F[AE25XO_p<Y625IWB=VLKTrýFU>E^%D6B=IW657Y^7Y^%ÎWY7Y^>v:=IW6257#8;7Yg64uN\AvDJIWB[N\g6<Y25\rGk ôNaF[7YôNaFU>EXY7TXY<YmY:;F[I

B=IW<?K#N\fN\^>ýDJICg6<Y625IWB;>DJ7?KM467YZWI,S6:;FNa5IW467YZWIxK|XY<Y7Y=4625798T<Y625IWB=SXb!AdF[VWB;>ý4uN\<?>O-

K#N\^>vXY<fN\:=7Y^[wS.&;([wS.&RV+(a\rGk0F[7YgG>x:=7Y46:3ôN4uN\:;F[mYD6Sa8=e\XfNg67?êu4625X98=NGrãx»dè Ò Á Ðî?ÀÀ£Ô+W&;U;e

XÊ=Q.S+W&ZOi#%^_ ,RV',"åS*ÊT+W ,[Q&"$ÎuÁIéK40Kj+Èì`0/+Èì`0/JS*ÊT+W ,[=

A ⊆ Xk! Ô[wS.&;([wS.&RV+Xm$R#*ST',XZ#%"Ð'ÀS*ÊT+Wd,[

Ac = X \ A. g rq#N\B=g6<YIx25:;F[I\F[467s8;7Y:;Ffbd>DuN\^25m?FON\\bwY7g6IWDc7Pi46257Y4625738;7Y:;FÓS*#%XÈQS.&g6IDJ7?KM467U8

D6B=<Y7Y:;F[B=<Y7Y462 r ¢IW46257?K#N\<IW<Y4uN\XY<Y7Y462]NAc46257xKR>E4625ANGbw8=N\AaNF[Ix8;7Y:;FD6B=<Y7Y:;F[B=<Y7Yb

F[B=<Y7YuNx<fNfKM:=<Y7<fN\g6uN\\bd>8=N\:=46IvKR>E4625ANCPiIxF[Iv<A\IW4CF[7YAC:;F[SD6B=IK#N\g6<YIW4d>EX_B=IW<=-K#N\fN\r ã 2]N\ZWB[N\^Âµ 7Y464uNg6]NF[7YZWIg6<Y2]NCPhN\462]NKR>EZW]e\guN4uN\:;F[mYD6Sa8=e\XYIËbs _ §p=m ª y uÈi ª zxz£,¥xzz£.y A′

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c

Ã

k257Y^>l8;S6\bRY7Q7Y57Y^7Y4dFON\^2@<Y625IWB=VLKÎWZded>EQ<Y625IWB;>Wr#kò:=<YXY<Y7YZWVW546IW=XY2ÎWY7Y^>vB=IW<?K#N\fN\T<Y625VWB#KM:=<?>E:=F[AC25XO_pDJIEg6<Y625IWB=VLK guN\467YZWI<Y625IWB=Srãx»dè Ò Á Ðî?ÀÄ£Ô+W&;U;e

AÊ=Q*ST+W&¯! ,Xc ,^lRV',"áS*ÊT+W ,[Q&"$ÎÅÆOìwé&)0/ÇéVæ=ëé*, S*ÊT+W ,[=

AR#*ST',XZ#%"Ð'ÀS*ÊT+Wd,[EXÈQST'*m),+WU;e P &i`! ÔV *S*ÊT+W ,[;d,XÈ¿cUQST',^l+

P(A) = B : B ⊆ A.N\SdK#N\?>E525=^>o8;S6\b Y7<Y625VWBTD6S6:;F9>x8;7Y:;FDcIEg6<Y625IWB=7Y^.AN\Yg67YZWI<Y625IWB=SQIWB[N\<

Y7MAaN\YgG><Y625VWBw8;7Y:;F :;K|IW25^ÊK@PhN\:=4d>E^ DcIEg6<Y625IWB=7Y^,rCNaF[7Y^Æg6]N@g6IK|IW5467YZWIT<Y625IWB=SAôN\^>

∅, A ⊆ P(A).

¸¹ºWá!ÓÈÇ Àu¼!áÖ r P(∅) = ∅ k¤Er P(∅) = ∅, ∅ kÚGr P(1, 2, 3) = ∅, 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3 r¡¢B=<Y7YuN<?KMB=VEXY253SCK#N\ZWm\bEY7sc7Y<@g6I\6B[7?ZWI<YB=IW<YS6^257Y462]N`B=VWY4625X?>DcIW^25mYg6<?>7Y57=-

^7Y4dF[7Y^ NDcIEg6<Y625IWB=7Y^ /hXY<?>E52RDcIW^25mYg6<?>4uN\57YY7Y46257Y^ Ný<fNfKM257YB[N\46257Y^1DcI\8;mYXY257<Y625IWB=SQDcI\F[mYZWIK|7YZWI,ÎWY7:=D6B[NLKM2]N\:=DJIWB=7D6B=IW657Y^>Wr h N\AdF9>EXY<Y46257\b d>E7Y57Y^7Y4C-F[7Y^µ<Y625IWB=SQDJI\F[mYZWILK#7YZWIp<Y625IWB=S

AIW<Y4uN\XY<fNpd>EvDcIEg6<Y625IWB=7Y^

ArH|<?>E^ 25464d>E^

4uNaF[IW^2]Na:;FH8;7Y:;F346DruC>EXY257TDcIEg6<Y625IWB=7Y^%<Y625I\B[SDcI\F[mYZWIK|7YZWI6r13257YXYIMK|XY<Y7Y=4625798 DJIW<Y4uN\525=^>@DuN\B=m 46257YS6DJIWB=<fe\g6A\IK#N\4ueM7Y57Y^7Y4CF[VLK

a2bbLAdF[VWB[e

IW<Y4uN\XY<fN\52[^>8=N\A\I a, b ruU:;F[I\F[467d>uPiI6bcY7TAWIW5798;46IW=T<fN\D625:=S8;798R7Y57Y^7Y4dF[VK46257ZWB[NCPhNB[I\52 b6XY<?>E52 a, b = b, a rc¡¢7YB[N\<cmYg6<Y257Y^>DcI\F[B=<Y7YJILKRN\52!2546467YZWIDJI\8;mYX?2]NDuN\B;>WbAdF[VWB=7|JmYg6<Y257|B=IW<YB=VWY462]NCPiI3A\IW5798;46IW=\rk F9>E^ KR>EDuNag6ACS46257 ^VLKM25^>|8;7Yg64uN\AIA\IW5798;46IW=XY2!7Y57Y^7Y4dF[VKTbuN\57ÐXÈVd*ºW[wS.Q%RV'OU;eEr

ùËò½ñ©¨ aª5³.ª ³ µ ª5³ µ ³x X ª5±E¬Y³O³´ 5Vñãx»dè Ò Á Ðî?ÀÈ¬Ðç&hèé&hr!íftoBé*Íç+¢íA&^_&"&RVd,X

a+b STR#!UQS*#%"Ð'Z[wS.&QS 〈a, b〉 Î g úÉc^_&"&RV

aR#*ST',XZ#%"Ð'ÊÈì`0/&K*,z)r!íË*,z4(.j&hry)t+¢íÌD«éK~&hry0)t+Èì onì`0/¶p¿DS*#.\î&^_&Ñ

"&RVb Í t&%èëBíÂ*,z(.j&hry)t+¢íÎ`+¢çz%æd~+Èì onì`0/¶ Ë#%[=' 〈a, b〉 ÎE .i#%Xc ,XZ#XÈº#.R *\*]Ó *),[Q&;\^# P a!UQ#ÔË#%[Q¯*V ,[wS*aO.)O ,XZ#%RaÒ

〈a, b〉 = 〈c, d〉 ⇔ a = c ∧ b = d. (∗)N\SCK#N\Y^>WbEY7@462578;7Y:;F#F[Ig67?êu462X98=NÒ*&Rn([=+WU abEDJI\46257?KRN\s46257sDcIKM257Yg6<?257=-

525=^>WbÔUQST',"µ8;7Y:;FxDuN\B[NlS6DJIWB=<feag6AWILK#N\4uNGb|FU>E5A\I P #.)pa-ôNK@PhN\:=46IW=\rMz\798o<fN\57?FOe8;7Y:;F@8;7Yg64uN\AQF[I6bHY7SdKR>ED6S6AC]NF[I6bHXYIýKæFU>E^ DJIa8;mYXY25Sp8;7Y:;F4uNa89K#N\Y4625798;:=<Y7\bXY<>E52B=IW<YB=VWY462]N\46257AWIW57U8;46IW=XY2 KM:=DJVEPiB=<YmYg64d>EX_r ¥@XY<?>CKM25=XY257\b¢ÎWY4uN,DuN\B=moS6DcIWB=<fe\g6AWI,-K#N\4ueKMD6B=ILK#N\g6<Y25254uNaXY<Y798?rJ¥sF[I4uN8;XY<YmY=XY25798M:=DJI\FU>EAN\4C>:=DJIW:=VWrãx»dè Ò Á Ðî?ÀlÏ çpË ¹À Ç ¾C¿ Å Ó!ÁÐ 〈a, b〉 = a, a, b Î1sN\57YfNCPiIWd>8;7Y:=<YXY<Y7:=D6B[NfKMg6<Y25\bcXY<?>xFON\Av<Yg67êu4625IK#N\4uNDuN\B[NS6DcIWB=<fe\g6AWILK#N\4uN

:=DJ7Pi462]NK#N\B=S6467YA(∗) <TD6257YB;KM:=<Y798Rg67?êu4625X98;2 b6XYIDJI\<YIW:;FONLKM2]N\^>8=N\A\I?KM25XY<Y7Y46257\rs4uN\5IWZW25XY<Y46257 ÎWY7Y^><Yg67?êu4625IK#N\|F[B=Va8;AWm#S6DcIWB=<fe\g6AWILK#N\4ue 〈a, b, c〉 baXY<?K#VWB=A\mS6DJIWB=<fe\g6A\ILKRNa4ue 〈a, b, c, d〉 2!IWZWVW54627 n-UAWmTS6DcIWB=<fe\g6AWILK#N\4ue 〈a1, a2, . . . , an〉

rc¢I,-g6IW646257|8=N\AoK KR>EDuN\g6ACSpDuN\B;>xS6DcIWB=<fe\g6AWILK#N\46798MÎWY7Y^>oF[I<YB=IW625T4uNgGK#N:=DJI,-:=IWd>Wr`FON\AQ46Dr¢F[B=Va8;AaNpS6DJIWB=<fe\g6A\IK#N\4uN 〈a, b, c〉 F[IN\5cIIW6257YAdFB=IW<YB=VWY462]N8=e\X?>:;K|I\8;7`F[B=<?>xAWI\5798;467`KM:=DJVEPiB=<YmYg6467

abb2cbcN\5JI 〈a, b, c〉 = 〈〈a, b〉, c〉 rHIa8;mYXY257|DuN\B;>S6DJIWB=<fe\g6A\ILKRN\46798c8;7Y:;F4uN\^DJI\F[B=<Y7Y6467#g6I@<Yg67?êu4625ILK#N\462]NsuN\B=g6<YI

K#N\Y467YZWIDJIa8;mYXY2]N255IEXY<?>E4CSAaN\B;F[7Y<98=N\6:=AC257YZWI6rãx»dè Ò Á Ðî?ÀÇÑ,ê(éup®r/+10/oVç&pæ0®rq,çÒ~zhonì¹S*ÊT+W ,[Qd,X

A+Bki"d,X¢+m"Ð'$&Q¸Ò ~&*étVèoppì`0xoVç&pæ0®rq,çÒ~zhonì ^lËÊÈV ï[Q *($ ~&*éutVèop%ì0oS*ÊT+W ,[Qd,XA+BmoR#*ST',XZ#%"Ð'

S*ÊT+Wd,[A×B = 〈a, b〉 : a ∈ A ∧ b ∈ B.oR#%^_ Q`%+WUQSTRV+W&Ð *),[Q&;\^#%"Ð'+m^_ *UQST',RI)p#%[=&QS P #KÓn;),+c! ,Xc ,^lR& P ;)O ®ÓUQS. ,R& P +m^_ *\*U+S*ÊT+W ,Ñ

[Qd,XÕÔA1 × A2 × · · · × An = 〈a1, a2, . . . , an〉 : a1 ∈ A1 ∧ a2 ∈ A2 ∧ . . . ∧ an ∈ An.

|<?>E52M255IEXY<?>E4AaN\B;F[7Y<98=Na6:[AC23<Y625IWB=VLKA2BF[IQ<Y625VWBg6IWA6PhN\g646257FU>EXO_DuN\B

S6DJIWB=<fe\g6A\ILKRNa4C>EXO_bGAdF[VWB;>EXO_oD6257YB;KM:=<fNTKM:=DJVEPiB=<YmYg64uNR8;7Y:;F|7Y57Y^7Y4dF[7Y^ <Y625IWB=SAbCN

g6B=S6ZdNMKM:=DcVEPiB=<YmYg64uN@7Y57Y^7Y4CF[7Y^ <Y625IWB=SBr\n,IWY4uN3:;FOe\gPhNaFUK|I3KR>CKM4625IW:=A\ILK#N\\b\Y7

8;7Y=52cAdF[VWB;>E#<Y7s<Y625IWB=VKA5S6

B8;7Y:;FRD6S6:;FU>WbEF[I

A×B = ∅ rEU:;F[I\F[46257\bCB=IW<YS6^Sa8=e\X46257|KMD6B=IW:;FfbWZWgG>Ed><Y625VWBA×B d>uPJ46257YD6S6:;FU>Wb\F[I`25:;F[462]NCPhN\d>DuN\B[N 〈a, b〉 ∈ A×B rs573KRF[7YgG>

a ∈ A 2 b ∈ B bGXY<?>E52wfN\g64d>EX_<sF9>EXO_<Y625IWB=VLK 46257 8;7Y:;FMD6S6:;FU>WbGKM6B=7?K<fNCPiIWY7Y4625Src¥sF[B=<?>EôN\4uN:=D6B=<Y7YXY<Y4uIW=A\IW6XY<?>g6ILK#VCgs üQxzu W §p=m ª y uÈi¢xz£,¥xzz£.y (a, b)

5Gù 74X ª5±E¬õU464ueuN\B=g6<YI25:;F[I\F[4ueTXY7?X_ue255IEXY<?>E4CSoAN\B;F[7Y<98=N\6:=AC257YZWIM8;7Y:;F 8;7YZWI46257YD6B=<Y7Y^257Y4C-

46IW=\rc1sNIWZWVEP¢JILKM257YÂ×B 6= B × A rw625VWB

A × AIW<Y4uN\X?<fN\^>lKj:=ACB=VEXY257

A2 2#4uN\<>CKRN\^>AdK#N\g6B[NaF[7Y^ /hAN\B;F[7Y<=-8=N\6:=AC25^1s<Y625IWB=S

Ar s4uN\5IWZW25XY<Y4627I\<Y4uN\XY<fN\^>

n- FOeDcI\F[mYZWm<Y625IWB=S

A/n > 1

1=bAn = A× · · · × A

︸︷︷︸

n

rcB=<?>L8;^Sa8;7Y^>vF[7Y\buY7A1 = A

r

¸¹ºWá!ÓÈÇ Àu¼!áÖ r 1, 2, 3 × 1, 2, 3 = 〈1, 1〉, 〈1, 2〉, 〈1, 3〉,

〈2, 1〉, 〈2, 2〉, 〈2, 3〉,〈3, 1〉, 〈3, 2〉, 〈3, 3〉 r

¤Er ∅ × 1, 2 = 〈∅, 1〉, 〈∅, 2〉 rÚGrRk :=<YA\IW57-DJI0KMD6B=IK#N\g6<Y7Y4625S4uNDJPhNa:[<?XY<?>C÷f46257AN\B;F[7Y<98=N\6:=AC257YZWI S6A6PhN\g6SKM:=DcVEPiB=<YmYg64C>EXO_-SGF[IWY:[N\^2]Nx:=25mD6S646AdF9>QDJPhN\:=<YXY<?>E<Y4C>l<oDuN\B[N\^2D/hS6DJIWB=<fe\gC-A\ILK#N\4C>E^2m1J525XY<Y`B=<Y7YXY<?>CK325:;FU>EX_rLk F[7Y4:=DcIW:=VWSGF[IWY:[N\^2]N\^>sDJPhN\:=<YXY<?>E<Y46m<Y7<Y625IWB=7Y^

R × R = R2 r ¢IEg6IW646257ÎW?7Y^>DJILKM257Yg6<Y257Y\b!Y7D6B=<Y7Y:;F[B=<Y7YF[B=Va89KR>E^2]N\B=ILK#NF[I<Y625VWBR3 r

ã 2]N\ZWB[N\^>Bµ7Y464uNGb6AdF[VWB=7@DJIWôNaZdNCP]>4uN\^æ<YIW6B[N\<YILK#N\`254dF[S625X98;73K D6B=<?>EDuN\g6ACSDcIWD6B=<Y7Yg64625X_g6<Y2]NCPhN\4uN@<Y625IWB[N\XO_baF[S46257|ôNa8=e3<fN\:;F[IW:=IK#N\462]NGrd¥@:;FONaF[462G<#DJILKR>E=-:=<?>EXO_ýD6B=<?>EA6PhN\g6VLK4uN\:=SdK#No4uN\^Ê8;7Yg64uN\A25464ueZWB[NaêuXY<?4ueoB=7YD6B=7Y<Y7Y4dFON\X98;m255IEXY<?>E4CSAN\B;F[7Y<98=N\6:=AC257YZWIgGK|VEXO_-<Y625IWB=VLKTbHDcIW57YZdNa8=e\Xfe,4uND6B=<Y7Yg6:;FONfKM257Y4625S8;7YZWI7Y57Y^7Y4C-F[VLK8=N\A\ID6S646AdF[VKDJPhN\:=<YXY<?>E<Y4C>WrNCPiVWY^>WbGY7

A ⊆ X2B ⊆ Y

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[Qd,X¢R *\*U+DÔk#m

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A ∩ A = Aki+(!&"oV ,&RV(R *\*]c[wS.&;),[Q P )m)Ù

kÊjmA ∪B = B ∪ A k[wS.&"Ð+W&RVR *\*]oC"Ð'm)ÙA ∩B = B ∩ A k[wS.&"Ð+W&RVR *\*]c[wS.&;),[Q P )m)Ù

kzUmA ∪ (B ∪ C) = (A ∪B) ∪ C k(ºa!UQSTR *\*]ÔC"Ð'm)ÙA ∩ (B ∩ C) = (A ∩B) ∩ C k(ºa!UQSTR *\*]2[wS.&;),[Q P )m)Ù

kmA ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)

ki[Q S**ST+W&^lR *\*]c[wS.&;),[Q P îXÆS;`%^_Q!&"C"Ð'm)Ù

kz&mA ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)

ki[Q S**ST+W&^lR *\*]oC"Ð'$XÆS;`%^_Q!&"[wS.&;),[Q P )m)Ù

kvdmA ∪ ∅ = A, A ∩ ∅ = ∅ Î

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³ ´

Û

ß à á

â ã äå

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A¿B¿C+D[w#%XZ*ST+mXc&oa$R#.Ñ

P a!U;&oS*#%^_&Q¸TR *\*U+(Îk#m

A ⊆ A ∪B ÙA ∩B ⊆ A

Ù

5C² 74X ª5±E¬õkÊjm

A ⊆ C ∧ B ⊆ C ⇒ A ∪B ⊆ CÙ

A ⊆ B ∧ A ⊆ C ⇒ A ⊆ B ∩ C ÙkzUm

A ⊆ B ∧ C ⊆ D ⇒ A ∪ C ⊆ B ∪D ÙA ⊆ B ∧ C ⊆ D ⇒ A ∩ C ⊆ B ∩D Î

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A8;7Y:;F 7Y57Y^7Y4C-

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A ⊆ B ∧ C ⊆ DôN\^>sDcIWAaN\<fN\\bLY7

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x ∈ Bb¢N8;7Y=52

x ∈ CbF[I

x ∈ Dr¢NaF[7Y^

x ∈ B5S6

x ∈ D2 DJIW46ILKM46257<

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[Qd,X¢R ,XZ#*¸TR&Ôk#m

A ⊆ BÙ

kÊjmA ∪B = B

ÙkzUm

A ∩B = AÎ

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A ∩ (A ∪B) = A2

A ∪ (A ∩B) = A.

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P a!U;&¯[Qd,X¢R *\*U+(Îk#m

A× (B ∪ C) = (A×B) ∪ (A× C)Ù

(B ∪ C)× A = (B × A) ∪ (C × A)Ù

kÊjmA× (B ∩ C) = (A×B) ∩ (A× C)

Ù(B ∩ C)× A = (B × A) ∩ (C × A)

ÙkzUm

A× (B \ C) = (A×B) \ (A× C)Ù

(B \ C)× A = (B × A) \ (C × A)Ù

kmA× (B 4 C) = (A×B)4 (A× C)

Ù(B 4 C)× A = (B × A)4 (C × A)

Ùkz&m

(A×B) ∩ (C ×D) = (A ∩ C)× (B ∩D)Î

ãx¾E¿ ¼ ã ]ND6B=<?>EA6PhN\g6SxDJIWAaN\Y7Y^>oD6257YB;KM:=<feXY<YmY=`DJIEg6D6S646AdF[S / NO1rG¢IW46257?KRNa8;7Y:;FF[I`B=VLKM46IW=M<Y62IWB=VKTbaKM25mYX¢8=N\A<?KR>EAC57RX_6XY7Y^>DJIWAN\<fN\\bdY7RIWuN@<Y625IWB;>ôN8=eF[7:[N\^77?57Y^7Y4CFU>WrwN\SdKRN\Y^>8;7Yg64uN\AcbJY7IWuNF[7<Y625IWB;>:[eo255ICXY<?>E4uN\^2!AaN\B;F[7Y<=-8=N\6:=AC25^2 bu<fNaF[7Y^25XO_p7Y57Y^7Y4CFON\^2!:[eoDuN\B;>S6DJIWB=<fe\g6A\IK#N\467\r13257YXO_<fNaF[7Y^ 〈x, y〉JmYg6<Y257Tg6ILK|IW54ueDuN\B[eS6DJIWB=<fe\g6A\ILKRN\4ueGr6k0F[7YgG>

〈x, y〉 ∈ A× (B ∪ C)⇔ x ∈ A ∧ y ∈ B ∪ C ⇔ x ∈ A ∧ (y ∈ B ∨ y ∈ C)⇔⇔ (x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C)⇔⇔ 〈x, y〉 ∈ A×B ∨ 〈x, y〉 ∈ A× C ⇔⇔ 〈x, y〉 ∈ (A×B) ∪ (A× C)

r

5C× 74X ª5±E¬õ A\IWB=<?>E:;FON\525=^><g67?êu4625X98;2@255ICXY<?>E4ES0AaN\B;F[7Y<98=N\6:=AC257YZWIå/hD6257YB;KM:=<fN-2@XY<?K#N\B;FON

B=VLKM46ILKRNaY46IW=*1bg67?êu4625X98;2 :=S6^>9/hg6B=S6ZdNp2 D62]eaFONB=VKM46ILK#N\Y46IW=*1IWB[N\<oB=IW<Yg6<Y257Y¨-46IW=XY2!A\IW4625S646AWX98;2KM<YZW5mYg67Y^N\F[7YB=4uNaFU>CKR>Wr

kN\Y467@8;7Y:;Ffb!C>ý46257A\IWD625ILKRN\J7Y<?^>E=546257DJIW<Y4uN\4d>EXO_QK#XY<?7Y[4625798`g6IK|IEg6VLKTr`gG>Ed>E=^><fN\XY<Ym?52N\SGF[IWôNaF9>EXY<Y4627IEg,913257YXO_

xJmYg6<Y257g6ILK|IW54d>E^7Y57Y^7Y4dF[7Y^,r

k0F[7YgG>x ∈ A× (B ∪ C) . . .

ObGF[Ig6I4625XY<Y7YZWIC>E=^>x46257`g6IW:=<Y52 r

ÒÙ; < Íàx'YÖ r D6B[NfKMg6<Y25\buAdF[VWB=7T<TDcIW4625Y798R<Yg67?êu4625ILK#N\4d>EX_p<Y625IWB=VLK :[e:=IW6257`B=VLKM467%bA1 = 1, 2, 3, 4, A2 = 1, 2, 3, 4, A3 = 1, 2, 3, 4,A4 = 1, 3, 2, 4, A5 = 3, 1, 4, 2, A6 = 1, 4, 3, 2,A7 = 2, 1, 3, 4, A8 = 1, 2, 3, 4, A9 = 4, 3, 2, 1 r

¤Er3¢IEguN\T7?57Y^7Y4CFU>x4uN\:;F[mYD6Sa8=e\X?>EXO_p<Y625IWB=VKEb∅, ∅, a, a, b, b, a, ∅,∅, ∅, ∅, ∅, ∅, ∅, ∅, 1, 2, ∅, 1, 1 bψ, ψ, ψ, ψ, ψ, ψ, 〈7, 10〉, 〈3, 3〉, x ∈ N : x2 < 7x ∈ N : x2 < 0, x ∈ N : x ≥ 0, x ∈ N : x2 = 4 bx ∈ Q : x2 = 4, x ∈ N : |3−x| < 3, x ∈ R : x2+4x+4 ≤ 0,x ∈ Q : (x+ 1)2 ≤ 0, x ∈ R : x2 + 1 ≥ 0.

ÚGr313257YXO_A = x, ∅, B = ∅, ∅, ∅, ∅, ∅.

/ NO1zWN\AC257Y^S<Y625IWB=ILKM2!^S6:=2C>ETB=VKR467xbud>x<fN\XO_6IEg6<Y2hPiI

A ∈ B 3/hB1,zWN\AC257Y^S<Y625IWB=ILKM2!^S6:=2C>ETB=VKR467xbud>x<fN\XO_6IEg6<Y2hPiI

A ⊆ B3

ñ r313257YXO_A = x, ∅, B = ∅, ∅, ∅, ∅, ∅, ∅ r

/ NO1zWN\AC257Y^S<Y625IWB=ILKM2!^S6:=2C>ETB=VKR467xbud>x<fN\XO_6IEg6<Y2hPiI

A ∈ B 3/hB1,zWN\AC257Y^S<Y625IWB=ILKM2!^S6:=2C>ETB=VKR467xbud>x<fN\XO_6IEg6<Y2hPiI

A ⊆ B3

Ý r313257YXO_x2yJmYgueB=VWY467IEgQ<Y625IWB=SQD6S6:;F[7YZWI6r¢zWN\AC25^<Y625IWB=IW^^S6:=<feC>E

B=VLKM467x2yb6C>x<fN\XO_6IEg6<Y2hP]>vDJIW4625Y:=<Y7TB=VKM46IW=XY2 r

/ NO1 x, y, ∅ = x, y, ∅ k/hB1 x, ∅, y = ∅ r|<?>x<fNLKM:=<Y7R8;7Y:;F3ÎWY52~K#7@DJICguN\46257ÌEg6DJILKM257Yg6<Y2m3

ùËò6587 ³d¯w³ µ ª5³ 5 öåGr3¢IEguN\TD6B=<?>EA6PhN\gp<Y625IWB=SF[B=Va8;7Y57Y^7Y4dFOILK|7YZWI

AFON\AC257YZWI6b6Y7AaN\YgG>8;7YZWI7Y57=-

^7Y4dF|8;7Y:;F`B=VKM46ICXY<Y7Y=46257s8;7YZWIxDJICg6<Y625IWB=7Y^ /iF[<Y4rg6]Nog6ILK|IW5467YZWIx^oN\^>

x ∈ A⇒ x ⊆ A1r

ì rRkQ>E<Y4uN\XY<?>E<Y625VWB x ∈ Z : ¬ϕ(x) buZWg6<Y257/ NO1

ϕ(x) =(

x > 0⇒ (x2 < 4 ∧ (x ≤ 3⇒ x > 0))) r

/hB1ϕ(x) = (x < 3⇒ (x 6= 7 ∧ x ≤ 0))

r×Gr ã ]NDJICguN\4d>EX_lyhS646A\X98;2 <YguN\4625IK#>EX_

ϕ(x)IWD625:[N\<Y625IWB;> x ∈ R : ϕ(x)/h:;F[IW:=Sa8=e\XT46DruIW<Y4uN\XY<Y7Y462]N4uND6B=<Y7Yg6<Y2]NCP]>WbuDJVEPiD6B=IW:;F[7T2F[Dr 1r

/ NO1x2 = 1

b/hB1

x = xb

/ X1x2 − 1 ≤ 0

b/hgB1

x < 7⇒ x = 0b

/ 71x2 < 0⇒ x2 = −1

b/iy;1

x2 = −1⇒ x = 7b

/hZ!1x2 < 0⇔ x > 0

b/h_B1

x2 = 4⇔ x = 1r

ó rRkQ>E525XY<?>Es<Y625IWB;>A∪B b A∩B b A \B b B \A 2 A4B

g6]NT4uN\:;F[mYD6Sa8=e\X?>EX_<Y625IWB=VLK

A2Br

/ NO1A = x ∈ N : x < 3, B = x ∈ N : x ≥ 3 b/hB1A = x ∈ N : x < 0, B = x ∈ N : x = 2 b/ X1A = x ∈ R : x < 1, B = x ∈ N : x < 1 b/hgB1A = x ∈ R : x < 1, B = x ∈ R : x < 2 b/ 71A = [0, 3], B = (1, 2]

rÖfø r313257YXO_D6B=<Y7Y:;F[B=<Y7Y462]eJmYg6<Y257v<Y625VWB

Z525XY<YlXfNCPiA\ILKM2F9>EXO_bNpDcIW4uN\gGF[I46257YX_

A = x ∈ Z : 2|x b B = x ∈ Z : ¬2|x b C = x ∈ Z : x < 20 rJ¥@D625:[N\<Y625IWB;>AcbA∪B b A∩B b A \B b A \C b B ∩C b Cc b A∩ (B \C)

bA4B

rÖ\Ö r3¢IWAaN\<fN\\b|Y7

A ∪ B 8;7Y:;Fx4uN8;^4625798;:=<?>E^t<Y625IWB=7Y^t<fNLKM257YB[N8=e\X?>E^ B=VLKM46I,-XY<Y7Y=46257<Y625IWB;>

A2Br yhIWB=^SJPiILK#N\x2#S6g6ILK|IEg64625N\4uN\5IWZW2XY<Y4d>ly N\AdFg6]N

D6B=<Y7YACB=Ia8;Sp2B=VWY4625X?>WrÖ ¤Er313257YXO_

A = 1, 3, 5 b B = 2, 4 2 C = 1, 5 rw4uN\57T÷Y<Y625VWB X bcFON\AC2HY7(A4X)4B = C

r

6nø 74X ª5±E¬õÖ ÚGr ã ]NguN\467YZWI<Y625IWB=S

AIWD625:[N\#8;7YZWI<Y625VWBRDcI\F[mYZWIKR> P(A)

r/ NO1

A = ∅, ∅, ∅ b/hB1A = 1, 2, 3, 1 b/hX*1A = α, β, ∅ b/hgB1A = b, bb, B b/h7*1A = δ b/iy;1A = σ, τ, ρ r

Ö ñ r D6B[NfKMg6<Y25\buXY<?><fN\XO_6ICg6<Y2 <fNfKM257YB[N\46257A ⊆ B

5S6B ⊆ A

r/ NO1

A = a, a, ∅, B = a b/hB1A = ∅, B = ∅, ∅ b/hX*1A = µ, ∅, B = µ, ∅ b/hgB1A = x ∈ N : x > 2, B = y ∈ N : y > 2 b/h7*1A = x ∈ R : x > 0, B = y ∈ N : y > 0 b/iy;1A = x ∈ Q : x2 ≤ 0, B = x ∈ Q : x6 + 7x5 − 3x = 0 r

ÖÝ r313257YXO_A = x ∈ Z : x2 < 7 b B = y ∈ N : |1− y| < 3 r

/ NO1okQ>E<Y4uN\XY<?>E4uN\:;F[mYD6Sa8=e\XY7<Y625IWB;>b2 rA4B

b252 r

((A ∩B)×B) \ (B × (A ∩B))b

25252 r P((A \B)× (B \ A))r

/hB1,|<?>25:;F[4625798;7`<Y625VWBC ⊆ Z FON\AC2 b6Y7 A4 C = B 4 C

3Ö åGr313257YXO_

A = x ∈ Z : 2|x ∧ |x− 1| ≤ 4 b B = x ∈ N : x2 < 9 r/ NO1okQ>E<Y4uN\XY<?>E4uN\:;F[mYD6Sa8=e\XY7<Y625IWB;>b

2 r(B \ A)× P(A \B)

b252 r P(A4B) ∩ P(B)

r/hB1,|<?>25:;F[4625798;7`46257YD6S6:;FU>x<Y625VWB

C ⊆ Z FON\AC2 bGY7 A× C = C ×B 3Öì r33g6ILK|IEg64625`DJIW<YIW:;FONCPi7TDJICg6D6S646AdF9>¡ KM257YB=g6<Y7Y4625No¤Er ÚGrÖ ×GrRkÜD6B=<Y7Y:;F[B=<Y7Y462

NB=IW<YDuNaF[B=<Y^>xDJICg6<Y625IWB;>

A = 1, 2, 6, 7, 8, B = 2, 3, 4, 7, 8 2 C = 4, 5, 6, 7, 8.U57B=VWY4C>EXO_<Y625IWB=VLK^IWY4uNQ<Y6S6g6ILK#N\p<fNlDcIW^IEXfeIWDJ7YB[N\X98;2 ∪,∩, c <Y7<Y625IWB=VLK

AbB2C3|<?>8;7Y:;FMKM=B=VEg,4625XO_,<Y62VWB 8 3

ùËò6587 ³d¯w³ µ ª5³ 6Bñ

Öfó r33g6ILK|IEg64625\bJY7g6]Ng6ILK|IW54d>EX_<Y625IWB=VLKAbB2C<fN\XO_6IEg6<feo4uN\:;F[mYD6Sa8=e\XY7

B=VLKM46IW=XY2 r/ NO1

A ∩ (B \ A) = ∅ b/hB1A ∪ (A ∩B) = A

b/ X1

A ∩ (A ∪B) = Ab

/hgB1(A ∪B) \ C = (A \ C) ∪ (B \ C)

b/ 71

A \ (B \ C) = (A \B) ∪ (A ∩ C)b

/iy;1A \ (B ∪ C) = (A \B) \ C b/hZ!1A4B = (A ∪B) \ (A ∩B)

r¤ ø rszWN\Av4uN8;D6B=IW=XY25798#DJIWAaN\<fN\\buY7

A4 (B 4 C) = (A4B)4 C3

¤ Ö r33g6ILK|IEg64625\b¢Y7og6]Ng6ILK|IW54C>EXO_DJICg6<Y625IWB=VLKA2BS6:;FON\5IW467D6B=<Y7Y:;F[B=<Y7Y462

X<fN\XO_6IEg6<fe4uN\:;F[mYD6Sa8=e\XY7B=VLKM46IW=XY2 r

/ NO1(Ac)c = A

b/hB1

A \B = A ∩Bc b/ X1

(A ∪B)c = Ac ∩Bc /hD6B[NLK|Ig67n,IWB=ZdN\4uNO1b/hgB1

(A ∩B)c = Ac ∪Bc /hD6B[NLK|Ig67n,IWB=ZdN\4uNO1b/ 71 ∅c = X

b/iy;1

Xc = ∅ r¤\¤Er33g6ILK|IEg64625\b¢Y7og6]Ng6ILK|IW54C>EXO_DJICg6<Y625IWB=VLK

A2BS6:;FON\5IW467D6B=<Y7Y:;F[B=<Y7Y462

X^oN\^>

A ⊆ B ⇒ Bc ⊆ Ac.

¤aÚGr D6B[NfKMg6<Y25\bRXY<?>g6]Ng6IK|IW54d>EXO_ <Y625IWB=VKAbB2C<fN\XO_6IEg6<fe-DJIW4625Y:=<Y7

B=VLKM46IW=XY2 rJz\7Y=5246257\buDcIEguN\TICg6DJILKM257Yg646257TD6B=<?>EA6PhN\gG>\r/ NO1

A \ (B ∪ C) = (A \B) ∩ (A \ C)b

/hB1A ∩ (A ∪B) = B

b/ X1

(A ∪B ∪ C) \ (A ∪B) = Cb

/hgB1A ∩ (B 4 C) = (A ∩B)4 (A ∩ C)

b/ 71

A ∪B = (A4B)4 (A ∩B)b

/iy;1(A ∪B)× C = (A ∪ C)× (B ∪ C)

b

66ù 74X ª5±E¬õ¤ ñ r ã N\467T:[e<Y625IWB;>

AbB2CbGFON\AC257\b6Y7

A ∪B = CrJwg67?êu4625ILK#N\T<Y625IWB;>

D2

EFON\AC257\b6Y7

D ⊆ AbE ⊆ B

bD ∩ E = ∅ 2 D ∪ E = C

r¤ Ý r ã N\467:[e<Y625IWB;>

AbB2Cr`wg67?êu462IK#N\<Y625VWB

DFONaAC2 bsY7

D ⊆ A2

D ∩B = D ∩ C = ∅ r¤\åGr ã N\467T:[e<Y625IWB;>

A2Brc¨MIW<?KM2]e\<fN\`B=VLKM4uN\46257

A ∪X = Br

¤ ì r33g6ILK|IEg64625`DJIW<YIW:;FONCPi7TDJICg6D6S646AdF9>¡ KM257YB=g6<Y7Y4625No¤Er ñ r¤\×Gr33g6ILK|IEg64625\bu?7

/ NO1 P(A) ∩ P(B) = P(A ∩B)b

/hB1 P(A) ∪ P(B) ⊆ P(A ∪B)r

¢IEguN\MD6B=<?>EA6PhN\gbCY7MKDJIEg6D6S646A\XY257Ó/hB1ÎWY7M46257Rd>E3<fNLKM257YB[N\462]NsK0g6B=S6Zde:;F[B=IW46m\r

¤ ó r D6B[NfKMg6<Y25\buXY<?>/ NO1 P(A \B) = P(A) \ P(B)

b/hB1 P(A4B) = P(A)4P(B)

rÚ ø r33g6ILK|IEg64625\bu?7

/ NO1A ⊆ B ⇒ P(A) ⊆ P(B)

b/hB1 P(A) ⊆ P(B)⇒ A ⊆ B

rÚ Ö r3¢IWAN\<fN\\bcY7Tg6]Ng6ILK|IW5467YZWI<Y625IWB=S

A^N\^>

A 6= P(A)r

Úd¤Er33g6ILK|IEg64625`DJIW<YIW:;FONCPi7TDJICg6D6S646AdF9>¡ KM257YB=g6<Y7Y4625No¤Er åGrÚWÚGr33g6ILK|IEg64625\bu?7g6]Ng6IK|IW54d>EXO_,<Y625IWB=VK

AbBbC2DôN\^>

/ NO1A ⊆ B ∧ C ⊆ D ⇒ A \D ⊆ B \ C b/hB1A ⊆ B ⇔ B = A ∪ (B \ A)

b/hX*1

A \B = B \ A⇒ A = Bb

/hgB1A×B = B × A⇔ (A = ∅) ∨ (B = ∅) ∨ (A = B)

b/h7*18;7Y=52

AbBbC2D:[es46257YD6S6:;F[7|F[I

A×B = C×D ⇒ A = C∧B = Dr

¢IWAaN\<fN\\buY7<fNCPiIWY7Y4627T46257YD6S6:;F[IW=XY268;7Y:;F325:;F[I\F[467\rÚ ñ r3¢IWAN\<fN\\b6Y7@46257 8;7Y:;FRD6B[NfKMgueGbGY7@g6]Ng6IK|IW54d>EXO_x<Y625IWB=VK

AbB2CôN\^>

(A×B) ∩ (C ×D) = ∅ ⇒ A ∩ C = ∅.

ùËò6587 ³d¯w³ µ ª5³ 6.5Ú Ý r ã ]N

A ⊆ X2B ⊆ Y

KR>ED6B=IK#N\g6<Y252S6g6ILK|IEg64625KM<YIWB;>ý4uNxg6IWDJ7Pi46257?462]N<Y625IWB=VLK

A× Y 2 A×B KM<YZ\5mYg67Y^D6B=<Y7Y:;F[B=<Y7Y462 X × Y rÚ\åGrs|<?> P(X × Y ) = P(X)× P(Y )?

Ú ì r313257YXO_XJmYg6<Y257oD6B=<Y7Y:;F[B=<Y7Y462]eGrn,VLKM25^>Wb Y7vDcIEg6<Y625IWB;>

A1bA2bA32A4F[798#D6B=<Y7Y:;F[B[<?7Y4u2 :[eKDcIEPiIWY7Y4625S,IWZWVW54d>E^,b8;7Y=52<fNfKM:=<Y7ôN\^>

A∗1 ∩ A∗

2 ∩ A∗3 ∩ A∗

4 6= ∅,ZWg6<Y257

A∗i = Ai

5S6A∗i = Aci

g6]Ni = 1, 2, 3, 4

r1sN\B;>E:=ILK#N\XY<?F[7YB;>ý<Y625IWB;>KDcIEPiIWY7Y4625SI\ZWVW54C>E^J/hF[<Y4rWDJIWD6B[NfKM4C>g62]N\ZWB[N\^µ 7Y464uNTg6]N@XY<?F[7YB=7YX_o<Y625I,-B=VLKo1r

Ú\×GrsN\SdK#N\Y^>WbWY7R255IEXY<?>E4AaN\B;F[7Y<98=N\6:=AC2ugGK#VCX_<Y625IWB=VLK<Yg67?êu4625IK#N\525=^>D6B=<?>DJIWÎCX?>DuN\B;>S6DJIWB=<fe\g6A\ILKRN\46798F[B[N\AdF[ILKRN\46798w8=N\AWIIWDc7YB[N\X98=NgGK|VEXO_o<Y^257Y4C-4d>EXO_rW¢IWAaN\<fN\\baY7|ÎWY4uNM<Yg67?êu4625ILKRN\|255ICXY<?>E4AN\B;F[7Y<98=N\6:=AC2G<Y625IWB=VLK

A2BD6B=<?>ýDJIWÎCX?>IEg6DcIKM257Yg64625Ixg6IW6B[N\46798syhS646A\X98;2<YguN\4625ILK#798?rï/ ÃÆÅ ÓEÀ6º ¿ÓGÀç

1sN\57Y?>v<fN\SCK#N\?>E\bcY7R8;7Y=52a ∈ A 2 b ∈ B b6F[I 〈a, b〉 ∈ P(P(A ∪B))

1rÚ ó rs|<?>255ICXY<?>E4AaN\B;F[7Y<98=N\6:=AC268;7Y:;F3IWDJ7YB[N\XU8=evPhe\XY<Y4ueO3

6C6 74X ª5±E¬õ

Ê++!éè

ê / NK %ëµ Ê)x

ã IF[798 DJIWB;>DJIW<Y4uN\525=^>B=VWY467s7Y57Y^7Y4dFU>`8;mY<?>EANôNaF[7YôNaFU>EXY<Y467YZWI6rEB=<?>25XO_DJIWÎCX?>vC>E525=^>K :;FON\46257`<fN\D625:[N\@yhIWB=ôN\54627sB=VWY467TôNaF[7YôNaFU>EXY<Y467`B=IW<YS6ÎK#N.-462]NGrWHIa8=NfKM2]NCP]>:=25mH8;7Yg64uNaAKR>EB[N\Y7Y462]N`FON\AC257¢8=N\AUg6]NsAaN\Yg67YZWIaObaUg6]NsKM:=<?>E:;F[AC25X_EObUg6]Ng6IK|IW5467YZWIaObCUg65NDJ7?KR467YZWIaObGU25:;F[4625798;7ObEAdF[VWB=7`46257`^2]NCP5>o:;K|IW25XO_:;>E^cIW525XY<=-4d>EX_,IEg6DJILKM257Yg64625A\VLKTru¡¢7YB[N\<TS6<YS6DJ7Pi4625^>vF[mT5S6A\m\r13257YX_

ϕ(x)cmYg6<Y257QyhS646A\X98=e<YguN\4625ILK#eI<fN\ACB=7Y:=257l<Y^257Y4646798

x ∈ [email protected]

A6PhN\guN\^>F[7YÌEgvF[798|DJIWB;>WbGY7X8;7Y:;FM<Y625IWB=7Y^æ46257YD6S6:;FU>E^,ruq|B[N\ACSa8=e\XY7`4uN\^æ:;>E^Ð-

JIW57@F[I

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ϕ(x)ObcUg6]Ng6IK|IW5467YZWI

x ∈ X/h<fN\XO_6IEg6<Y2m1ϕ(x)

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ϕ(x)Or

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XY<?>CFON\^>TU25:;F[4625798;7x ∈ X bag6]N3AdF[VWB=7YZWIÒ/h<fN\X_6ICg6<Y2m1 ϕ(x)

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ϕ(x)Or

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(∀x ∈ X)2(∃x ∈ X)

),XZ#%RV('ìÈ), P akiX¢+(a*¸*am<Y^257Y464uexr

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(∀x ∈ X)ϕ(x)8;7Y:;F8;S6T<YguN\46257Y^,ruz\7Y:;F3IW46ID6B[NfK2-

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KR>EB[N\Y7Y46257(∃x ∈ X)ϕ(x)

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x0

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• wguNa46257 (∀x ∈ X)ϕ(x)8;7Y:;F3D6B[NfKMg6<Y2K#7 ⇔ x ∈ X : ϕ(x) = X

b

• wguNa46257 (∃x ∈ X)ϕ(x)8;7Y:;F3D6B[NfKMg6<Y2K#7 ⇔ x ∈ X : ϕ(x) 6= ∅ r

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∀xϕ(x) (5S6

(∀x)ϕ(x))2 ∃xϕ(x) (

5S6(∃x)ϕ(x)).

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A = B ⇔ ∀x(x ∈ A⇔ x ∈ B).

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A ⊆ B ⇔ ∀x(x ∈ A⇒ x ∈ B).

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(∀x ∈ R)x2 ≥ 0.

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F[mYD6Sa8=e\XYI∃x(x2 − 2x− 1 = 0).

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6 Å

• (∀x ∈ 1, 2, 3)ϕ(x)⇔ ϕ(1) ∧ ϕ(2) ∧ ϕ(3)b

• (∃x ∈ 1, 2, 3)ϕ(x)⇔ ϕ(1) ∨ ϕ(2) ∨ ϕ(3)r

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x ∈ X +RV+W&;U;e

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• (∀x ∈ A)ϕ(x)⇔ ∀x(x ∈ A⇒ ϕ(x))¿

• (∃x ∈ A)ϕ(x)⇔ ∃x(x ∈ A ∧ ϕ(x))Î

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A ⊆ B ⇔ (∀x ∈ A)x ∈ B.¤Ersw625IWB;>

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(∀ ε)(ε > 0⇒ (∃δ)(δ > 0 ∧ (∀x)(|x− x0| < δ ⇒ |f(x)− f(x0) < ε|))),XY<?>E52!g6S6YIZWIWB=<Y798?r

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U:=XO_6IK#N\4625SEDJIWD6B=<Y7Yg64625AaNM25^D625AN\X98;2 bfNR:=<YXY<Y7YZWVEPiILKR>AdKRN\4dFU>CêuAaNaF[IWB¢IWZWB[N\4625XY<YIW4d>4uNU:=XO_6IK#N\4625SED6257YB;KM:=<Y7YZWI XY<PiIW4CSA\IW4625S646A\X98;2 r`13257ÎWY4uN<fNaF[7Y^ S6D6B=IW=XY25KR>EB[N\Y7Y ∃x(x ∈ A⇒ ϕ(x))

N\462 ∀x(x ∈ A ∧ ϕ(x))r

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A ⊆ Xb\g6I

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b <fN\25^D62AaN\X98=N

x ∈ A⇒ ϕ(x)8;7Y:;F#D6B[NfKMg6<Y2KRN46257Y<fN\57YY46257MIEgF[7YZWI6bCXYIDJIEg6:;FONfKM25^>

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(∃x ∈ A)ϕ(x)8;7Y:;F3<fNfKM:=<Y7Ty NCPi:=<?>CK#7\r

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ϕ(x, y) = (x < y), x, y ∈ R ruk0F[7YgG>KR>EB[N\Y7=-46257∀x(x < y)46257T8;7Y:;F<YguN\46257Y^,b!F9>E5A\I46ILKRe,yhS646A\X98=e<YguN\4625IK#ep<Y^257Y4646798

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∀x(. . . . . . . . . . . . . . .︸︷︷︸<fN\:=25mYZ ).

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x2 ≥ 0/ N0g6B=S6ZW257YZWIIEXY<?>CKM25=XY257

x < 01r

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(∀x)(∀y > x)(∃z)(x < z ∧ z < y)

6 ö

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(∀x)(∀y)(y > x⇒ (∃z)(x < z ∧ z < y)).

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z<Y^257Y464ueK|IW54ueGr

¤ErRkÜKR>EB[N\Y7Y462S∃x∃y(x > 5) ∧ y > 1

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• ¬(∃x ∈ A)ϕ(x)⇔ (∀x ∈ A)¬ϕ(x)Î

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∀x(ϕ(x) ∨ ψ(x))⇒ ∀xϕ(x) ∨ ∀xψ(x)Or

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∀x(ϕ(x) ∨ ψ(x)) ∧ ¬(∀xϕ(x) ∨ ∀xψ(x)),

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∀x(ϕ(x) ∨ ψ(x)) ∧ (∃x)¬ϕ(x) ∧ (∃x)¬ψ(x).

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ST"Ð+W&RVR& Px ∈ X +¢RV+W&;U;e

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bYFON\AC257Y7 <YguN\46257ϕ(x0)∧ψ

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ϕ(x0)2ψrc¡¢Iv<A\IW57Y2

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<YguN\46257(∃xϕ(x)) ∧ ψ b6XYIA\IW6XY<?>g6IK|VEgr

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∀x(x > 7 ∨ y < 3)2y < 3 ∨ ∀x(x > 7)

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¿oÁh»C¹a¼ ºd» Ò Áh» Ì Õ£Ô+W&;U;eϕ(x, y)

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x ∈ X, y ∈ Y ÎÔY&Q%'k#m ∀x∀y ϕ(x, y)⇔ ∀y∀xϕ(x, y)k[wS.&"Ð+W&RVR *\*]o),XZ#%RV('ìÈ)p#% ,[Qd,X ;`!d,^lRË'!U;e)m)ÙkÊjm ∃x∃y ϕ(x, y)⇔ ∃y∃xϕ(x, y)k[wS.&"Ð+W&RVR *\*]o),XZ#%RV('ìÈ)p#% ,[Qd,XQS.UQS.&i`!d*º ,X¢'OU;ehmÙkzUm ∃x∀y ϕ(x, y)⇒ ∀y∃xϕ(x, y)

Î

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bR<fN\D6B[NfKMg6<Y2K|IW=IW6Sp<YguN\ ∃x∃y ϕ(x, y)

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Dϕ 6= ∅r ã I

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yôN

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(∀x > 1)(∀n ∈ N)xn > 12

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yr

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I = N5S6

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⋃

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i∈IAi = x : (∀i ∈ I)x ∈ Ai.

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x ∈⋃

i∈IAi ⇔ (∃i ∈ I)x ∈ Ai,

s »m£TW.y ¥Q¡m£.y Æi ª y Æ.þ~ £.y z£.y B¡m .¥xzy Wa¨©W|= £.y «|=þT¢.z*¥y ï©W ª .© ¡myhec§,;W©Wx xwv*xzy .sWsÆwwlpc;;¡m «£.y w¥2*þ*v*£.y (¡ x yC¥xzu Wc.z@=£ £,=xz £,cx.y ~©x.y ©W~;¢s ½Æ ª ¨v*£,Æa¨©W|= £,v*jc,£.y ¥ ¡mucy £,v* ª* ;=£.(¡©W*v*xzy £Ôxj.y ©W~; puwv*xy Z§%.v.=£,ù£, Wuz§)æ£|©Wxwv*xzy =

9!× ð ²s³ µn: õñ@¶W³ : ±E¬õXY<?>E52 b|N\4uN\5I\ZW25XY<Y462578=N\AKD6B=<?>EDuN\g6ACSgGK#VCX_0<Y625IWB=VLKTb C>E7Y57Y^7Y4dF[7Y^ :=S6^>IW<Y4uN\XY<fNd>ET7Y57Y^7Y4CF[7Y^jAdF[VWB=7YZWIWM<#8;798R:=A6PhN\g64625A\VLKTrJ¢ICg6IW646257\b

x ∈⋂

i∈IAi ⇔ (∀i ∈ I)x ∈ Ai,

XY<?>E52wd>E`7?57Y^7Y4CF[7Y^æD6B=<Y7?ACB[Ia8;SxIW<Y4uN\XY<fNd>E@7Y57Y^7Y4CF[7Y^ AaN\Yg67YZWI< 8;7YZWI:=A6PhN\gC-4625A\VLKTrJN\SdK#N\Y^>Wb6Y7#8;7Y=52

I = 1, 2 b6F[I⋃

i∈IAi = A1 ∪ A2

IWB[N\< ⋂

i∈IAi = A1 ∩ A2.

NaF[7Y^DJIa8;mYXY2]Nx:=S6^>2D6B=<Y7YACB=Ia8;SQB=IEg6<Y254d><Y625IWB=VLKS6IWZWVW5462]N8=eoDJIa8;mYXY2]Nv:=S6^>2D6B=<Y7YACB=Ia8;S,gGK|VEXO_p<?625IWB=VKK F[7Y4,:[N\^j:=DJIW:=VWb6K8=NaAC2!DJIa8;mYXY2]NAdKRN\4dFU>CêuAaNaF[IWB=VLK:=<YXY<Y7YZWVEPiILK|7YZWIo2!IWZWVW5467YZ\IS6IWZWVW462]Na8;eDcI\8;mYXY2]NN\F[7YB=4uNaFU>CKR>x225^D652AaN\XU8;2 r`gG>

I = N^IWY7Y^>vS6?>CKRN\TF[7YTIW<Y4uN\XY<Y7Y

∞⋃

n=0

AnIWB[N\< ∞⋂

n=0

An.

¸¹ºWá!ÓÈÇ Àu¼!áÖ r313257YXO_

I = NIWB[N\<

An = [−n, n]g6]N

n ∈ N rGk0F[7YgG>∞⋃

n=0

An = RIWB[N\< ∞⋂

n=0

An = 0.

¤Er313257YXO_I = N+ IWB[N\< Bn = (0, 1

n)g6]N

n ∈ N+ rGk0F[7YgG>∞⋃

n=1

Bn = (0, 1)IWB[N\< ∞⋂

n=1

Bn = ∅.

ÚGr313257YXO_I = N

IWB[N\<Cn = (0, 2− 2

n+2]g6]N

n ∈ N r6k0F[7YgG>

∞⋃

n=1

Cn = (0, 2)IWB[N\< ∞⋂

n=1

Cn = (0, 1).

1sNpg6IK|IW5467oB=IEg6<Y254d>l<Y625IWB=VLK ^IWY7Y^>F[7YvS6IWZWVW54625DcI\8;mYXY257oB=IW<Phe\XY<Y46IW=XY2 rk6B=7?K DcIW<YIWB=IW^æ8;7Yg64uN\Acb 46257:=AWIWB=<?>E:;FON\^>ýKÆFU>E^ KR>EDuN\g6ACS<DJIa8;mYXY2]NxS6IWZWVW¨-4625IW467YZWID6B=<Y7YACB=I\8;Sr

5Vòù¼¨ aª5³.ª ³ µ ª5³ ? ±Côyónó µ ª5± µ ® µ ³x X ª5±E¬Y³O³´ 9 öãx»dè Ò Á Ðî?À .S+mRÔS*ÊT+W ,[Qd,X Ai : i ∈ I R#*ST',XZ#%"Ð'B&*éyr/wíp®r/+¢í ki^lËÊ@ç&*ç ì&*éyrKwíp®r/+¢íKm*¿ïÒ%'C ,Xc ,^lR&Í%XZ# P & P [Qd¸TR&Ó&^_&"&RV('Óa[Q S=ºa!UQSTR&;¿cUQST',^l+ÆÒ%'

(∀i, j ∈ I)(Ai 6= Aj ⇒ Ai ∩ Aj = ∅).kÙ:=<YXY<Y7YZWVW546IW=XY2 b|B=IEg6<Y254uN Bn : n ∈ N+ <g6B=S6ZW257YZWIlD6B[<>EA6PhN\g6S4uNDcI,-D6B=<Y7Yg64625798R:;F[B=IW46257`46257|8;7Y:;F3B=IEg6<Y254ueB=IW<Phe\XY<Y4ueGrHICg6IW646257`8=N\AKÆKR>EDuN\g6ACS<?KR>EA6P]>EX_IWDJ7YB[N\X98;2 :=S6^>2 D6B=<Y7YACB=Ia8;Sb¢ÎWY7Y^>

^VLKM25ÌK@PhN\:=46IW=XY2]N\XO_,IWDJ7YB[N\X98;2!S6IWZWVW54625I\4C>EXO_r ¿oÁh»C¹a¼ ºd» Ò Áh» Ì ï£Ô+W&;U;e Ai : i ∈ I ,[w#*S Bi : i ∈ I Ê=QOaf[Q *ST+mR#%"Ð+V *S*ÊT+W ,[Qd,X¾Oi#%^_ ,R& P [wS.&;([wS.&RV+

XÎEY&Q%'

k#m ( ⋃

i∈IAi

)c

=⋂

i∈IAci ,

( ⋂

i∈IAi

)c

=⋃

i∈IAci

k[w#%XZ#!& ,[WÒ#%R#n^#*ST+(#.º#KÓAV Q`!d,^lRV+W ,RV'OU;e)m)ÙkÊjm ⋃

i∈I(Ai ∪Bi) =

⋃

i∈IAi ∪

⋃

i∈IBi;

kzUm ⋂

i∈I(Ai ∩Bi) =

⋂

i∈IAi ∩

⋂

i∈IBi;

km ⋃

i∈I(Ai ∩Bi) ⊆

⋃

i∈IAi ∩

⋃

i∈IBi;

k&m ⋂

i∈IAi ∪

⋂

i∈IBi ⊆

⋂

i∈I(Ai ∪Bi);

ãx¾E¿ ¼ z\7Y:;FýF[I0:;FON\46guN\B=g6ILK|7B=IW<YS6ÎLKRN\46257lIWDuN\B;F[7I Na:ONag6m-§AC:;F[7Y46:h8;I,-4uN\546IW=XY2 bJNKR>EA\IWB=<?>E:=F[Sa8=e\XY7g67?êu4625X98;mg6<?2]NCPhN\S6IWZWVW5462IW4C>EXO_f/h2¢<?KR>EA6P]>EX_B1sIWB[N\<IEg6DcIKM257Yg646257`D6B[NfK|IB[N\XO_ES646ACS,AdKRN\4dFU>CêuAaNaF[IWB=VLK /h¡ KM257YB=g6<Y7Y46257ÚGrØ¤p1r

¨MIW<?KRN\fN:=25m@B=VLKM46257Y@B=ICg6<Y254d>o<Y625IWB=VLK2546g67YAC:=ILKRN\467@S6A6PhN\guN\^2J2546g67YAC:=VKTru4uN\:=<Y7YZWID6S646AdFOSýKM25g6<Y7Y462]Nx4uN8;25:;F[I\F[4625798;:=<Y7:[eDcIEgGK|V\8;462572546g67YAC:=ILKRN\467B=IEg6<Y254d><Y625IWB=VLKTb\FON\AC257¢8=N\A Ai,j : i ∈ I∧j ∈ J /hZWg6<Y257 I 2 J :[e`S6:;FON\5IW4C>E^2G46257YD6S6:;FU>E^2<Y625IWB[N\^2w2546g67YAC:=VLKo1 g qr1sNTDJICgGK#Va8;46257M2546g67YAC:=ILK#N\4C>EXO_vB=ICg6<Y254uN\X_<Y625IWB=VKÎWY7Y^>KR>EA\IW4d>CKRN\sDcI,-gGK|V\8;467g6<Y2]NCPhN\462]N|S6IWZWVW54625I\467\r?kQ>E^N\ZdN|F[I8;7Yg64uN\AsACB=V\F[AC257YZWI#KR>L8=N\=46257Y462]NGrL13257YXO_s ,©W|= £.y .v*§%;y v.åWÂy £,v* ª* ;=£.y y .¥xz£.z| ª =©(Wzx(¡ Þ (ª y | I × J

¬Wwv*©Wxz@=w;£,¯©W*v*xzy £,¯WA〈i,j〉 : 〈i, j〉 ∈ I × J

WÈ¥xzy ¤W¥y ni ª y@xw=§.y ¡m £.y zþ.v*£Ty uw¥È©i¥xz(¡£.y ¡m «.z@=£T%xw z|xw=|2y Ai,j

§.y xzz|Z Aij

²Cø ð ²s³ µn: õñ@¶W³ : ±E¬õ

Ai,j : i ∈ I ∧ j ∈ J JmYg6<Y2573DJICgGK#Va8;46257M2546g67YAC:=IK#N\4ueB=IEg6<Y254ue`<Y625IWB=VKTrWk0F[7YgG>g6]NAaN\Yg67YZWIi ∈ I ÎWY7Y^>x<Yg67?êu462IK#N\T<Y625VWB

Bi =⋂

j∈JAi,j = x : (∀j ∈ J)x ∈ Ai,j,

N4uN\:;F[mYD646257T<Y625VWB⋃

i∈I

⋂

j∈JAi,j =

⋃

i∈IBi.

NaF[7Y^jôN\^>

x ∈⋃

i∈I

⋂

j∈JAi,j ⇔ (∃i ∈ I)(∀j ∈ J)x ∈ Ai,j.

k N\4uN\5IWZW25XY<Y4d>:=DcIW:=VWg67?êu4625Sa8;7Y^>2546467RDJIEgGK|Va8;467Mg6<Y2]NCPhN\462]N@S6IWZWVW5462IW467\b\XY<?>E52⋂

i∈I

⋂

j∈JAi,j,

⋃

i∈I

⋃

j∈JAi,j

2 ⋂

i∈I

⋃

j∈JAi,j .

¸¹ºWá!ÓÈÇ Àu¼t13257YX_I = J = N+ IWB[N\< An,m = [ 1

n,m]

rGk0F[7YgG>∞⋃

n=1

∞⋂

m=1

An,m = (0, 1].

U:;F[I\F[46257\b6g6]Ng6ILK|IW5467YZWIn ∈ N+ ôN\^>

Bn =∞⋂

m=1

An,m =[ 1

n, 1

]

2:=S6^Sa8=e\X@KM:=<?>E:;F[AE257<Y625IWB;>Bn

I\F[B=<?>E^Sa8;7Y^>vXfNCP]>xIEg6XY25467YA(0, 1]

r¡HN\ACY7MDcIEgGK|V\8;467Rg6<Y2]NCPhN\462]NsS6IWZWVW5462IW467|ôNa8=esDJ7?KM467RK@PhN\:=46IW=XY2 bW652:[A\I@<?KM2]e.-

<fN\467T<TD6B[NfK#N\^2B[N\XO_ES646ACSpAdK#N\4dFU>CêuAaNaF[IWB=VLKTr ¿oÁh»C¹a¼ ºd» Ò Áh» Ì ð £Ô+W&;U;e Ai,j : i ∈ I ∧ j ∈ J Ê=Q*ST+W&D[Q .S+mRa¯S*ÊT+W ,[Qd,XZÎY&Qp'

k#m ⋃

i∈I

⋃

j∈JAi,j =

⋃

j∈J

⋃

i∈IAi,j;

kÊjm ⋂

i∈I

⋂

j∈JAi,j =

⋂

j∈J

⋂

i∈IAi,j;

kzUm ⋃

i∈I

⋂

j∈JAi,j ⊆

⋂

j∈J

⋃

i∈IAi,j .

5Vòù¼¨ aª5³.ª ³ µ ª5³ ? ±Côyónó µ ª5± µ ® µ ³x X ª5±E¬Y³O³´ ²Vñ

ãx¾E¿ ¼ ¨MIW<YS6ÎLK#N\462]N3:[e`N\4uN\5IWZW2XY<Y467|g6IsF9>EXO_<#¡ KM257YB=g6<Y7Y462]N`ÚGr åGbFU>E^B[N\<Y7Y^8;7Yg64uN\AoKAC5S6XY<YILKR>E^ÎW^7Y46XY257@AWIWB=<?>E:;FON\^><`¡ KM257YB=g6<Y7Y462]NÚGr Ý r

`gG>I = J

bF[I,A\IWB=<?>E:;FONa8=e\X<F[7YZWIFUKM257YB=g6<Y7Y462]NÎWY7Y^>S6D6B=IW=XY2546I\FON\X98;m/hDJICg6IW646257#8=N\AKD6B=<?>EDuN\g6ACSpAdKRN\4dFU>CêuAaNaF[IWB=VLKo1rwN\^2]Na:;F

⋃

i∈I

⋃

j∈JAi,j

2 ⋂

i∈I

⋂

j∈JAi,j

JmYg6<Y257Y^>xD625:[N\⋃

i,j∈IAi,j

2 ⋂

i,j∈IAi,j.

ã IF[798#DcIWB;>v^VLKM2]e\X@Ig6<Y2]NCPhN\462]N\XO_vS6IWZWVW54625IW4d>EXO_bGB=IW62552=^>F[IK IEg646257Y:=257=-4625Sg6I2546g67YAC:=ILK#N\4d>EX_0B=IEg6<Y254<Y625IWB=VLK /hJIlFON\Ap8;7Y:;FoKR>EZWIEg64625798Q1r#wguN\B=<fNQ:=25m8;7Yg64uN\A,XY<fN\:[N\^2 bJY7ôN\^>g6IxXY<?>E46257Y462]No<B=IEg6<Y254ueo<Y625IWB=VLKTbJAdF[VWB[Nv46257M8;7Y:;F`254C-g67YAC:=ILKRN\4uNGrH13257`8;7Y:;FF[Ipg6S6Y7vSGF[B=S6g646257Y46257\b ZWgG>Eog6<Y2]NCPhN\462]NxS6IWZWVW54625IW467Î\Y4uN<Yg67?êu4625ILK#N\g6]Ng6IK|IW54d>EXO_,B=IEg6<Y254<Y625IWB=VLKTrãx»dè Ò Á Ðî?À¤£Ô+W&;U;e A Ê=Q*ST+W&¯RV+W&Oia$[; *ST+mRaÀS*ÊT+W ,[Qd,XZÎÔY&Q%'

⋃

A = x : (∃A ∈ A)x ∈ A

,[w#*S⋂

A = x : (∀A ∈ A)x ∈ A.

¸¹ºWá!ÓÈÇ Àu¼ 3<fN\:[N\g64625^>Wb6Y7⋃

P(A) = A.

kÜFU>E^%XY7Y5S,S6g6ILK|IEg64625^>Wb6Y7T<fN\X_6ICg6<fegGK#N<fNfKM257YB[N\462]NGr/ ⊆ 1M3:;FON\5^>xg6ILK|IW5467 x ∈ ⋃P(A)

rwQg67?êu4625X98;2 :=S6^>S6IWZWVW54625IW46798#IW<Y4uN\XY<fNF[I6bY725:;F[4625798;7

B ∈ P(A)buFON\AC2 Y7

x ∈ B rw|<?>E52 ôN\^> x ∈ B ⊆ Abc<fNaF[7Y^

x ∈ A bXYI^257Y2[^>vDcIWAaN\<fN\\r/ ⊇ 1@kl7T÷?^>ýF[7YB[N\<g6ILK|IW5467 x ∈ A r¢¢IW46257?K#N\ A ∈ P(A)

b <fNFO7?^µDcIEg6<Y625IWB=7Y^<Y625IWB=S

Abd;KM2]N\g6XY<fe\X?>E^ ITFU>E^,bdY7

x ∈ ⋃P(A)8;7Y:;F|:[N\Î

AbdXYIA\IW6XY<?>g6ILK|VEgr

²Gù ð ²s³ µn: õñ@¶W³ : ±E¬õ;Ù; < Íàx'YÖ rsN\D625:[N\@DJIW4625Y:=<Y7T<YguN\462]N2yhS646A\X98;7<YguN\4625ILK|7\r

/ NO1z\7Y=52J525XY<Yd>x2y:[eB=VWY467\bCF[IN\5JI

x8;7Y:;F#^4625798;:=<Y7sICg

ybEN\5JI

x8;7Y:;F

^4625798;:=<Y7TICgyr

/hB113257ÀaN\YguN525XY<YuN46257YDuN\B=<?>E:;FON@8;7Y:;F3DJIEg6<Y257Y4uND6B=<Y7Y<ÚGr/hX*1©MKRN\g6B[NaFMAaN\Yg6798R525XY<Yd>vB=<Y7YXY<?>CK325:;F[7988;7Y:;F3525XY<Yueg6IEguNaF[462]eGr/hgB1,z\7Y=52!AdK#N\g6B[NaF3525XY<Yd>

x46257|8;7Y:;F3525XY<Yueg6IEguNFO4625eGbGF[I

x8;7Y:;F3<Y7YB=7Y^,r

/h7*1,z\7Y=52n8;7Y:;F25XY<Yue4uNaF[S6BONa54uepKM25mYAC:=<feIEg¤EbHF[I46257x25:;F[4625798=e525XY<Yd>

4uNaF[S6B[N\5467x, y, z

FON\AC257\b6Y7xn + yn = zn

r/iy;1x©MKRN\g6B[NaFMfN\g646798R525XY<Yd>vKR>E^27YB=46798R46257|8;7Y:;F3B=VLKM4C>,¤Er/hZ!1okN\B;F[IW= J7Y<?KM<YZW5m?g64uNsAaN\Yg6798 525XY<Yd>`B=<Y7YXY<?>CK325:;F[798u8;7Y:;FHB=VLKM4uNMF[798!525XY<=-

6257`5S6525XY<Y6257@D6B=<Y7YXY2KM46798?r/h_B1,z\7Y=52

n8;7Y:;F525XY<Yue,4uNaF[S6B[N\54ueGb!F[IDcIDJIEg646257Y:=257Y4625S8;798g6Ig6IK|IW546798

DJI\F[mYZW2!4uNFOS6B[N\46798#I\F[B=<?>EôN\^>v525XY<YJm`4uNaF[S6B[N\54ueGr/h2m1vU:;F[4625798;7T525XY<YuNXfNCPiAWILKM2FONGb6AdF[VWB[NDJIEg6<Y257Y5IW4uND6B=<Y7Y<ÚguN8;7TB=7Y:=<?F[m Ö r/Ø8Q1x©TN\YgG>x<Y625VWBR46257YD6S6:;FU>ôNXYI4uN89KR>EY798MgGK#N7Y57Y^7Y4dF9>Wr/hAn1 Ö 8;7Y:;F37Y57Y^7Y4CF[7Y^j467YSGF[B[N\54C>E^%^46IW?7Y462]N525XY<YB=<Y7YXY<?>CKM25:;FU>EX_r/hm1vU:;F[4625798;7T7Y57Y^7Y4dFM467YSGF[B[N\54C>v^46IWY7Y462]N525XY<YxB=<Y7YXY<?>CK325:;FU>EXO_!r/h^1vU:;F[4625798;7T46257Y:=A\IW6XY<Y7Y46257TKM257Y57`525XY<Yx4uNaF[S6B[N\54C>EXO_r/h4B1xU:;F[4625798;7T46257Y:=A\IW6XY<Y7Y46257TKM257Y57`525XY<Yx46257YDuN\B=<?>E:;F9>EXO_r/hI!1vU:;F[4625798;7T525XY<YuND6B=<Y7YXY2KM4uNg6I525XY<Yd>vB=<Y7YXY<?>CK32:=F[798

xr

/hDB1 ã ]NÀN\Yg6798525XY<Yd>B=<Y7YXY<?>CKM25:;F[798 B=VWY46798ICg<Y7YB[NT25:;F[4625798;7#525XY<YuN@g6IT4625798IEgGKMB=I\F[4uNGr

/n1,z\7Y=52A8;7Y:;F3<Y625IWB=7Y^%46257YD6S6:;FU>E^,b6F[IôNDcIEg6<Y625VWB#K@PhN\=XY2KR>Wr

/hB;1x©TN\YgG>x<Y625VWBRôN4uN\g6<Y625VWB#K@PhN\=XY2KR>\r/h:;1,z\7Y=52

x8;7Y:;F525XY<Yuep46257YSa8;7Y^4ueGb¢F[I

x8;7Y:;FKM25mYAC:=<Y7IEg-AN\Yg6798525XY<Yd>

Sa8;7Y^46798?r/iF=1x13257`25:;F[4625798;7`4uN89KM25mYAC:=<fN525XY<?uNB=<Y7YXY<?>CKM25:;FONGr/hSB1 ã ]NAaN\Yg67YZWI<Y625IWB=S 46257YD6S6:;F[7YZWI25:;F[4625798;7ý<Y625VWBYb#AdF[VWB;>l8;7Y:;F,K4625^

<fNfKRN\B;FU>x5S68;7Y:;F3<T4625^jB=IW<Phe\XY<Y4d>Wr/ Ë1©TN\YgG>p46257YD6S6:;FU>pDJICg6<Y625VWBs<Y625IWB=Sp525XY<YpB=<Y7YXY<?>CKM25:;F9>EXO_ôN46257YD6S6:;F9>

DJIEg6<Y62VWBRKsPhN\=XY2KR>Wr

5Vò6587 ³d¯w³ µ ª5³ ²y5/iKo1x9:;F[4625798;7`4uNa89KM25mYAC:=<fN46257Yg6IEguNaF[462]N525XY<YuNXfNCPiA\IKR2FONGr/V1 ã ILK|IW54d>x<Y625VWBH8;7Y:;F3D6S6:;FU>5S6ôND6S6:=FU>DcIEg6<Y625VWBYr©@F[VWB=7`<`FU>EX_KR>EB[N\Y7Yp:[e<YguN\462]N\^2½3T©@F[VWB=7`:[eD6B[NfKMg6<Y2K#7*3

¤Ersw4uN\57T÷YT<Y^257Y46467TKR>E:;F[mYD6Sa8=e\XY738=N\AWIK#IW5467T2¢<?KR2]e\<fN\467K DJIW4625Y:=<?>EXO_pKR>O-B[N\Y7Y462]N\XO_r/ NO1

(∀x (x > 3))⇒ y < 5k

/hB1(∃x (x > 3))⇒ x > 3

k/ X1 ∃x∀y (x > 3⇒ y < 5)⇒ ∀x (x 6= 7)

k/hgB1 ∃x∃y (x 6= y ∨ z ≤ 0)

k/ 71 ∀x (x+ y = z)

k/iy;1 ∀x∀y (x+ y = z)

k/hZ!1 ∃x∀y (x+ y = z)⇒ x− y 6= z

k/h_B1 ∀x∀y (x+ y = z) ∧ ∃z (x− y 6= z)

k/h2m1 ∀x (x+ y = z)⇒ ∃z (∃y (x− y 6= z) ∧ ∀z (x · y ≥ z))

k/Ø8Q1 ∃x (x < y ∨ x < z)

k/hAn1 ∃x∀y (x < y ⇒ x < z ∧ z < y)

k/hm1 ∀x∃y (x < y) ∨ x < z

rÚGrs|Iº Ò ÀuÐWº,DcIW4625Y:=<Y7<YguN\462]NO3

/ NO1(∀x ∈ R)(∃n ∈ N)x < n

k/hB1 ¬(∃x ∈ R)(∀y ∈ R)x ≤ y

k/ X1

(∃x ∈ N)(∀y ∈ N)x ≤ yk

/hgB1(∀k ∈ Z)(∃l ∈ Z) k + l = 0

k/ 71

(∀x ∈ R)(x ≥ 0⇒ (∃y ∈ R)x = y2)k

/iy;1(∀x, y ∈ R)(x < y ⇒ (∃q ∈ Q)x < q < y)

rñ r33g6ILK|IEg64625!7YôNaFMÚGr Ö rÝ r33<YS6DJ7Pi4625Tg6ILK|VEg,¡ KM257YB=g6<Y7Y462]NÚGrØ¤EråGr3¢IEguN\RD6B=<?>EA6PhN\gbdY7#K¡ KM257YB=g6<Y7Y4625SÚGrØ¤Ë/h7*1H4627RÎWY4uN@<fN\:;FOedD625#KR>E4625AN\462]NB=VLKM46ILKRN\Y46IW=XY2]eGr

ì r33g6ILK|IEg64625T¡ KM257YB=g6<Y7Y46257ÚGr ÚGr

²!6 ð ²s³ µn: õñ@¶W³ : ±E¬õ×GrsN\467YZWILK#N\DJIW4625Y:=<Y7KR>EB[N\Y7Y462]NÍ/iF[<Y4rL<fN\D625:[N\25XO_T467YZdN\X98;7 c7Y< S6?>EXY2]NM:;>E^Ð-cIW5S467YZdN\X98;2m1r/ NO1

(∀x ∈ R)(∃n ∈ N)x < nk

/hB1(∃x ∈ N)(∀y ∈ N)x ≤ y

k/hX*1

(∀x ∈ R)(x ≥ 0⇒ (∃y ∈ R)x = y2)k

/hgB1(∀x, y ∈ R)(x < y ⇒ (∃q ∈ Q)x < q < y)

k/h7*1 ∃x∀y(y > 0⇒ ∃z(x = y · z)) k/iy;1 ∃x(x < 3 ∨ ∀y(y > x⇒ y ≥ 3))

k/hZ!1 ∀x(x 6= 2⇒ ∀z(z ≥ x)) ∨ ∃y(y = 2⇒ ∃t(y > t))

k/h_B1 ∀x(x 6= 0 ∨ ∃y(x · y 6= x))

ró r3©@F[VWB=7o<xDJIW4625Y:=<?>EXO_KR>EB[N\Y7YguN\g6<fe:=25mvS6D6B=IW=XY25xD6B=<Y7Y<xS6?>EXY257xAdK#N\4dF9>O-êuANaF[IWB=VLKIWZWB[N\4625XY<YIW4d>EX_B3/ NO1 ∀x(x ≥ 0⇒ ∃y(y2 = x))

k/hB1 ∃x∀y(y ≥ x ∧ ∃z(z > y))

k/hX*1 ∀x∃y(y 6= 0⇒ y ≥ x)

k/hgB1 ∀A∃B(∀x(x ∈ B ⇒ x ∈ A) ∧B 6= ∅) r

ÖLø r33<YS6Dc7Pi4625Tg6IK|VEg¡ KM257YB=g6<Y7Y462]NÚGr ñ rÖWÖ r3B=<Y7YD6B=ILK#N\g6<Y25oyhIWB=^oN\54d>g6ILK|VEgbHY7oAdKRNa4CFU>CêuAaNaF[IWB;>IWZWB[N\462XY<YIW467\b¢AdF[VWB=7

AdK#N\4dFU>CêuACS\8=eDcI46257Y<fN\57YY4d>EXO_ <Y^257Y464d>EXO_:[eD6B=<Y7Y^257Y46467 / ¥@XY<?>CKM25=XY257XO_6IEg6<Y2!IAdKRN\4dFU>CêuAaNaF[IWB;>xF[7YZWI:[N\^7YZWIF9>ED6SB1r

Ö ¤Er3¢IEguN\D6B=<?>EA6PhN\gG>yhS646AWX98;2 <YguN\4625IK#>EX_ϕ(x)

2ψ(x)

Ip<fN\ACB=7Y:=257<Y^257Y4646798x ∈ R DJIWAN\<YSa8=e\XY7\bEY7sDJIW4625Y:=<Y7s<YguN\462]N139§:[eD6B[NLK#N\^2cB[N\XO_CS646ACSxAdK#N\4C-FU>CêuAaNFOI\B[VLKTr/ NO1 ∃x(ϕ(x)⇒ ψ(x))⇒ (∃xϕ(x)⇒ ∃xψ(x))

k/hB1

(∀xϕ(x)⇒ ∀xψ(x))⇒ ∀x(ϕ(x)⇒ ψ(x))k

/hX*1(∃xϕ(x)⇒ ∃xψ(x))⇒ ∀x(ϕ(x)⇒ ψ(x))

rÖ ÚGr3¢IEguN\æD6B=<?>EA6PhN\gG>µyhS646A\X98;2<YguN\4625ILKR>EXO_

ϕ(x, y)I<fN\ACB=7Y:=257j<Y^257Y4646798

x, y ∈ R DJIWAN\<YSa8=e\XY7\bRY7DcIW4625Y:=<Y7<YguN\462]N139§j:[eD6B[NfK#N\^2sB[N\X_CS646ACSAdK#N\4dFU>CêuAaNaF[IWB=VLKTr/ NO1 ∃x∃y ϕ(x, y)⇒ ∃xϕ(x, x)

k

5Vò6587 ³d¯w³ µ ª5³ ².9/hB1 ∀ϕ(x, x)⇒ ∀x∀yϕ(x, y)

rÖ ñ r ã ]NDcIEguN\4d>EX_,B=IEg6<Y2542546g67YAC:=ILKRN\4d>EXO_ Ai : i ∈ I KR>E<Y4uN\XY<?>E

⋃

i∈IAi

IWB[N\< ⋂

i∈IAi.

/ NO1I = R, Ai = x ∈ R : i ≤ x k/hB1I = N, Ai = x ∈ R : |x− 3| < 1

i+1 k/ X1

I = R, Ai = x ∈ R :F[Z

(x) = i k/hgB1I = N, Ai = x ∈ R : 1− 1

i+1≤ x ≤ 2 + 1

i+1 k/ 71

I = N+, Ai = (1− 2i, 1 + 1

i)k

/iy;1I = N+, Ai = x ∈ R : 1− 1

i≤ x < i+ 1

i k

/hZ!1I = N+, Ai = x ∈ R : 1− (−1)i

i< x ≤ (2 + 1

i)2 k/h_B1

I = N+, Ai = x ∈ R : 1− 1i≤ x < 2− 1

i2 k/h2m1

I = N+, Ai = [i, i2]k

/Ø8Q1I = (0,+∞), Ai = 〈x, y〉 ∈ R2 : y ≤ i · x k/hAn1I = (0,+∞), Ai = 〈x, y〉 ∈ R2 : y = i · x r

ÖLÝ r313257YXO_ Ai : i ∈ I cmYg6<Y257T2546g67YAC:=IK#N\4ueB=IEg6<Y254ueDcIEg6<Y625IWB=VLKD6B=<Y7?:=F[B=<Y7Y462YIWB[N\<

A ⊆ Yrc3g6ILK|IEg64625`DJIW4625Y:=<Y7`K@PhN\:=46IW=XY2 r

/ NO1 ⋂

i∈IAi ⊆

⋃

i∈IAi;

/hB1(∀i ∈ I)(Ai ⊆ A)⇒

⋃

i∈IAi ⊆ A;

/hX*1(∀i ∈ I)(A ⊆ Ai)⇒ A ⊆

⋂

i∈IAi.

Ö åGr313257YXO_ Ai : i ∈ I 2 Bi : i ∈ I cmYgue-2546g67YAC:=ILK#N\4C>E^2@B=IEg6<Y254uN\^2<Y625IWB=VLKTruHIWAN\<fN\\bcY7/ NO1

(∀i ∈ I)(Ai ⊆ Bi)⇒⋃

i∈IAi ⊆

⋃

i∈IBi;

/hB1 ⋃

i∈IAi ∩

⋃

i∈IBi =

⋃

i∈I

⋃

j∈I(Ai ∩Bj);

/hX*1 ⋂

i∈IAi ∪

⋂

i∈IBi =

⋂

i∈I

⋂

j∈I(Ai ∪Bj).

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I = 1, 2 b6F[I⋃

i∈IAi = A1 ∪ A2

IWB[N\< ⋂

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Ö ×Gr3¢IWAN\<fN\\bY7 ⋃

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i∈I Ai8;7Y:;F`4uN8-

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(f) ⊆ Yb6F[I:=D6B[NfKMg6<Y7Y46257B=VLKM46IK#N\Y46IW=XY2 IW6SFU>EXO_pg67?êu4625X98;2

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yhS646A\X98;m\rJN\4625^8;7Yg64uN\AvD6B=<Y798;g6<Y257Y^>vg6IF[7YZWI<fN\ZdN\g646257Y462]NGbE<fN\SCK#N\Y^>Wb6Y7TDJIa8;m=-XY257@g6<?257Yg6<Y254C>yhS646A\X98;2E8;7Y:;FM46257YB=IW<Y7YB;K#N\546257s<?KM2]e\<fN\467@<@DJI\8;mYX?257Y^æyhS646A\X98;2 bG25464d>E^2:PiILKR>WbL8;7Y=52u<Y4uN\^>yhS646A\X98;m\bCF[I<?4uN\^>@8;798g6<Y257Yg6<Y2546m\rGz\7Y:;F FOI25:;F[I\F[467M:=DJIW:;F[B=<Y7YY7=-46257\b8=N\A\IY7,K:=<YA\IW57pXY<YmY:;F[IB=IW<Yg6<Y257Y]N:=25mF[7pgGK#NlDJIa8;mYXY2]NGb XYI<,yhIWB=ôN\467YZWID6S646AdF[SKM25g6<Y7Y462]Ns8;7Y:;F346257YDJIWD6B[NfKM467\r¸¹ºWá!ÓÈÇ Àu¼ ~kQ>E<Y4uN\XY<?>Eg6<Y257Yg6<Y2546m@yhS646AWX98;2

f(x) = 1x

r ¡HN\A:;yhIWB=^SJPiILK#N\467DJIW57YXY7Y46257A/hXY<YmY:;F[7oK :=<YA\IW54d>EX_<Y625IWB[N\XO_l<fN\guN\B1T<D6S646AdF[SKM25g6<Y7Y462]N,4uN\:=<Y798g67?êu4625X98;2!8;7Y:;Fc7Y<Y:=7Y46:[ILKM467WbHJIp^>ýF[798`yhS646AWX98;2 DJIpD6B=IW:;F[S46257<Y4uN\^>Wr N\Î<fN\guN\46257IEXY<?>CKM25=XY27ôN:=7Y46:Yb¢N\57DcIWD6B[NLKM467DJIW57YXY7Y46257DJILKM254646I6B=<Y^257Y%bu~kQ>E<Y4uNaXY<?>EôN\AC:;>EôN\54d>,DJICg6<Y625VWBs525XY<YB=<Y7YXY<?>CKM25:;FU>EX_

D ⊆ R bJg6]NAdF[VWB=7YZWIKR>EB[N\Y7Y46257f(x) = 1

x

/hZWg6<Y257x ∈ D 1|ôN:=7Y46:3525XY<YJILKR>dOr

zWN\AK FON\AC25^æB[N\<Y257@IWD625:;>CKRNa`yhS646A\X98;m/iKM257Y^>T8;S6\b6Y7T:[N\^æKM<YVWBR46257sKR>E:;FON\B-XY<?>Ë1w31sNa8;XY<YmY=XY25798#FON\ABb

f : X → Y, f(x) = przepis.

² ö

z\7Y:;FRF[I4uN8;XY<YmY=XY25798#KR>EA\IWB=<?>E:;F9>CKRN\4d>WbcXO_6IE`46257|8;7YgG>E4d>x:=DJIW:=VWr N\^ [wS.&+_T4uNIWZWVEPW8;7Y:;FKM<YIWB=7Y^,bWXO_6IERd>CK#Na8=eùN\B=g6<Y25798H:=A\IW^D6525AWILK#N\467#:;>CF[SuN\X98;7\bd4uN`D6B=<?>EA6PhN\gyhS646A\X98=Ns8;7Y:;FRIWD625:[N\4uNAC255A\IWôNTK#N\B=S646AaN\^2 rG13257YAC257YgG>K <?KM2]e\<YACS<sFU>E^jDcI\8=NfKM2]N:=25mA\IW46257YXY<Y46IW=o:=D6B[NfKMg6<Y7Y462]NYV ;[w#%X¢R *\*U+!&Dì¢RV+WU P +MyhS646A\X98;2 b!XY<?>E52S6DJ7?KM46257Y462]N:=25m\bWXY<?>D6B=<?>EDuN\g6AC257Y^Æ46257#D6B[<>ED625:[N\525=^>R8;7Yg6467Y^SN\B=ZWS6^7Y4CF[ILKM2GgGK#VCXO_B=VWY4d>EX_K#N\B;F[IW=XY2 r¸¹ºWá!ÓÈÇ Àu¼¥@ACB=7Y=5^>vyhS646A\X98;m

f : R→ RKM<YIWB=7Y^

f(x) =

x2 8;7Y=52

x ≤ 0x+ 1

8;7Y=52x ≥ 0

.

N\SdKRN\Y^>WbCY73462578;7Y:;F F[IDJIWD6B[NfKM4d>IWD625:YbCZWgG>E3<sD6257YB;KM:=<Y7YZWIKRN\B=S646ACSoKR>E4625ANGbY7f(0) = 0

/hXY<?>E52 〈0, 0〉 ∈ f 1bcN<g6B=S6ZW257YZWI6b6Y7 f(0) = 1/hXY<?>E52 〈0, 1〉 ∈ f 1buXYI8;7Y:;F3:=D6B=<Y7YXY<Y467<Tg67?êu4625X98=eyhS646AWX98;2 ru¢IWD6B[NLKM4d>I\D625:M^VWZEPiC>oKR>EZW]e\guNaT46Dr6FON\ABb

f(x) =

x2 8;7Y=52

x < 0x+ 1

8;7Y=52x ≥ 0

.

k0F[7YgG> K#N\B=S646AC2F[7%g67?êu4625Sa8=eyhS646AWX98;m%4uN B=IW<Phe\X?<Y4C>EXO_t<Y625IWB[N\XO_(−∞, 0)2

[0,+∞)b6KM25mYXT:;>CF[SuN\X98=NFON\ANGb\8=N\AoDJIWD6B=<Y7Yg64625I6bu46257`ÎWY7`^257Y`^257U8;:[XYNGr

¨MIW<YDuNaF[B=<?>E^>oF[7YB[N\<TAC255AaNK#N\Y4C>EXO_pD6B=<?>EA6PhN\g6VK yhS646AWX98;2 r¸¹ºWá!ÓÈÇ Àu¼!áÖ r h S646A\X98=N25g67Y4dF9>EXY<?46IW=XY25IK#NGrã ]Ng6ILK|IW5467YZWI46257YD6S6:;F[7YZWI<Y625IWB=S

XÎWY7Y^>oIWACB=7Y=525syhS646A\X98;mT25g67Y4dFU>EXY<=-

46IW=XY25ILK#e4uN<Y625IWB=<Y7X/h25g67Y4dF9>EXY<Y46IW=4uN

X1|4uN\:;F[mYD6Sa8=e\XYIËb

idX : X → X, idX(x) = x.

¤Er h S646A\X98=NXO_uN\B[N\AdF[7YB;>E:;F9>EXY<Y4uNGr13257YXO_XJmYg6<Y257 S6:;FON\5IW4ueMD6B=<Y7Y:;F[B=<Y7Y462]eGr ã ]NMg6IK|IW5467YZWI3DJICg6<Y625IWB=S

A ⊆ Xg67?êu4625Sa8;7Y^>vyhS646A\X98;mXO_uN\B[N\AdF[7YB;>E:;FU>EXf<Y4ue<Y625IWB=SA4uN\:;F[mYD6Sa8=e\XYIËb

χA : X → 0, 1, χA(x) =

18;7Y=52

x ∈ A08;7Y=52

x 6∈ A .

ÚGr3¨M<YSGFfrã ]Ng6ILK|IW54C>EXO_,46257YD6S6:;FU>EXO_p<Y625IWB=VK

X2YÎWY7Y^>x<?g67?êu4625IK#N\`yhS646AWXU8;m

πX : X × Y → X, πX(〈x, y〉) = x,AdF[VWB[e4uN\<?>CK#N\^>B=<YSGF[7Y^4uN

X/hB=<YSGF[7Y^4uND6257YB;KM:=<fNIW;1?rR¥@XY<?>CKM25=XY257

N\4uN\5IWZW2XY<Y46257@ÎWY7Y^>vIWACB=7Y=525`B=<YSGF34uNg6B=S6ZdeIWYr

Å ø ?µ ¶C³[®ñ rs|2]edZ525XY<YcIKR>Wr¥@AN\<YSa8;7 :=25m\bLY7 g6IW6B=<Y7 <Y4uN\4d>TIW6257YAdFfbLXY<?>E52dXY2]eCZM525XY<YJILKR>@F[7Y!8;7?:=F¢D6B=<?>EA6PhN.-g67Y^yhS646A\X98;2 r¢13Dr g6ILK#IW54uNxyhS646A\X98=N

a : N → RF[I,DJIpD6B=IW:;F[SlXY2]edZ,525XY<Y

B=<Y7YXY<?>CKM25:;FU>EX_î4oKR>E:;FON\B=XY<?>IW<Y4uN\XY<?>Ea(n) = an

bud>vF[I<fN\SCK#N\?>E\r|2]edZp525XY<YJILKR>QIW<Y4uN\XY<fNa^>Q4uNpIWZWVEP 〈an : n ∈ N〉 5S6 〈an〉n∈N

/iK s4uN.-525<Y27npNaF[7YôNaFU>EXY<Y46798`S6?>CK#N:=25m

(an)n∈N

1r ã 7?êu4625Sa8=e\XA\IW46ACB=7?F[4d>ýXY2]eCZx4uNIWZWVEP#KR>E:;FON\B=XY<?>-DcIEguN\xg67?êu4625Sa8=e\X?>ZWIKM<YVWB /h46Dr

an = 1n

1E4Qg6IW^>E=546257<fN\A6PhN\guNa^>Wb6Y7#8;7Y:;F3IW4,IWACB=7Y=5IW4d>xg6]NKM:=<?>E:;F[AC25X_p525XY<Yx4uNaF[S6B[N\54C>EXO_rklN\B;F[IR8;7Y:=<YXY<Y7s<fN\SdKRN\?>E\bCY73XY2]eCZ 〈an : n ∈ N〉 8;7Y:;F XY<?>E^ 25464d>E^Æ4625M<Y625VWBan : n ∈ N =

B=46Z(a)r¢U:;F[I\F[46257\b¢B=IW<?K#N\Y^>QXY2]eCZ

an = (−1)nr k0F[7YgG>

〈an : n ∈ N〉 8;7Y:;F|IW627YAdF[7Y^ÊU46257Y:=AWIW6X?<YIW4C>E^À/iyhIWB=ôN\54627%bW46257Y:=A\IW6XY<YIW4d>E^<Y625IWB=7Y^jDuN\BRS6DJIWB=<fe\g6A\IK#N\4d>EXO_1bcDJICg6XY<fN\:3ZWgG> an : n ∈ N = −1, 1 rÝ rMU46g67YAC:=ILK#N\4uNB=IEg6<Y254uN<Y625IWB=VLKTr¥@XY<?>CKR25=XY257\buB=IW<?K#N\fN\`ÎWY7Y^>o46257sFU>E5A\IXY2]edZW2w525XY<YcIK|7\r6©TN\YguNyhS646A\X98=NGbAdF[VWB=798|g6<Y257Yg6<Y254ue38;7Y:;FM<Y625VWB#525XY<Yv4uNaF[S6B[N\54C>EXO_8;7Y:;FMXY2]eCZW257Y^,r <YXY<Y7YZWVW54d>E^D6B=<?>EA6PhN\g67Y^jÎWZdeC>E@XY2]eCZW2J<Y625IWB=VLKTr63IWZWVW54627Y46257Y^ <fN\RXY2]edZWVK<Y625IWB=VLK:[e2546g67YAC:=IK#N\467B=ICg6<Y254C>v<Y625IWB=VLKTrk DJIWD6B=<Y7Yg64625^B=IW<Yg6<Y2]N\57|B=IW<?KRNafN\525=^>:=S6^>2GD6B=<Y7YACB=Ia8;7R2546g67YAC:=IK#N\4d>EX_B=ICg6<Y254<Y625IWB=VLKTbwF[B[N\AdF[Sa8=e\XF[IDcI\8;mYXY257254dF[S625X?>L8;46257\bwXYI,d>uPiI,<YS6Dc7Pi46257KR>O-:;FON\B=XY<fN8=e\XY7\r ã IDJ7?KM4d>EXO_<fN\:;F[IW:=ILKRN\F[B=<Y7YuN@8;7#8;7Yg64uN\Av:=D6B=7YX?>E<YILKRN\\r13257YXO_

IcmYg6<Y257<Y625IWB=7Y^ 46257YD6S6:;FU>E^ /h4uN\<?>CK#N\^>ýZWIx<Y625IWB=7Y^ 2546g67YAC:=VKo1b

NXg6IK|IW54d>E^ <Y625IWB=7Y^,r U46g67YAC:=ILKRN\4ueB=IEg6<Y254uelDcIEg6<Y625IWB=VLK<Y625IWB=S

X4uN\<?>CK#N\^>g6ILK#IW54uesyhS646A\X98;mA : I → P(X)

/h254dF[S625X?>L8;46257Z4TU4ES6^7YB=Sa8;7Y^>dDc7?KM467DcIEg6<Y625IWB;>x<Y625IWB=S

X7Y57Y^7Y4dFON\^2w<Y625IWB=S

I1?ruHICg6IW646257#8=N\AoK KR>O-

DuN\g6ACSXY2]eCZWSbW246g67YAC:=IK#N\4ueTB=IEg6<Y2546mR<Y625IWB=VK<fN\D625:=Sa8;7Y^>s8=N\A\I 〈Ai : i ∈ I〉5S6 〈Ai〉i∈I r¢¢IEg6IW64627F[7Y`8=N\ApKR>EDuN\g6ACSXY2]eCZWSXY<?>E^µ25464d>E^j8;7Y:;F2546g67YAp-:=ILK#N\4uNB=ICg6<Y254uNp<Y625IWB=VLK 〈Ai : i ∈ I〉 bHNpXY<?>E^ 25464d>E^µB=IEg6<Y254uNp<Y62IWB=VKAi : i ∈ I rN\SdK#N\Y^>8;7Yg64uN\AcbY7xB=VWY4625XYNpFON46257vICg6ZWB;>CKRNB=IW52 D6B=<?>g67?êu4625ILK#N\4625SS6IWZWVW54625IW46798p:=S6^>2D6B[<?7YACB=I\8;SÊB=ICg6<Y254C><Y625IWB=VKTr`q|mYg6<Y257IW4uN25:;F[I\F[4uNlg6IWD6257YB=ID6B=<?>g67?êu4625ILKRN\4625S0S6IWZWVW5462IW467YZWI255ICXY<?>E4ES0AN\B;F[7Y<=-8=N\6:=AC257YZWIGr

k 8;7Yg64d>E^<DJILKR>EY:=<?>GXO_D6B=<?>EA6PhN\g6VLKÊB=IW<?K#N\fN\525=^>xyhS646A\X98;mB=<YSGF[SbJIWACB=7Y-5IW4ueM4uNs255IEX?<?>E46257 AaN\B;F[7Y<98=N\6:=AC25^

X×Y rWz\7Y:;FF[IsD6B=<?>EA6PhN\gyhS646A\X98;2GgGK|VEXO_<Y^257Y4C-4d>EXO_rE¥@ZWVW54627\bIv;CRn)OU P aÓ%Xcd*U;eÍST"Ð+W&RVRV'OU;e4uN\<?>CK#N\^>g6IK|IW54ue3yhS646AWX98;mfIWACB=7Y-

5IW4ueQ4uNl255IEXY<?>E46257,AaN\B;F[7Y<98=N\6:=AC25^ gGK|VEXO_<Y62IWB=VKTr ã ]NlS6D6B=IW:=<YXY<Y7Y462]N<fN\^2]N\:;Ff(〈x, y〉) D625:=<Y7Y^>vKRF[7YgG> f(x, y)

r6k N\4uN\5IWZW2XY<Y4C>v:=DcIW:=VWpÎWY7Y^>x<Yg67êu4625IK#N\yhS646A\X98;mTF[B=<Y7YXO_bcXY<?F[7YB=7YXO_p2!IWZWVW54627@g6IK|IW546798#:=A\IW6XY<YIW46798M255IW=XY2<Y^257Y464d>EXO_r

6Bò½ñÒ7 ª ³ 8µ ±y×=³aª ?µ ¶C³Oª Å ñ

¸¹ºWá!ÓÈÇ Àu¼ h S646A\X98=egGK#VCX_x<Y^257Y464d>EXO_8;7Y:;F|yhS646AWX98=N plus : R×R→ R<fN\guN\4uN

KM<YIWB=7Y^plus(x, y) = x+ y

rklB=7Y:=<YXY257\b <YguN\B=<fN:=25m\bY7B=IW<YDuNaF[B=Sa8;7Y^>yhS646A\X98;7\b AdF[VWB;>EXO_lD6B=<Y7YXY2KMg6<Y27Yg6<Y254ue

8;7Y:;F@255IEXY<?>E4pAaN\B;F[7Y<98=N\6:=AC2HgGK|VEXO_Ã/h5S6,KM25mYXY798Q1M<Y625IWB=VLKTrJHIa8=NfKM2]N:=25mKRF[7YgG>pDcI,-8;mYXY257`:=A6PhN\g6IK|798RyhS646A\X98;2 rãx»dè Ò Á Ðî?À £Ô+W&;U;e

A,B ,[w#*S

CÊ=QOa ! ,Xc ,^lRV',"Ð+ÍS*ÊT+W ,[w#%"Ð+ÐRV+W&O(',"Ð+_¿ÐS*#.\

f : A→ B × C ! ,Xc ,^lRaÆv;CRC)OU P a!ÎDY&Q%'n^#À)p#*¸*!&i`! a ∈ A +_(RV+W& P a&^_&"&RV('

f1(a) ∈ B+f2(a) ∈ C

¿Di#.),+W&Ó¸.&f(a) = 〈f1(a), f2(a)〉

ÎÐ &R iB **dpÊ *),[;&;\=Ñ^l+m^l+_\"Ð'v;CRn)OU P &

f1 : A→ B+f2 : A→ C

Î~£¯#*ST',XZ#%"Ð' P &xzhoywçfté*, ì§kz =V ,ÑX¢+W&Q%RV+W Ô+W&[=XÈQS*aÒ+Z%[=p`Oamv;CRC)OU P +

fÎ

1sN0<fN\A\IW6XY<Y7Y46257F[7YZWIDJIEg6B=IW<Yg6<?2]NCPiSS6IWZWVW54625^> DcI\8;mYXY257255ICXY<?>E4ES AaN\B;F[7Y<=-8=N\6:=AC257YZWI6rãx»dè Ò Á Ðî?À£Ô+W&;U;e 〈Ai : i ∈ I〉 Ê=Q*ST+W&À+mR!&;)** ,XZ#%Ra[Q *ST+mRaÍB *S*ÊT+W ,[Qd,XÃS*ÊT+W ,[=XÎoY&Q%'ÒèÈé,ëf(Ëê>+Èìwé+~ ìê(éup®r/+10/¾oVç&pæ0®rq,çÒ~zhonì & P [Q *ST+mRV'ÐR#*ST',XZ#%"Ð'

S*ÊT+Wd,[∏

i∈IAi = f ∈ XI : (∀i ∈ I)f(i) ∈ Ai.

N\SCK#N\Y^>WbJY7M8;7Y=52¢g6]NAN\Yg67YZWIi ∈ I ^oN\^> Ai = X

bcF[IoKRF[7YgG>pI\F[B=<?>E^SC-8;7Y^> ∏

i∈I Ai = XI bXYI<YZdN\g6<fNý:=25mx<DcI\F[IEXY<Y4ue24CF[7YB=D6B=7?FON\X98=eDJI\F[mYZWILKRN\462]N2;KM2]N\g6XY<?>IFU>E^,b6Y7D6B=<?>L8;m?F[7TIW<Y4uN\XY<Y7Y462]N46257`:[eD6B=<?>EDuN\g6A\IK|7\r

YÙTØ ¡W YÝ§Ø|'! %Ûx&§fw'kÜFU>E^%B[I\<Yg6<Y2]N\57`DJICguN\^>vDuN\B=m`g67?êu4625X98;2!<Y255S6:;F[B=ILKRN\4d>EXO_,D6B=<?>EA6PhN\guN\^2 r

ãx»dè Ò Á Ðî?À¤£Ô+W&;U;ef : X → Y

Î¥ Î d,X¢+m"Ð'*¿¸.&Õv;CRn)OU P #

fP &;&)(y-/+ïé*Íç&pæQéppìwé*Íç ++_QS.&"Ð'

f : X −→1−1 YP &;\^l+[Qd¸TRV'," #%[i`%C"&RV ," [wST';V ,[wS*aO.)O ,X¢ P &À ,R#$[Qd¸TR&¯XZ#%[= *\*U+_¿2UQST',^l+

(∀x1, x2 ∈ X)(x1 6= x2 ⇒ f(x1) 6= f(x2)).CRn)OU P ¯i#.)paR#*ST',XZ#%"Ð'Ò&Q¸Àì+.qK0Kopq,íð^lËÊÀ"d,X¢+m"Ð'*¿È¸.& P &;#"$"VÎ« Î d,X¢+m"Ð'*¿Ô¸.& v;CRn)OU P #

fP &;#%/+¢ç'&ù+2+_QS.&"Ð'

f : X −→na YP &;\^l+o)p#*¸*%'

&^_&"&RV P & P [wS.&;U+mXZ.S+W&;ST+mRV' P &;ZXZ#%[= *\*U+(av;CRn)OU P +_¿2UQST',^l+(∀y ∈ Y )(∃x ∈ X) y = f(x).

Å ù ?µ ¶C³[®CRn)OU P ¯i#.)paR#*ST',XZ#%"Ð'Ò&Q¸§z,è4&`qK0Kopq,íÎ

° Î¯ÏV&;\^l+.v;CRn)OU P #fP &;ï[Qd¸TR ,XZ#%[= *\*U+W ,XZ#Ð+)(TR#+* ÒR#*ST',XZ#%"Ð' P a©*úr!ç)qK0/¯+Èì`0©qK0ht+ïéyr/+¢çp®r/+¢íf+Æ+_QS.&"Ð'

f : X −→1−1na Y

ÎoÖÂi#.),+W& P v;CRn)OU P +Z"d,X¢+m"Ò'&Q¸=¿È¸.& P &;\Oì¸qK0Kopq.íÎ

N\SdKRN\Y^>Wb Y7,yhS646A\X98=Nf8;7Y:;FU4uNxKRF[7YgG>2RFU>E5A\IQKRF[7YgG>Wb|AC257?gG>ý8;798<Y625VWB

K#N\B;F[IW=XY2d8;7Y:;F#XfNCPheD6B=<Y7YXY2KMg6<Y257Yg6<Y254ueGr6z\7Y=52JôN\^>g6ILK#IW54ue`yhS646AWX98;mf : X → Y

bF[IyhS646A\X98=N

f ′ : X → B=46Z(f)

<fN\guN\4uNK#N\B=S646AC257Y^f ′(x) = f(x)

8;7Y:;FRU4uNEjLrJz\7Y:;FF[IÓ÷YB=VCgJPi7Y^æDJ7?KM4d>EXO_F[B=S6g646IW=XY2JK <YB=IW<YS6^27Y4625SF[7YZWIDJIa8;mYXY2]N¯446257 8;7Y:;F#IW46IFON\A4uNaF[S6B[N\5467s8=N\ADJIa8;mYXY257B=VWY46IK#N\B;F[IW=XY25ILK|IW=XY2 r ¢IWD6B[NLKM46257<Yg67?êu4625ILK#N\4uNyhS646A\X98=N/hB=IW<?K#N\fN\^>yhS646A\X98;7RK:=7Y46:=257RD6257YB;KM:=<Y798g67?êu4625X98;2m1¢DJILKM25464uN@^257Y¢8;7Yg646IW<Y4uN\XY<Y4627IWACB=7Y=5IW4uesD6B=<Y7YXY2KMg6<Y257?g6<Y2546m\bWXYI@S6ÎWY52KM25NM:=D6B[NLKMg6<Y7Y4627\bWXY<?>M8;7Y:;FIW4uNs:=S6Bh8;7YA\X98=eGr¸¹ºWá!ÓÈÇ Àu¼!áÖ r3¨MIW<?K#N\Y^>xyhS646AWX98;m

f : R → R<fN\guN\4ueKM<YIWB=7Y^

f(x) = x2 rJ13257#8;7Y:;F@IW4uNB=VWY46ILK#N\B;F[IW=XY25ILKRNGbWJIf(1) = f(−1) = 1

rC13257H8;7Y:;F IW4uN`F[7Y#U4uNObWZWgG>EM4625725:;F[4625798;7525XY<YuNB=<Y7YXY<?>CK325:;FON

x ∈ R FON\AaNGbJY7 f(x) = x2 = −1rw1sNaF[IW^2]N\:;F

yhS646A\X98=Nf ′ : R → [0,+∞), f ′(x) = x2 8;7Y:;F@:=S6Bh8;7YA\X98=eGbwZWgG>Eg6]NAaN\Yg67YZWI

y ∈ [0,+∞)25:;F[4625798;7

x =√y ∈ R b6FON\AC257`Y7 f ′(x) = (

√y)2 = y

r¤Er313257YXO_yhS646AWX98=N

g : (0, 1) → (0, 2)JmYg6<Y257p<fN\guN\4uNKM<YIWB=7Y^

g(x) = 2xr

z\7Y:;F#F[IyhS646A\X98=NB=VWY46ILKRN\B;F[IW=XY25ILK#NGbGcIg6]Ng6ILK#IW54d>EXO_x1, x2 ∈ (0, 1)

8;7Y=52x1 6= x2

F[Ig(x1) = 2x1 6= 2x2 = g(x2)

r¢z\7Y:;FIW4uNvFON\ACY7U4uNObwZWgG>Eg6]Ng6ILK|IW5467YZWI

y ∈ (0, 2)8;7Y:;F

x = y2∈ (0, 1)

b6FON\AC2!Y7g(x) = 2 · y

2= yrJ|<?>E52

g8;7Y:;F362 8;7YAWX98=eGr

ÚGr313257YXO_π1 : R×R→ R, π1(x, y) = x

cmYg6<Y257`B=<YSGF[7Y^æ4uND6257YB;KM:=<feIWYruzWN\APhNaFUK|I:=D6B[NLKMg6<Y25#8;7Y:;FMF[I:=S6Bh8;7YA\X98=NGbJN\57`46257|8;7Y:;F3254a8;7YA\X98=NGr

ñ r3¨MIW<?K#N\Y^>yhS646A\X98;mh : N × N → N

IWD625:[N\4ueKM<YIWB=7Y^h(n, k) = n2 + k2 r

h S646AWX98=NsFON@46257¢8;7Y:;FU4uNObaJI`46DrW525XY<YuN ì 46257¢8;7Y:;F :=S6ôe@gGK|VEXO_AdKRN\g6B[NaF[VLK525XY<?4uNaF[S6B[N\54d>EX_ru13257|8;7Y:;FMF[7Y1− 1

b6JI46Drh(0, 1) = h(1, 0) = 1

rãx»dè Ò Á Ðî?À¤£Ô+W&;U;e

f : X → YÎ

¥ Î,£Ô+W&;U;eZ ⊆ X

Î-, Kppì`hppì`0/Çv;CRn)OU P +f! S*ÊT+W ,[=

ZR#*ST',XZ#%"Ð''v;CRn)OU P

f Z : Z → Y, (f Z)(x) = f(x)Î

« Î.CRn)OU P g : Z → Y

R#*ST',XZ#%"Ð'§~&hry0)tè-y0/+Èì`0/Îv;CRn)OU P +fP &;\^l+

X ⊆ Z ,[w#*SÓn^#Ð)p#*¸*!&i`! x ∈ X "$#%"Ð'

f(x) = g(x)Î

@E§..£ ª WEy v*xzz£.y 2v*©W.þy (¡v*jc,£.y ¥ ¡mya¨.£ ª ¥ ¡my.¡m W./m\z;a¨.£ ª ¥ ¡ .

6Bò½ñÒ7 ª ³ 8µ ±y×=³aª ?µ ¶C³Oª Å 5¸¹ºWá!ÓÈÇ Àu¼!áÖ r3¨MIW<?K#N\Y^>,DJIW46ILKM46257yhS646A\X98;m

f : R → RIWACB=7Y=5IW4ueKR<YIWB=7Y^

f(x) = x2 rk0F[7YgG>oIWJXY25mYXY2573F[798 yhS646AWX98;2wg6I<Y625IWB=S[0,+∞)

8;7Y:;F#yhS646A\X98=eB=VWY46ILKRN\B;F[IW-XY25ILK#eGr h S646AWX98=N f 8;7Y:;FMF[7YTD6B=<Y7YgJPiS6Y7Y46257Y^%yhS646A\X98;2 f [0,+∞)

r¤Ers¥@JXY25mYX?257yhS646A\X98;2

h : N × N → N<fN\guN\46798MKM<YIWB=7Y^

h(n, k) = n2 + k2 g6I<Y625IWB=S 0 × N 8;7Y:;F8;S6T254a8;7YA\X98=eGr

ãx»dè Ò Á Ðî?À £Ô+W&=U;ef : X → Y

+g : Y → Z

Î§ÅÕQéy-y0/+Èì`0/¾v;CRn)OU P +f+gR#*ST',XZ#%"Ð'Pv;CRn)OU P

g f : X → ZS*#OO#%Ra$XÆS. ,[Q&"

(g f)(x) = g(f(x))Î

¸¹ºWá!ÓÈÇ Àu¼!áÖ rRk :=<YA\IW57 :=DJI\FU>EAaNÊ:=25mXY<fN\:=7Y^ DJIW57YXY7Y46257%b kQ>E<Y4uN\XY< <PiIWY7Y46257 yhS646A\X98;2f(x) = x2 − 1

2g(x) =

√xOruk257Y^>8;S6\bcY7M8;7Y:;FsF[IoDJIW57YXY7Y462746257YDc7Pi467\r

`gG>Ed>E=^>D6B=<?>L8;mY52 b!Y7f : R → R

2g : [0,+∞) → R

b!F[I,IWAaN\<YSa8;7:=25m\bY7K;KM257?F[57DJILKR>EY:=<Y7;8sg67?êu4625X98;2¢<PiIWY7Y46257FU>EX_pyhS646AWX98;2¢46257M8;7Y:;F@ÎWY52K|7\rFU>E^ D6B=IW657?^7Y^ ÎWY4uN:=IW6257xIEXY<?>CKM25=XY257DcIWB[N\g6<Y25\b4uNýD6B=<?>EA6PhN\g<fN.-A6PhN\guN8=e\X\bY7

f : R \ (−1, 1) → [0,+∞)rU464ueÎWY52K|IW=XY2]e8;7Y:;F46257YXYI

2546467<Yg67?êu4625ILKRN\46257<PiIWY7Y462]NGb!<YZWICg646257<g6B=S6Zdexg67?êu4625X98=eDcI\8;mYXY2]NvyhS646A\X98;2 bXY<?>E^46257`JmYg6<Y257Y^>x:=25m`<fN8;ÎK#N\Tqfr

¤Er313257YXO_A,B

2CcmYgueý<Y625IWB[N\^2 46257YD6S6:;F9>E^2 b

f : A → B × Cg6IK|IW54ue

yhS646A\X98=eGb Nf1 : A → B

2f2 : A → C

8;798:=A6PhN\g6IKR>E^2À/hIEg6DcIKM257Yg64625ID6257YB;KM:=<fex2Hg6B=S6ZdeO1r¢IW4uN\gGF[I46257YXO_

π1 : B × C → B2π2 : B × C → CJmYgueoB=<YSGFON\^2 ICg6DJILKM257Yg64625Io4uNoD6257YB;KM:=<feo2 g6B[S6ZWeoIWYrck0F[7YgG>

f1 = π1 f2f2 = π2 f

rkN\B;F[Iv<fN\SdKRN\?>E\bY7:=A6PhN\guN\46257yhS646A\X98;2J8;7Y:;FPhe\XY<Y467\bXY<?>E52u8;7Y=52

f : X → Yb

g : Y → Z2h : Z → W

bcF[I(h g) f = h (g f)

r ã ILK|VEgýDJI\<YIW:;FONLKM2]N\^>8=N\A\I46257?F[B=S6g6467`?KM25XY<Y7Y46257\rãx»dè Ò Á Ðî?ÀåÏV&;\^l+

f : X −→1−1na Y

¿ n è4+bopqKÒét*,&*éVæj+¢íY! bv;CRn)OU P +f!&Dì¢RV+m P &"Ð'

R#. P a!U; /Ôf−1 : Y → X

,[w#*Sf−1(y) = x⇔ f(x) = y

ÎHIWD6B[NfKM46IW=-DJILKR>EY:=<Y7;8g67?êu462X98;2KR>E4625AN<y N\AdF[Sb`Y7yhS646A\X98=N

f8;7Y:;FQ62¨-

8;7YA\X98=eGr!n,IWY4uNF[mg67?êu4625X98;mB=IW<Y:=<Y7YB=<?>Eg67?êu4625Sa8=e\XyhS646A\X98;mIEgGKMB=I\F[4ueog6IoyhS646A\X98;2½Æ 2xw=y £TWz©W Q@=£¥ 10Q¡mw¤( y f y g W ; ,;cfa¨.£ ª ¥ ¡ =|2y_.W

g f = 〈x, z〉 : ∃y(〈x, y〉 ∈ f ∧ 〈y, z〉 ∈ g).

Å 6 ?µ ¶C³[®B=VWY46ILK#N\B;F[IW=XY25ILK#798

fr k0F[7YgG>cmYg6<Y257Y^>^257Y2

f−1 :B=46Z

(f) → Xb K#N\B=S6467YA

g67?êu4625Sa8=e\X?>DJIW<YIW:;FON8;7MJ7Y<M<Y^2]N\4rCn,IWY4uN`F[7YM<Yg67?êu4625ILKRN\RyhS646A\X98;m3IEgGKMB=I\F[4ueTg6IB=VWY46ILK#N\B;F[IW=XY25ILK#798|yhS646AWX98;2

f<YZWIEg646257`<Tg6B=S6Zdeg67?êu4625X98=eDcI\8;mYXY2]NyhS646A\X98;2 r r

¸¹ºWá!ÓÈÇ Àu¼13257YX_f : R→ R

JmYg6<Y257guN\4uNKM<YIWB=7Y^f(x) = x3 rwzWN\Ax46257?F[B=S6g646I<fN\SdK#N\?>E\b!8;7Y:;FF[IQ62 8;7YA\X98=NGrU:;F[4625798;7<fNaF[7Y^ yhS646A\X98=NIEgGKMB=I\F[4uN

f−1 : R → R<fN\guN\4uNvD6B=<Y7Y<K#N\B=S6467YAf−1(y) = x ⇔ x3 = y

rws57x3 = y ⇔ x = 3

√yr!|<?>E52

f−1(y) = 3√y5S6b6<fN\^257Y462]Na8;e\X@IW<Y4uN\XY<Y7Y46257T<Y^257Y4646798?b

f−1(x) = 3√xr

YÙ 2 ¢xT câ '|xH #"P|'$áÝ ¢xT câ

ãx»dè Ò Á Ðî?À¤£Ô+W&;U;ef : X → Y

,[w#*SA ⊆ X

+B ⊆ Y

Î¥ Î3, &ç.ry0/ S*ÊT+W ,[=

AkiXÆS;`%^_Q!&"Àv;CRn)OU P +

fmÒR#*ST',XZ#%"Ð'ÀS*ÊT+Wd,[

f [A] = y ∈ Y : (∃x ∈ A)y = f(x) = f(x) : x ∈ A.

« Î,¬§&hry0hppì*Àé&ç.ry0/¹S*ÊT+W ,[=BkiXÆS;`%^_Q!&"Àv;CRn)OU P +

fm$R#*ST',XZ#%"Ð'ÀS*ÊT+Wd,[

f−1[B] = x ∈ X : f(x) ∈ B.

kF9>E^Æ^25798;:=XYSA\IW46257YXY<Y4678;7Y:;F|AC255AN`SCK#N\ZI46I\FON\X98;2 rd1sNTIW<Y4uN\XY<Y7Y46273IW6B[N\<YS<Y625IWB=S:;F[IW:=Sa8;7:=25mKÆ52F[7YB[NaF[S6B=<Y7^oNFO7?^oNaF9>EXY<Y46798@B=VWY467:;>E^JIW57\r ¥@D6B=VEXY<F[7YZWIDcIEguN\467YZWIDJILKR>EY798 4uNa8;X?<YmY=XY25798 :=DJIaF9>EAN:=25m

f(A)2 ~f(A)

rCHICg6IW646257MKK#>EDuN\g6ACSD6B=<Y7YXY2K|IW6B[N\<YSx<Y625IWB=SoDJI\8=NfKM2]N8=e:=25ms:;>E^JIW57

f−1(B)2 ~f−1(B)

r6¢IW46257?K#N\ 8;7YgC-4uN\AD6B=<?>L8;m?F[7@D6B=<Y7Y<`4uNa:#IW<Y4uN\XY<Y7Y462]N:[e<sDJ7?KM4d>EX_xKM<YZW5mYg6VLK4uNa89KR>EZWICg64625798;:=<Y7@246257`D6B=ILK#N\g6<fe\XY7g6I46257YDJIWB=IW<YS6^257YbEKM25mYX`F[IIW467TJmYgueIWcIKM2]e\<YSa8=e\XY7\rk KR>EDuN\g6ACSQD6B=<Y7YXY2K|IW6B[N\<YSQ<Y625IWB=Sý462574uN\57Y?>^>E52IWD625:=Sa8=e\XY7YZWIZWIx:;>E^Ð-

cIW5S<p:;>E^JIW57Y^ yhS646A\X98;23IEgGKMB=I\F[46798Ð4:[eQF[IlB=VWY467pB=<Y7YXY<?>WrRB=<?>L8;m?FU>:=DJIW:=VW<fN\D625:=SDJIW<?K#N\]No<YB=7Y:=<?FOexSdKRN\Y467Y^SQ|<?>CF[7Y54625AWILKM2H4uNv25XO_B=IW<YB=VWY46257Y46257\rzW7?gG>E4uN:;>CF[SuN\X98=NGb|AdF[VWB[NQKR>EguN8;7:=25m6S6g6<?25pK#eaF[D652K#IW=XY2 b F[IlDC>CFON\46257\b XY<?>

f−1[B]IW<=-

4uN\XY<fND6B=<Y7YXY2K#IW6B[N\<<Y625IWB=SBKM<YZW5mYg67Y^%yhS646A\X98;2

fXY<?>,IW6B[N\<T<Y625IWB=S

BKM<YZW5m=-

g67Y^.yhS646A\X98;2f−1 rHz\7Y:;FF[I8;7Yg64uN\AF[B[S6g646I\[DJIW<YIWB=4uNGrH¥@AaN\<YSa8;7o:=25mcIKM257Y^b Y7IWuNF[7v<?4uN\XY<Y7Y462]N:=25moDcIWACB;>CKRN8=ef/iKR>EAaN\<fN\46257oF[7YZWIDcIW<YIW:;FONfKM2]N\^>v8=N\A\ID6B=IW:;F[7

?KM25XY<Y7Y462571?r ½Æjc,£.y ¥ ¡ ci2þ v.Z£, Wuz§.=¡ ¥40

f−1 = 〈y, x〉 : 〈x, y〉 ∈ f.

6Bòù65,X ¬Y³d*õQª > ¬Y®,³aª²s± X ¬Y³d*õ Å 9ã ]NS6D6B=IW:=<YXY<Y7Y462]N<fN\^2]N\:;FD625:[N\

f [a] 2 f−1[a] JmYg6<Y257Y^>D625:[N\ f [a]2

f−1[a]rJ¥@XY<?>CKM25=XY257\b

f [a] = f(a) r¸¹ºWá!ÓÈÇ Àu¼!áÖ rsz\7Y=52

f : R → R8;7Y:;Fo<fN\guN\4uNýKM<YIWB=7Y^

f(x) = x2 IWB[N\< A = [−1, 2]2

B = (1, 2)bGFOI

f [A] = [0, 4]2f−1[B] = (−∞,−1) ∪ (1,+∞)

r¤Er313257YXO_

g : N × N → NcmYg6<Y257oIWD625:[N\4uNxKR<YIWB=7Y^

g(n, k) = n2 + k2 IWB[N\<46257YXO_A = N× 0 2 B = 0, 1, 2 r6k0F[7YgG>

g[A] = g(n, k) : 〈n, k〉 ∈ N× 0 = n2 + k2 : n ∈ N ∧ k = 0 == n2 : n ∈ N rýA\IW57Y2

g−1[B] = 〈n, k〉 ∈ N2 : g(n, k) ∈ 0, 1, 2 == 〈n, k〉 ∈ N2 : n2 + k2 = 0 ∨ n2 + k2 = 1 ∨ n2 + k2 = 2 == 〈0, 0〉, 〈0, 1〉, 〈1, 0〉, 〈1, 1〉 r

B=<Y7Yg6:;FONLKM25^>xF[7YB[N\<XfNCPhe:=7YB=25mTFUKM257YB=g6<Y7Yg6I\FU>EXY<fe\X?>EXO_ýIW6B[N\<YVLK 2 D6B=<Y7YXY2K2-IW6B[N\<YVLK<Y625IWB=VLKTr ¿oÁh»C¹a¼ ºd» Ò Áh»,é Í £Ô+W&;U;e

f : X → YÎ ^#ù! ,Xc ,^lRV'OU;eAS*ÊT+W ,[Qd,X

A1, A2 ⊆ X+

! ,Xc ,^lR& P [Q *ST+mRV' Ai : i ∈ I V **S*Ê+W %[Qd,X87 S*#!U;eV *S*aðXcd,XcUQS*#.RB#,= P a!U;&S*#%^_&Q¸TR *\*U+(Îk#m

f [A1 ∪ A2] = f [A1] ∪ f [A2]¿ Q`!d,^lRV+W& P

f [⋃

i∈I Ai] =⋃

i∈I f [Ai]Ù

kÊjmf [A1 ∩ A2] ⊆ f [A1] ∩ f [A2]

¿ Q`!d,^lRV+W& Pf [

⋂

i∈I Ai] ⊆⋂

i∈I f [Ai]Ù

kzUmf [A1] \ f [A2] ⊆ f [A1 \ A2]

Îãx¾E¿ ¼ 3g6ILK|IEg64625^>F9>E5A\I-gGKM257ý:=DJIW=B=VEg DcIKR>EY:=<?>EX_ <fN\57YY46IW=XY2 b#B[7?:[<FOmDJIW<YIW:;FONfKM2]N8=e\X|8=N\AWI?KM25XY<Y7Y46257\r/ NO113257YXO_ Ai : i ∈ I cmYg6<Y257MB=IEg6<Y254ue`DcIEg6<Y625IWB=VLK<Y625IWB=S£rd3:;FON\5^>g6ILK#IW5467y ∈ Y r6k0F[7YgG>x^oN\^>o4uN\:;F[mYD6Sa8=e\X?>XY2]edZB=VLKM46IK#N\Y46IW=XY2Wb

y ∈ f [⋃

i∈I Ai]⇔ (∃x)(x ∈ ⋃

i∈I Ai ∧ y = f(x))⇔⇔ (∃x)(∃i ∈ I)(x ∈ Ai ∧ y = f(x))⇔⇔ (∃i ∈ I)(∃x)(x ∈ Ai ∧ y = f(x))⇔⇔ (∃i ∈ I)y ∈ f [Ai])⇔⇔ y ∈ ⋃

i∈I f [Ai]b

D6B=<?> XY<?>E^ D6257YB;KM:=<fN2XY<?K#N\B;FONB=VLKM46ILKRN\Y46IW=KR>E4625AaN8=e0<g67?êu4625X98;2IW6B[N\<YS<Y625IWB=Sb|g6B=S6ZdN2sD62]eaFONB=VLKM46ILK#N\Y46IW=<ýg67?êu4625X98;2@S6IWZWVW54625IW46798:=S6^><Y625IWB=VLKTb

Å ² ?µ ¶C³[®<fN\#F[B=<Y7YXY2]N /hAC5S6XY<YILK#NO1#B=VLKM46ILK#N\Y46IW=<`¡ KM257YB=g6<Y7Y462]NÚGr Ý /hB1ru¢ILK|IEPhN\46257`:=25m`4uNN\:[N\g6mT§AC:;F[7Y46: 8;IW4uN\546IW=XY2!AWIW6XY<?>g6ILK|VEgr/hB1M13257YXO_

A1, A2 ⊆ XrJ3:;FON\5^>g6ILK|IW5467

y ∈ Y ruk0F[7YgG>pD6B[NfKMg6<Y2K|7M8;7Y:;F@4uN\:-F[mYD6Sa8=e\XY7TB=IW<YS6ÎLKRN\46257%b

y ∈ f [A1 ∩ A2]⇔ (∃x)(x ∈ A1 ∩ A2 ∧ y = f(x))⇔⇔ (∃x)(x ∈ A1 ∧ x ∈ A2 ∧ y = f(x))⇒⇒ (∃x)(x ∈ A1 ∧ y = f(x)) ∧ (∃x)(x ∈ A2 ∧ y = f(x))⇔⇔ y ∈ f [A1] ∧ y ∈ f [A2]⇔⇔ y ∈ f [A1] ∩ f [A2]

bD6B=<?>`XY<?>E^D6257YB;KM:=<fNM2\F[B=<Y7YXY2]NRB=VKM46ILK#N\Y46IW= KR>E4625ANa8=eR<g67?êu4625X98;2dIW6B[N\<YST<Y625IWB=Sbg6B=S6ZdN2wXY<?K#N\B;FONB=VLKM46IK#N\Y46IW=`<@g67?êu4625X98;2wD6B=<Y7YACB=Ia8;Sx<Y625IWB=VKTbE<fN\|KR>E4625AaN\46257s<¡ KM257?B[g6<?7Y462]NxÚGrØ¤Ë/h7*1r|<?>E52H<YZWICg646257<g67?êu4625X98=eo<fNfKM257YB[N\462]N<Y625IWB=VLKg6ILK#VCgo8;7Y:;F<fN\A\IW6XY<YIW4d>Wr

N\SdKRN\Y^>Wb6Y7TKDJILKR>EY:=<?>G^FUKM257YB=g6<Y7Y4625SKDJICg6D6S646AdFON\X_f/hB1|2Z/hX*1#46257TguN:=25m<fNa:=FOedD625<fNfKM257YB[N\462]NxB=VLKM46IW=XY2]eGr B=<?>EXY<?>E4uN,F[7YZWI8;7Y:;FuN\B=g6<YI,DJIEg6IW64uN8=N\AKKR>EDuN\g6ACSvB=IW<?DuNaF[B=>CK#N\4d>EXO_oKDJIWD6B=<Y7Yg64625^ÆB=IW<Yg6<Y2]N\57RD6B[NfKB[N\X_CS646ACSoAdK#N\4dF9>O-êuANaF[IWB=VKTraU:;F[I\F[46257\bK-g6B=S6ZW25798¢XY<?mY=XY26DJILKR>EY:=<f7YZWI`g6IK|IEg6SA\IWB=<?>E:;FON\525=^>T<#X?<YmY-XY25ILK|798vB=IW<Yg6<Y257Y546IW=XY2@AdK#N\4CFU>CêuAaNaF[IWB[N-7YZW<?>E:;F[7Y46X98=N\5467YZWI-KM<?ZW5mYg67Y^ AWIW4625S646A\X98;2 rk257Y^>WbJY7DJ7Pi4uNoB=IW<Yg6<Y257Y546IW=K F9>E^KR>EDuN\g6ACS46257<fN\XO_6IEg6<Y2 bwNoF[I8;798MKR>EôN.-ZdNCPid>v7?K#7Y4dF[SuN\54d>g6ILK|VEg,B=VLKM46IW=XY2!KDcIEg6D6S646A\XY257/hB1r¥@6:=7YB;K#N\X98=NQFONýKR>L8=N\=462]ND6B=<?>EXY<?>E4um\b 462578;7Y:;Fo<fN\g6IK|IEg67Y^ y N\AdF[Sb Y7,46257

guN:=25m<fN\:;FOedD625<fNfKM257YB[N\462]NlB=VKM46I\[X?2]eGr ã IK|VEgKR>EôN\ZdNlKM:=AN\<fN\462]NA\IW46ACB=7?Fz-46798yhS646A\X98;2#2|A\IW46ACB=7?F[4C>EXO_<Y625IWB=VLKTbH46Dr

f : R → R, A1, A2 ⊆ Rb¢FON\AC25XO_Y7

f [A1]∩f [A2] 6⊆ f [A1∩A2]/h2uN\4uN\5IWZW25XY<Y4627|KKR>EDuN\g6ACSoDJIW<YIW:;FONCP]>EXO_<fNLKM257?BONaB1?b

8;7Yg64d>E^ :PiILK|7Y^ KR>EôN\ZWNKM:=AaN\<fN\462]NA\IW4dF[B=D6B=<?>EA6PhN\g6Sb XYIQDcIW<YIW:;FONLKR2]N\^>8=N\AWI?KM25XY<Y7Y46257ar¥@AaN\<YSa8;7@:=25m 8;7Yg64uN\Acb6Y7`D6B=<?>og6IEguNaF[A\IKR>E^j<fNCPiIWY7Y4625SxIyhS646AWX98;2

f¡ KM257YB=g6<Y7=-

46257 ñ r Ö ÎWY4uNKM<YÎCXY4625\r ¿oÁh»C¹a¼ ºd» Ò Áh»,é 5Þ £Ô+W&;U;e

f : X → YÊ=Q*ST+W&Bv;CRn)OU P aÃ[Qd¸TR ,XZ#%[= *\*U+W ,XZapÎ ^#

! ,Xc ,^lRV'OU;e$S*ÊT+W ,[Qd,XA1, A2 ⊆ X

+ï! ,X2 ,^lR& P [Q *ST+mRV' Ai : i ∈ I V *S*ÊT+W ,[Qd,X97"$#%"Ð'ÒX¢&Q%'k#m

f [A1 ∩ A2] = f [A1] ∩ f [A2]¿ Q`!d,^lRV+W& P

f [⋂

i∈I Ai] =⋂

i∈I f [Ai]Ù

kÊjmf [A1] \ f [A2] = f [A1 \ A2]

Îãx¾E¿ ¼ ¢IWAaN\Y7Y^>oFU>E5AWIs8;7Yg64ueB=VKM46IW=\b6DJIW<YIW:;FONCPi7@DJIW<YIW:;FONfKM2]Na8=e\X@g6I:[N\Î,-g6<Y257Y5467YZWIS6g6IK|IEg646257?462]NGr

6Bòù65,X ¬Y³d*õQª > ¬Y®,³aª²s± X ¬Y³d*õ ÅCÅ

/ NO1 ã <Y25mYAC2d¡ KM257YB=g6<Y7Y4625S ñ r Ö KR>E:;FON\B=XY<?>TDJIWAN\<fN\FU>E5A\If [A1]∩f [A2] ⊆ f [A1∩A2]

r3:;FON\5^>xg6ILK|IW5467

y ∈ Y r6k0F[7YgG> y ∈ f [A1] ∩ f [A2]⇔ y ∈ f [A1] ∧ y ∈ f [A2]r

|<?>E52H<g67?êu4625X98;2HIW6B[N\<YS<Y625IWB=S25:;F[4625798=ex1 ∈ A1

2x2 ∈ A2

bcFON\AC257Y7<fN\XO_6IEg6<Y2f(x1) = y = f(x2)

rGs573yhS646A\X98=Nf8;7Y:;F#B=VWY46IK#N\B;F[IW=XY25ILK#NGbGXY<?>E2w^S6:=2wd>E

x1 =x2 = x

ruk0F[7YgG>8;7Yg64uN\Ax ∈ A1 ∩ A2

2 DJIW46257?K#N\y = f(x)

buF[Iy ∈ f [A1 ∩ A2]

bXYIA\IW6XY<?>g6ILK#VCgr

1sN\:;F[mYD6467TF9KR257YB=g6<Y7Y46257Tg6I\F9>EX?<?>KsPhN\:=46IW=XY2 D6B=<Y7YXY2K|IW6B[N\<YVK<Y625IWB=VLKTr ¿oÁh»C¹a¼ ºd» Ò Áh»,é 5Ì £Ô+W&;U;e

f : X → YÎ ^# ! ,Xc ,^lRV'OU;eIS*ÊT+W ,[Qd,X

B1, B2 ⊆ Y+

! ,Xc ,^lR& P [Q *ST+mRV' Bi : i ∈ I V *S*ÊT+W ,[Qd,X;:9S*#!U;eV .SaðXcd,XcUQS*#.RB#,= P a!U;&S*#%^_&Q¸TR *\*U+(Îk#m

f−1[B1∪B2] = f−1[B1]∪f−1[B2]¿ Q`!d,^lRV+W& P

f−1[⋃

i∈I Bi] =⋃

i∈I f−1[Bi]

ÙkÊjm

f−1[B1∩B2] = f−1[B1]∩f−1[B2]¿ Q`!d,^lRV+W& P

f−1[⋂

i∈I Bi] =⋂

i∈I f−1[Bi]

ÙkzUm

f−1[B1 \B2] = f−1[B1] \ f−1[B2]Î

ãx¾E¿ ¼ ¢IW46ILKM46257@D6B=<?>EA6PhN\g6ILK|IS6g6ILK|IEg64625^>`8;7Yg64ueB=VLKM46IW=\bGDJIW<YIW:;FONCPi7@DcI,-<YIW:;FONfKM2]Na8=e\X|8=N\A\I?KM25XY<Y7Y46257\r/hB1@1M257YX_

B1, B2 ⊆ Yr3:;FON\5^>,g6ILK#IW5467

x ∈ X rwk0F[7YgG>ý^oN\^>,4uN\:;F[mYD6Sa8=e\X?>XY2]eCZB=VLKM46IK#N\Y46IW=XY2Wb

x ∈ f−1[B1 ∩B2]⇔ f(x) ∈ B1 ∩B2 ⇔⇔ f(x) ∈ B1 ∧ f(x) ∈ B2 ⇔⇔ x ∈ f−1[B1] ∧ x ∈ f−1[B1]⇔⇔ x ∈ f−1[B1] ∩ f−1[B2]

bD6B=<?> XY<?>E^÷D6257YB;KM:=<fN2TF[B=<Y7YXY2]NB=VKM46ILK#N\Y46IW=lKR>E4625AaN8=e<g67?êu462X98;2D6B=<Y7YXY2K2-IW6B[N\<YST<Y625IWB=S`<fN\ g6B=S6ZdNR2WXY<?K#N\B;FONRB=VKM46ILK#N\Y46IW= < g67?êu4625X98;2WD6B=<Y7YACB=Ia8;S<Y625IWB=VLKTr¢IK|IEPhN\46257`:=25m`4uNoN\:[N\g6mT§AC:;F[7Y46: 8;IW4uN\546IW=XY2 A\IW6XY<?>g6ILK|VEgr

N\SCK#N\Y^>25:;F[I\F[4ueýB=VW?4625XYmDcIW^25mYg6<?>¡ KM257YB=g6<Y7Y46257Y^ ñ r Ö Ný¡ KM257YB=g6<Y7Y46257Y^ñ r Ú¯4D6B=<Y7YXY2K|IW6B[N\<?>v<Y625IWB=VLKU<fN\XO_6IKMSa8=e:=25ms57YD625798 IEgvIW6B[N\<YVLK <Y625IWB=VKTrCkQ>O-4625AaNF[Io<Ty N\AdF[SbJY7Kg67?êu4625X98;2¢IW6B[N\<YSp<Y625IWB=S,KR>E:;F[mYD6Sa8;7AdK#N\4dF9>CêuANaF[IWB@7YZW<?>E:-F[7Y46X98=N\54d>WbAdF[VWB=7YZWI6B[N\A K g67?êu4625XU8;2oD6B=<Y7YXY2K|IW6B[N\<YS <Y625I\B[Sr¥@J7YXY46IW=F[7YZWIAdK#N\4CFU>CêuANaF[IWB[N`UDcIWZdN\B=:=<fN#:;>CF[SuN\X98;m\bCXYIPhNaFUK|IT<YIWuN\XY<?>E^>DJIWB=VLKM4CSa8=e\XMg6ILK|IEgG>IW6SFUKM257YB=g6<Y7YrNa8;^27Y^>s:=25mF[7YB[N\<KM<fN8;7Y^4d>E^2d<fN\57YY46IW=XY2]N\^2WDcIW^25mYg6<?>@IW6B[N\<fN\^2dNMD6B=<Y7=-

XY2K|IW6B[N\<fN\^2<Y625IWB=VLKTr

Å × ?µ ¶C³[® ¿oÁh»C¹a¼ ºd» Ò Áh»,é éÈ£Ô+W&;U;e

f : X → Y ,[w#*S

A ⊆ X+B ⊆ Y

ÎEY&Q%'k#m

A ⊆ f−1[f [A]]Ù

kÊjmf [f−1[B]] ⊆ B

Îãx¾E¿ ¼ / NO113257YX_

A ⊆ Xr|3:;FON\5^>-g6IK|IW5467

x ∈ Ar k0F[7YgG><pg67?êu4625X98;2

IW6B[N\<YS<Y625IWB=SôN\^>f(x) ∈ f [A]

rds57R<Mg67?êu4625X98;2cD6B=<Y7YX?2K#IW6B[N\<YS<Y625IWB=SôN\^>

f−1[f [A]] = x ∈ X : f(x) ∈ f [A],XY<?>E52

x ∈ f−1[f [A]]buXYIA\IW6XY<?>g6ILK#VCgr

/hB1313257YXO_B ⊆ Y

rw3:;FON\5^>g6ILK#IW5467y ∈ Y rck0F[7YgG>D6B[NfKMg6<Y2K|7s8;7Y:;F`DJI\4625Y:=<Y7B=IW<YS6ÎLK#N\46257%b

y ∈ f [f−1[B]]⇔ (∃x)(x ∈ f−1[B] ∧ y = f(x))⇔⇔ (∃x)(f(x) ∈ B ∧ y = f(x))⇒ y ∈ B b

D6B=<?>XY<?>E^ D6257YB;KM:=<fNýB=VLKM46IK#N\Y46IW=xKR>E4625ANý<vg67?êu4625X98;2#IW6B[N\<YS<Y625IWB=SbHg6B=S6ZdN<g67?êu4625X98;2sD6B=<Y7YX?2K#IW6B[N\<YS0<Y625IWB=Sb|<YN\x25^D62AaN\X98=Nx8;7Y:;FxIEXY<?>CKM25:;FON /iKyhIWB=^S657(∃x) y ∈ B ÎWY7Y^>ÌWD6S6=XY25 AdK#N\4dF9>CêuANaF[IWBYbZWgG>E 4625XY<?7YZWI3IW446257AdK#N\4CFU>CêuACSa8;7*1r

N\SdKRN\Y^>WbWY7M<Ry N\AdF[Sf(x) ∈ f [A]

46257#KR>E4625AaNGbdY7x ∈ A rCn,IWY7M:=25mMcIKM27Y^<YguN\B=<?>E\baY7

x 6∈ A IWB[N\< Y7 8;7Y:;F x′ ∈ A bLFON\AC257 Y7 f(x) = f(x′)r\|<?>E52Eg6ILK|VEgDJIEgC-

D6S646AdF[SÃ/ NO1M46257guNv:=25mDJIWD6B[NfKM25\bJd>I\F[B=<?>EôNaB=VLKM46IW=\rw¡HN\ACY7KDJICg6D6S646AWXY257/hB146257guN:=25mICgGKMB=VEXY25IW:;FONaF[462579825^D652AaN\X98;2 bHZWgG>E,ÎWY74625725:;F[46257Y

x ∈ XFON\AC257\b|Y7y = f(x)

rM¥@XY<?>CKM25=XY257\bRg6ILK|VEgb#Y7ýK DJILKR>EY:=<?>E^ F9KM257YB=g6<Y7Y4625S46257ôNB=VLKM46IW=XY2!KR>EôN\ZdNð/hDcIEg6IW646257#8=N\AoK D6B=<?>EDuN\g6ACSp¡ KM257YB=g6<Y7Y462]N ñ r Ö 1|DJIEguN\462]NA\IW4dF[B=D6B=<?>EA6PhN\g6Sr¢IEg6IW646257FON\ACY7@8=N\ApKÆD6B=<?>EDuN\g6ACS¡ KM257YB=g6<Y7Y462]N ñ rØ¤EbwKM<YÎEXY46257Y46257<fNCPiIWY7Y

DcIW<?KRN\]NS6<?>E:=AN\KM25mYXY798?r ¿oÁh»C¹a¼ ºd» Ò Áh»,é Õ£Ô+W&;U;e

f : X → Y ,[w#*S

A ⊆ X+B ⊆ Y

ÎEY&Q%'k#mfÏV&;\^l+v;CRn)OU P # P &;

1− 1 A = f−1[f [A]]

ÎkÊjmfÏV&;\^l+v;CRn)OU P # P &;<(TR#+*

B = f [f−1[B]]Î

ãx¾E¿ ¼ HIW46257?K#N\DcIEguN\467<fNCPiIWY7Y462]NvDJIW<?K#N\]Na8=eoKR>W7Y525^2546ILK#N\IWD625:[N\467DJI,-KR>EY798|D6B=IW657Y^>\bEg6ILK#VCgxDJIW7YZdNa8=e\X?>4uNKM<YÎCXY46257Y4625Svg6ILK|IEg6Sx¡ KM257?B[g6<?7Y462]N ñ r ñDcIW<YIW:;FONLKM2]N\^>|<?>CF[7Y54625A\ILKM2 r

6Bò6587 ³d¯w³ µ ª5³ Å öYÙ; < Íàx'YÖ r313257YXO_

f, g : R→ RrJNaD625:[N\`4uN\:;F[mYD6Sa8=e\XY7<YguN\462]NIyhS646A\X98=N\X_

f2gr

/ NO1 h S646AWX98=N f 8;7Y:;F3IWACB=7Y:=ILK#NGr/hB1 h S646AWX98=N f 8;7Y:;F3IWZWB[N\4625XY<YIW4uNGr/ X1 ã I@<Y625IWB=SIWACB=7Y:=VK-yhS646A\X98;2f4uN\57Yfesg6ILK|IW546257 ^oNCPi7|525XY<Yd>g6ICguNaF[46257\r

/hgB1v9:;F[4625798=eg6ILK|IW546257g6S6Y7oN\B=ZWS6^7Y4CFU>Wb!g6]NAdF[VWB;>EXO_lK#N\B;F[IW=yhS646AWX98;2f8;7Y:;Fs^4625798;:=<fNIEgK#N\B;F[IW=XY2yhS646A\X98;2

gr

/ 71HICXY<feLKM:=<?>IEgDJ7?KM467YZWI@^25798;:=XfNRyhS646A\X98=NfD6B=<?>L8;^Sa8;7 F9>E5A\I3K#N\B;F[IW=XY2

46257YSa8;7Y^467\r/iy;1w625VWBR^25798;:=X@<Y7YB=IKR>EXO_,yhS646AWX98;2

f8;7Y:;F346257YIWZWB[N\4625XY<YIW4d>vIEgg6IEPiSr

/hZ!1xHICXY<feLKM:=<?>,ICg,Dc7?KM467YZWIo^25798;:=XfNyhS646A\X98=Nf8;7Y:;F3:;FONCPhNGr

¤Er3¨MIW<?K#N\Y^>yhS646A\X98;mf : R → R

rHN\D625:[N\:;>E^JIW525XY<Y46257DcIW4625Y:=<Y7o<YguN\462]N/hJ7Y<TS6?>EXY2]N:;>E^JIW5S467YZWN\X98;2 ¬ 1r/ NO1x!25XY<YuN Ö 46257|8;7Y:;F3IWACB=7Y:=7Y^%yhS646AWX98;2

fr

/hB1x!25XY<YuN Ö 46257|8;7Y:;F3IWZWB[N\4625XY<Y7Y46257Y^jZWVWB=4d>E^jyhS646AWX98;2fr

/ X1 h S646AWX98=N f 46257|8;7Y:;F3IWZWB[N\4625XY<YIW4uN<Tg6IEPiSr/hgB1 h S646AWX98=N f 46257|8;7Y:;FMyhS646A\X98=eB=IW:=4ue\XfeGrÚGrsN\D625:[N\34uN\:;F[mYD6Sa8=e\XY7@<YguNa462]NIXY2]edZWSo525XY<YoB=<Y7YXY<?>CKM25:;FU>EX_ 〈an〉n∈N

rCklIW546IS6?>CK#N\:;>E^cIW52

an4uNIW<Y4uN\XY<Y7Y46257`KR>EB[N\<YVKXY2]eCZWSr

/ NO1|2]eCZ 〈an〉n∈N

8;7Y:;F3IWZWB[N\4625XY<YIW4d>Wr/hB1|2]eCZ 〈an〉n∈N

8;7Y:;F3^IW46I\F[IW4625XY<Y4d>Wr/ X1,|2]eCZ 〈an〉n∈N

8;7Y:;F3B=IW<Y6257YY4d>xg6I+∞ r/hgB1x13257Y:=AWIW6XY<Y7Y46257`KM257Y57@KR>EB[N\<YVLK XY2]eCZWS 〈an〉n∈N

8;7Y:;F3g6ICguNaF[4625X_r/ 71,¥@g,DJ7?KM467YZWI^25798;:=XfNXY2]eCZ 〈an〉n∈N

8;7Y:;F3B=IW:=4ue\X?>Wrñ r313257YXO_

f : R→ RrJN\D625:[N\`:;>E^JIW525XY<Y46257@DJIW4625Y:=<Y7T<Y625IWB;>Wr

/ NO1w625VWBFU>EXO_-525XY<YB=<Y7YXY<>CK325:;FU>EX_bg6]N,AdF[VWB;>EX_yhS646AWX98=NfD6B=<?>L8;^Sa8;7

KRN\B;F[IW=XY2!Sa8;7Y^467\r/hB1w625VWBR^25798;:=X@<Y7YB=IKR>EXO_,yhS646AWX98;2

fr

/ X1,w625VWBpIWACB=7Y:=VLK yhS646A\X98;2f/ MKRN\ZdN.^@¥@ACB=7Y:yhS646A\X98;2M8;7Y:;F¼ ¾!¼ À ÇaÒ Á

525XY<YueB=<Y7YXY<?>CKM25:;FOeO1r

×Cø ?µ ¶C³[®/hgB1,w625VWB#IWZWB[N\4625XY<Y7YZWVWB=4C>EXO_yhS646AWXU8;2

fr

Ý r3¨MIW<?K#N\Y^>QXY2]eCZ 〈an〉n∈N

525XY<YlB=<Y7YXY<?>CKM25:;FU>EX_-IWB[N\<yhS646A\X98;mf : R → R

rN\D625:[N\@:;>E^cIW525XY<Y46257@DJIW4625Y:=<Y7T<YguN\462]NIWB[N\<T<?625IWB;>Wr/ NO1x©TN\YgG>vKR>EB[N\<XY2]edZWS 〈an〉n∈N

8;7Y:;FMKRN\B;F[IW=XY2]eyhS646A\X98;2fr

/hB1vkN\B;F[IW=XY2yhS646A\X98;2fg6]N N\B=ZWS6^7Y4dF[VLKtcmYgue\X?>EXO_ KR>EB[N\<fN\^2XY2]edZWS

〈an〉n∈N

:[eg6IEguNaF[46257\r/hX*1,w625VWB!^25798;:=X<Y7YB=ILKR>EX_yhS646AWX98;2

fJmYgue\X?>EXO_KR>EB[N\<fN\^2dXY2]eCZWS 〈an〉n∈N

r/hgB1,w625VWBTIWZWB[N\4625XY<Y7Yg6IW54d>EXO_lyhS646A\X98;2

f46257JmYgue\X?>EXO_-IWZWB[N\462XY<Y7Y462]N\^2

g6IW54d>E^2XY2]eCZWS 〈an〉n∈N

råGr3¢IWAN\<fN\B=VLKM46ILKRN\Y46I\[gGK|VCX_,g67?êu4625X98;2!DJIa8;mYXY2]NyhS646A\X98;2 rì r D6B[NfKMg6<Y25\buXY<?>DcIW4625Y:=<Y7<Y625IWB;>v:[eyhS646A\X98=N\^2Z/iK:=7Y46:=257Tg6B=S6ZW25798#g67?êu4625X98;2DcI\8;mYXY2]NyhS646A\X98;2m1r!z\7Y=52HFON\AY4,DcIEguN\25X_g6<Y257Yg6<Y2546m2H<Y625VWB3KRN\B;F[IW=XY2 rz\7Y=5246257o4oS6<fN\:[N\g64625Tg6]N\XY<Y7YZWI6r/ NO1 x ∈ R : (∃y ∈ R) y = x2 b/hB1 〈x, y〉 ∈ R2 : y = x2 b/hX*1 〈x, x2〉 : x ∈ [0,+∞) b/hgB1 〈x, y〉 ∈ R2 : y ≤ x2 ∨ y ≥ x2 b/h7*1 〈x, y〉 ∈ R2 : x < −1 ∧ y = x2 r

×Gr ã ]NDJIW4625Y:=<?>EXO_,yhS646AWX98;2KR>E<Y4uN\XY<?>E25XO_,<Y625VWB|KRN\B;F[IW=XY2 r/ NO1

f : R→ (0,+∞), f(x) = exb

/hB1f : R→ R, f(x) = ex

b/hX*1

f : Z→ Z, f(x) = x2 b/hgB1

f : (0,+∞)→ R, f(x) = 1x

b/h7*1

f : N→ N, f(n) = n+ 1b

/iy;1f : Z→ Z, f(n) = n+ 1

b/hZ!1

f : Q→ Q, f(q) = 2b

/h_B1f : P(N)\∅ → N, f(A) = minA (=

4uNa8;^4625798;:=<?>7Y57Y^7Y4dF <Y625IWB=SA1b

/h2m1f : P(P(N))→ P(N), f(A) =

⋃A r

6Bò6587 ³d¯w³ µ ª5³ ×Vñ

ó r33<fN\:[N\g64625\bY7 ∏

i∈1,2Ai2A1 × A2

:[eyhIWB=^oN\54627vB=VWY4d>E^2RIW6257YAdFON\^2 r¢IWAaN\<fN\\bcY7^IWY4uN<Y4uN\57T÷Y62 8;7YA\X98;mDJIW^25mYg6<?>vFU>E^2!<Y625IWB[N\^2B- F[7Y4,y N\AdFDJIW<?K#N\]N4uN\^8;7T<Y7T:=IWueSGF[IWY:[N\^2]N\\r

Öfø r313257YXO_f : X → Y

JmYg6<Y257`254a8;7YA\X98=eGru¢IWAaN\<fN\\buY7@yhS646AWXU8=Nf ′ : X → B=46Z

(f)<fN\guN\4uNKM<YIWB=7Y^f ′(x) = f(x)

8;7Y:;Fs62 8;7YA\X98=eGrÖ\Ö r ã ]NsguN\46798¢yhS646A\X98;2

f : R→ R<YuN\guN\\b\XY<?>

f8;7Y:;FB=VWY46IK#N\B;F[IW=XY25ILK#N@2dU4uNOr

z\7Y=52f462578;7Y:;F

1−1bfKM:=AaN\<fN\

x1

2x2

FON\AC257\bLY7x1 6= x2

bN\57f(x1) = f(x2)

rz\7Y=52

f46257|8;7Y:;FRU4uNObEKR>E<Y4uN\XY<?>GB=46Z

(f)r

/ NO1f(x) = 2x − 7

b/hB1

f(x) = x4 + 1b

/ X1f(x) = 2x + x

b/hgB1

f(x) = x3 − x2 b/ 71

f(x) = sin xb

/iy;1f(x) = |x| − 1

rÖ ¤Er313257YXO_

f : (−∞, 0]→ [0,+∞)JmYg6<Y257<fN\guN\4uNxKM<YIWB=7Y^

f(x) = x2 r 3g6I,-K|IEg64625\bu?7TyhS646A\X98=Nf8;7Y:;F3B=VWY46IK#N\B;F[IW=XY25ILK#N2JU4uNOr

Ö ÚGr3¢IWAaN\<fN\\bCY73yhS646A\X98=Nπ1 : R×R→ R, π1(x, y) = x

8;7Y:;F|:=S6Bh8;7YA\X98=eGbGN\57M462578;7Y:;F3254a8;7YA\X98=eGr

Ö ñ r3¢IWAaN\<fN\\b Y7yhS646A\X98=Nf : [0, 2π] × [0,+∞) → R2 <Yg67?êu4625ILKRN\4uNKM<YIWB=7Y^

f(ϕ, r) = 〈r cosϕ, r sinϕ〉 8;7Y:;F3:=S6Bh8;7YA\X98=eGbJN\57@46257#8;7Y:;F3254a8;7YA\X98=eGrÖLÝ r33g6ILK|IEg64625\bLY78;7Y=52WyhS646A\X98=N

f : A→ B×C 8;7Y:;F¢:[S6Bh8;7YA\X98=eGbLF[I8;798!:=A6PhN\g6ILK|7f1 : A→ B

2f2 : A→ C

F[7Y:[e:=S6Bh8;7YA\X98=N\^2 rJ|<?>pN\4uN\5IWZW2XY<Y4d>vy N\AdF 8;7Y:;FD6B[NfKMg6<Y2KR>WbuZWgG>vyhS646AWX98=N

f8;7Y:;F3254a8;7YA\X98=eO3

Ö åGr33g6ILK|IEg64625\buY7#8;7Y=52f : X → Y

bg : Y → Z

2h : Z → W

bGF[I

(h g) f = h (g f).

ÖLì rsw4uN\57T÷Y@yhS646A\X98;7IEgGKMB=I\F[467Tg6I4uN\:;F[mYD6Sa8=e\X?>EXO_pyhS646A\X98;2 r/ NO1

f : R→ R; f(x) = x3 + 1b

/hB1g : (0,+∞)→ R; g(x) = ln x

b/ X1

h : Z→ Z; h(n) = n+ 5r

×Gù ?µ ¶C³[®Ö ×Gr ã ]NsDJIEguNa4C>EXO_DJIW4625Y798¢yhS646A\X98;26:=D6B[NLKMg6<Y25\bWXY<?>:[e`B=VW?46IK#N\B;F[IW=XY25ILK#7#2CU4uN

IWB[N\<@KR>E<Y4uN\XY<?>EIW6B[N\<f [A]

2D6B=<Y7YXY2K#IW6B[N\<f−1[B]

r/ NO1

f : R→ R; f(x) = x2 + 2; A = [−1, 2); B = (2, 3]b

/hB1f : R→ R; f(x) = 2x; A = (−1, 1); B = [1, 4]

b/hX*1

f : N2 → N; f(n, k) = n+ k; A = 1 × N; B = 3 b/hgB1f : N2 → N; f(n, k) = nk; A = N×2; B = 2n+1 : n ∈ N b/h7*1f : Z2 → Z; f(n, k) = n2k; A = Z× 0, 1; B = 0 b/iy;1f : Z2 → Z; f(n, k) = n2 − k; A = 1 × Z; B = 0 b/hZ!1f : Z→ Z2; f(n) = 〈2n,−2n〉; A = 2n : n ∈ Z; B = N×0 b/h_B1f : R2 → R2; f(x, y) = 〈x+y, x−y〉; A = R×0; B =

I\OX

rÖLó r3¢IWAN\<fN\\bcY7g6]Ng6ILK#IW546798#62 8;7YAWXU8;2

f : X → YIWB[N\<T<Y625IWB=S

B ⊆ YIW6B[N\<

<Y625IWB=SBKM<YZW5mYg67Y^.yhS646AWX98;2

f−1 8;7Y:;FB=VLKM4d>QD6B=<Y7YXY2K|IW6B[N\<YIKM2 <Y625IWB=S BKM<YZWmYg67Y^%yhS646A\X98;2fr

¤ ø r33g6ILK|IEg64625`DJIW<YIW:;FONCPi7TDJICg6D6S646AdF9>¡ KM257YB=g6<Y7Y4625N ñ r Ö r¤ Ö r3¢IEguN\TD6B=<?>EA6PhNag,yhS646A\X98;2

f : R→ R2!<Y62IWB=VK

A,B ⊆ R FON\AC25XO_b6Y7/ NO1

f [A] ∩ f [B] 6⊆ f [A ∩B]k

/hB1f [A \B] 6⊆ f [A] \ f [B]

r D6B=VWcIK#N\ý<Y4uN\57T÷YýD6B=<?>EA6PhN\gG>25464d>EXO_yhS646A\X98;2Ò/ I\ACB[7?[IW4C>EXO_ 4uN2464C>EXO_<Y625IWB[N\XO_b646Dru:=A\IW6XY<YIW4C>EXO_B1bJôN8=e\X?>EX_DJILKR>EY:=<Y7K@PhN\:=46IW=XY2 r

¤W¤Er33g6ILK|IEg64625`DJIW<YIW:;FONCPi7TDJICg6D6S646AdF9>¡ KM257YB=g6<Y7Y4625N ñ rØ¤Er¤\ÚGr313257YXO_

f : X → Yrc3g6ILK#ICg64625\buY7

/ NO18;7Y=52Hg6]Nvg6ILK|IW54C>EXO_A,B ⊆ X

ôN\^>f [A ∩ B] = f [A] ∩ f [B]

bJF[IyhS646A\X98=N`8;7Y:;F3B=VWY46ILK#N\B;F[IW=XY25ILKRNnk

/hB18;7Y=52g6]Ng6ILK#IW4C>EXO_A,B ⊆ X

ôN\^>f [A \ B] = f [A] \ f [B]

bF[IyhS646A\X98=N`8;7Y:;F3B=VWY46ILK#N\B;F[IW=XY25ILKRNGr

¤ ñ r33g6ILK|IEg64625`DJIW<YIW:;FONCPi7TDJICg6D6S646AdF9>¡ KM257YB=g6<Y7Y4625N ñ r ÚGr¤ Ý r3¢IEguN\TD6B=<?>EA6PhNag,yhS646A\X98;2

f : R→ R2!<Y62IWB=VK

A,B ⊆ R FON\AC25XO_b6Y7/ NO1

f−1[f [A]] 6⊆ Ak

/hB1B 6⊆ f [f−1[B]]

r

6Bò6587 ³d¯w³ µ ª5³ ×y5 D6B=VWJILK#N\<?4uN\57T÷YD6B=<?>EA6PhN\gG>025464d>EX_0yhS646A\X98;2$/hIWACB=7Y=5IW4d>EXO_ 4uN25464d>EXO_<Y625IWB[N\XO_b646Dru:=AWIW6XY<YIW4d>EX_B1bcôNa8=e\X?>EXO_DJILKR>EY:=<Y7K@PhN\:=46IW=XY2 r

¤aåGr313257YXO_f : X → Y

ru3g6IK|IEg64G25\buY7/ NO18;7Y=52g6]Nog6IK|IW5467YZWI

A ⊆ XôN\^>

f−1[f [A]] = AbwF[IvyhS646AWX98=N8;7Y:;F

B[V\Y46IK#N\B;F[IW=XY25ILKRNGr/hB18;7Y=52Hg6]Nog6IK|IW5467YZWI

B ⊆ YôN\^>

f [f−1[B]] = BbJF[IvyhS646A\X98=N8;7Y:;F

U4uNOr¤ ì r33g6ILK|IEg64625T¡ KM257YB=g6<Y7Y46257 ñ r Ý r¤a×Gr313257YXO_

f : X → Yr 13257YXO_

A,A1, A2 ⊆ X2B,B1, B2 ⊆ Y

r D6B[NfKMg6<Y25\bXY<?>/ NO1

f [A14 A2] = f [A1]4 f [A2]b

/hB1f−1[B14B2] = f−1[B1]4 f−1[B2]

b/ X1

f [A ∩ f−1[B]] = f [A] ∩B r¤ ó r313257YXO_

f : X → YIWB[N\<

A ⊆ X, B ⊆ Yru3g6IK|IEg64625\b6Y7

A ∩ f−1[B] ⊆ f−1[f [A] ∩B]

IWB[N\<lDJICguN\D6B=<?>EA6PhN\g4uNF[I6b3Y7KµDJILKR>EY:=<Y>E^ÜKM<YIWB=<Y7ÎWY746257ld>EB=VLKM46IW=XY2 r

Ú ø r313257YXO_f : X → Y

2g : Y → Z

IWB[N\<A ⊆ X

2B ⊆ Z

ru3g6IK|IEg64625\b6Y7/ NO1

(g f)[A] = g[f [A]]b

/hB1(g f)−1[B] = f−1[g−1[B]]

rÚ Ö r313257YXO_

f : X → YIWB[N\<

g : Y → Zr|3g6ILK|IEg64625\b Y78;7Y=52

g f 8;7Y:;FB=VWY46ILK#N\B;F[IW=XY25IK#N2f8;7Y:;FRU4uNObGF[I

g8;7Y:;F3B=VWY46ILK#N\B;F[IW=XY25ILKRNGr

ÚW¤Er313257YXO_f : X → Y

IWB[N\<g : Y → Z

rc3g6ILK#ICg64625\bcY7R8;7Y=52g f 8;7Y:;FMU4uN`2

g8;7Y:;F3B=VWY46ILK#N\B;F[IW=XY25IK#NGbGF[I

f8;7Y:;FRU4uNOr

Ú\ÚGr3¨MIW<?K#N\Y^>vyhS646A\X98;mf : Z→ N

guN\4ueKM<YIWB=7Y^f(x) = x2 + x

r/ NO1wg67?êu4625IK#N\yhS646A\X98;m

g : Z → ZFON\Acb!d>yhS646A\X98=N

f g C>uPhNxB=VWY46I,-KRN\B;F[IW=XY25ILK#NGr/hB1|<?>25:;F[4625798;7@yhS646A\X98=N

h : Z→ ZFON\ANGbud>vyhS646AWX98=N

f h d>uPhNU4uNz3

×!6 ?µ ¶C³[®Ú ñ r3¨MIW<?K#N\Y^>vyhS646A\X98;m

f : N2 → NguN\4ueKM<YIWB=7Y^

f(n, k) = n2 + k2 r/ NO1x¢IEguN\RD6B=<?>EA6PhN\g46257Y:=A\IW6XY<YIW467YZWI<Y62IWB=S

C ⊆ N2 FON\AC257YZWI6b\Y7#yhS646A\X98=Nf C

8;7Y:;F3B=VWY46ILK#N\B;F[IW=XY25IK#NGr/hB1¢IEguN\D6B=<?>EA6PhN\gý<Y625IWB=S

D ⊆ N IWB[N\<TyhS646AWX98;2 g : N → DFON\AC25XO_bJY7

yhS646A\X98=Ng f 8;7Y:;FRU4uNOr

Ú Ý r313257YXO_f : X → Y

2g : Y → X

rG¢IWAN\<fN\\bGY7 8;7Y=52g f = idX

bCF[IyhS646A\X98=Nf8;7Y:;F3B=VWY46ILK#N\B;F[IW=XY25IK#NGbuNyhS646AWX98=N

g8;7Y:;FRU4uNOr

Ê++!>=

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Or¤Er313257YXO_

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U+W ,Xc ÔV ,[wS*aO.), P &T<Y625V\BXr

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N+ <fN\guN\4ueKRN\B=S646AC257Y^

nRm⇔ (∃k ∈ Z)m = kn⇔ n|m,XY<?>E52DcID6B=IW:;F[SB=7Y]N\X98;m@DJICg6<Y257Y546IW=XY2 j ruz\7Y:;FMF[IB=7Y]N\X98=N<?KMB=I\F[4uNGb6cI<fNfKM:=<Y7

n|n rz\7Y:;FMF[7YD6B=<Y7YXO_6IEg6462]NGbuJI`8;7Y=52nRm

2mRp

b6F[I<Y4uN\XY<?>WbcY7m = k1n

2p = k2mg6]NTDJ7?KM4d>EX_

k1, k2 ∈ ZrEs57MKRF[7YgG>

p = k1k2n2c525XY<YuN

k1k2

FO7? 8;7Y:;F#XfNCPiA\ILKM2FONGb<fNaF[7Y^

nRprdz\7Y:;FKMB=7Y:=<YXY257R:PhN\cITN\4dF9>E:;>E^7?F[B;>EXY<Y4uNGbWJI|8;7Y=52

m = k1n2n = k2mg6]NDJ7?KM4d>EXO_

k1, k2 ∈ ZbGF[IôN\^>

n = k1k2nbGXY<?>E52

k1k2 = 1rGs57 8;7Y=52J255IEXY<?>E4

gGK|VEXO_525XY<YXfNCPiA\ILKM2F9>EXO_0KR>E46IW:=2 Ö b F[I<Y4uN\XY<?>WbRY7ýN\5JIk1 = k2 = 1

b#N\5JIk1 = k2 = −1

r!¡HNxg6B=S6ZdNÎWY52~K|IW=s8;7Y:;FR8;7Yg64uN\ApKR>EAC5S6XY<YIW4uNGb JI<Y4uN\XY<?>uPiIWC>F[I6bdY7

n = −m bdXYIR8;7Y:;F|46257YÎWY52~K#7\bWcIIW6257#F[73525XY<Yd>:[NTg6IEguNaF[46257\rENaF[7Y^Æ^S6:=2d>En = m

buXYIAWIW6XY<?>g6ILK|VEgrN\SdKRN\Y^>WbCY7 8;7Y=52cB[I\<YDuNaF[B=<?>E^>FON\A:[N\Î<Yg67?êu4625ILK#N\4ueB=7Y]N\X98;m34uN<Y62IWB=<Y7

525XY<Y XfNCPiA\ILKM2FU>EX_Zb#FOI46257QcmYg6<Y257lIW4uN-XY<YmY=XY25ILKR>E^÷DJIWB=<fe\g6AC257Y^,bRZWgG>El46257

cmYg6<Y257@:PhN\JIN\4dFU>E:;>E^7?F[B;>EXY<Y4uNGruU:;F[I\F[46257\bEôN\^>46Dr1| − 1

2 −1|1 b6N\57 1 6= −1r

B=<?>EA6PhN\g67Y^ XY<YmY=XY25ILK|7YZWI3DJIWB=<fe\g6ACS#8;7Y:;F F[7YHU<?KR>EA6PhN46257YB=VLKM46IW= ≤ 4uNR<Y625I.-B[N\XO_ý525XY<YcIKR>EXO_NXY<?>

Rr!N\SdKRN\Y^>WbJY7<Y7KM<YZW5mYg6Sý4uNoK#N\B=S6467YAp<?KMB=I\F[46IW=XY2

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KM:=<?>E:;F[AC25XO_0DJIEg6<Y625IWB=VLKg6ILK|IW5467YZWIl<Y625IWB=SXr¨M7Y]N\X98;7MXY<YmY=XY25ILK|7YZWIDcIWB=<fe\g6ACSvXY<?mY:;FOIIW<Y4uN\XY<fN:=25mM:;>E^JIW57Y^ ≤ 5S6:;>E^JI,-]N\^2cDJIEg6IW64d>E^2ï/h46Dr b ≤X XY<?> E

1ruNaF[7Y^ ≤ Î\Y7`^257Ys2546467@<Y4uN\XY<Y7Y46257T4625F[I6b\g6ITAdF[VWB=7YZWI`:=25mRD6B=<?>E<?KR>EXY<LN\25525=^>WrWq >46257#D6B=IK#N\g6<Y25#g6IT46257YDJIWB=IW<YS6^257YbaZWgG>cmYg6<Y257Y^>ýB=IW<?K#N\fN\<?625IWB;>ý525XY<YJILK|7

N, Z, Q, R/h225XO_QDcIEg6<Y625IWB;>Ë1bF[I:;>E^Ð-

cIW ≤ JmYg6<Y257^2]NCPM<fNfKM:=<Y7p:;K#Ia8;7,:;FON\46guN\B=g6ILK|7,<Y4uN\XY<Y7Y46257\rkä25464d>EXO_8;7Yg64uN\A:;>CF[SuN\X98=N\XO_8;7YZWI<Y4uN\XY<Y7Y46257TJmYg6<Y257T<fN\57YfNCPiIICg,D6B=<?>L8;m?F[798Mg67?êu4625X98;2 r¬ |Á§,v ª Ð|2þ y *¤(|ïÀ©W~;£.;=z£.y xwv*jc,£.y ;@Ig nRm ⇔ (∃k ∈ N)m = kn

tuwv*xzy z|ï y u@¡mwv*£, ª W©Wxz|Ig¡mwv*£.(¡v*jc,£*yl¥m¡my |Õ% | i= £.(¡ xwv*xzy = PO

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baF[I^VLKM25^>Wb\Y7

xP &;«"ÐRV+W& P QST'©kÊ;aORQjm¯[Qd,X¢RV'

y5S6Y7

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b F[IQK<Y625IWB=<Y7pXY<YmY=XY25ILK|IS6DJIWB=<fe\g6A\ILK#N\4C>E^〈X,〉 ôN\^>\b6Y7¤38;7Y:;F3^4625798;:=<Y7TB=VLKM467 Ö rk 4uNaF[S6B[N\54d>:=DcIW:=VWpKMD6B=ILKRN\g6<YN\^>¼ *([wav46257YB=VLKM46IW=\rn,VLKM25^>x^2]N\46IKM2l-XY257\b6Y7

xP &;ï"ÐRV+W& P QST'oICg

y2!D625:=<Y7Y^>

x < yb6ZWgG>

x ≤ y2x 6= y

r qz\7Y=52

x ≤ y5S6

y ≤ xF[I,^VLKM25^>WbY7

x2y:[eV ,[Qd,X¢RV',XZ#%^lR&?r§57Y^7Y4dF9>\b

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XôND6B=<?>E4uN8;^4625798`gGK#N,7Y57Y^7Y4dFU>2B=IW<?K#N\Y^>B=7Y]N\X98;m254C-

AC5S6<98;2s4uN<Y625IWB=<Y7 P(X)r#13257YX_

x, y ∈ X, x 6= yr k0F[7YgG>0ICXY<?>CKM25=XY257

x 6⊆ y 2 y 6⊆ x rc|<?>E52!7Y57Y^7Y4dFU> x 2 y :[e46257YDcIWB=VKM4d>CK#N\5467\r¤Er3¨MIW<?K#N\Y^>B=7Y]N\X98;m3DcIEg6<Y257Y546IW=XY2c4uN<Y625IWB=<Y7

N+ rdk F9>E^ KR>EDuN\g6ACSxÎ\Y4uNKM:=AaNa<fN\,g6S6YIDuN\B7Y57Y^7Y4dF[VLKj46257YDcIWB=VKM4d>CK#N\54C>EXO_b46DrÚ2R×GrU:;F[I\F[46257\bN\462!Ú46257|8;7Y:;F3g6<Y257Y54625AC257Y^j×GbuN\462!4uNIEgGKMB=V\Ffr

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XÎÏV&;\^l+

! ,Xc ,^lR&Ð%XZ# &^_&"&RV('XaÀV ,[Qd,X¢RV',XZ#%^lR&ÍXÆS;`%^_Q!&" [Q&^#!U P + ≤ ¿DUQST',^l+ P &;\^l+ ≤P &;¢Vd P R#.¿¢ Ò"d,X¢+m"Ð'*¿Æ¸.& ≤ P &;Zê½ì+Èìwé*, é&hr!íftonì`0/ S*ÊT+W ,[=

XÎ

z\7Y=52 ≤ 8;7Y:;F525462IK#>E^ DJIWB=<fe\g6AC257?^ <Y625IWB=SXbHF[IDuN\B=m 〈X,≤〉 4uN\<?>CKRN\^>S*ÊT+W ,[Q&"^l+mRV+W ,Xc Ó*V ,[wS*aO.)O ,XZ#%RV',"æ2w^VKM25^>WbEY7@B=7Y]N\X98=N ≤ ^l+mRV+W ,Xc oV ,[wS*aO.), P &<Y625VWB

Xr

ã IK|VEg,DcIW4625Y:=<Y7YZWIF9KM27YB=g6<Y7Y462]NDcIW<YIW:;FONLKM2]N\^>8=N\A\I?KM25XY<Y7Y46257\r ¿oÁh»C¹a¼ ºd» Ò Áh»Õ ÕÂÏV&;\=^l+ ≤ UQS.;\*U+W ,Xc Bk *=V ,X¢+W&Q%RV+( /ÔV^l+mRV+W ,Xc mcV ,[wS*aO.), P &ïS*ÊT+Wd,[X+Y ⊆ X

ù[Q&^#!U P # Q`%[w#pRË+WUQS. ,R# ≤ Y UQS.;\*U+W ,Xc |kz =V ,X¢+W&Q%RV+W /Ô2^l+mRV+W ,Xc mV ,[wS*aO.), P &oS*ÊT+Wd,[

YÎ

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L ⊆ XP &;wçÒ1ppè~pK÷0/ kiX

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V ,[wS*aO.), P &L¿UQST',^l+«Ò%'

(∀x, y ∈ L)(x ≤ y ∨ y ≤ x).

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A ⊆ XP &;Óç+æjwçÒ1ppè~p®÷0/SkiX

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2B2

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N+ rk F9>E^KR>EDuN\g6ACSPhN\6XYS6XO_67Y^æ8;7Y:;FT46Dr<Y625VWBL = 2n : n ∈ N bNN\4dFU>O-PhN\6XYS6XO_67Y^<Y625VWBR525XY<?D6257YB;KM:=<?>EXO_r

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ó r33g6ILK|IEg64625`kl4625IW:=7YA Ý r ñ rÖfø r ã ]NDJIEguN\4d>EXO_,<Y625IWB=VLK

X2B=7Y]N\X98;2

RKX:=D6B[NfKMg6<Y25\buXY<?>

R8;7Y:;F3B=7Y]N\X98=e

B=VLKM46ILKRN\Y46IW=XY2 r/ NO1

X = N; xRy ⇔ 5|(x− y) b/hB1X = Z; xRy ⇔ 2|(x+ y)

b/ X1

X = Z; xRy ⇔ 3|(x+ y)b

/hgB1X = R; xRy ⇔ x+ y = 1

b/ 71

X = R; xRy ⇔ x− y ∈ N b/iy;1X = R2; 〈x1, y1〉R〈x2, y2〉 ⇔ x1 = x2

b/hZ!1

X = R2; 〈x1, y1〉R〈x2, y2〉 ⇔ x1 = y2

b/h_B1

X = P(N) \ ∅; ARB ⇔ A ∩B 6= ∅ b/h2m1X = N2; 〈x, y〉R〈s, t〉 ⇔ x+ t = y + s

b/Ø8Q1

X = x ∈ Z : 0 < |x| ≤ 4; xRy ⇔ xy > 0r

ã ]NFU>EX_B=7Y]N\X98;2RbJAdF[VWB=7:[eoB=7Y]N\XU8=N\^2 B=VLKM46ILKRN\Y46I\[X?2 bwIWD625:[N\<Y625VWB3255I,-

B[N\<YILKR> X/Rr

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FU>E^%DJICg6<Y2]NCPiIW^,r/ NO1

X = N; S = P,N bJZ\g6<Y257 P 8;7Y:;F@<Y625IWB=7Y^525XY<Y,DuN\B=<?>E:;F9>EXO_b!N N<Y625IWB=7Y^%525XY<Yx46257YDuN\B=<?>E:;FU>EX_b/hB1

X = R; S = [k, k + 1) : k ∈ Z b/hX*1X = R2; S 8;7Y:;F3B=IEg6<Y254ueD6B=IW:;FU>EXO_pDJIW:;FON\XY2 y = x+ b

g6]Nb ∈ R b/hgB1

X = R2; S 8;7Y:;F3B=IEg6<Y254ueIWACB=mYZWVLKDJIW:;FON\XY2 x2 + y2 = bg6]N

b ≥ 0r

Ö ¤Er ã N\467T:[e4uN\:;F[mYD6Sa8=e\XY7TB=7Y]N\X98;7T4uN<Y625IWB=<Y7N+ r

• xRy ⇔ (∃k ∈ Z)x− y = 4kb

• xSy ⇔ x|y ∨ y|x,• xTy ⇔ x+ 3 ≥ y

r/ NO1 D6B[NfKMg6<Y25\bwXY<?>:[eF[IvB=7Y]N\X98;7B=VLKM46ILKRN\Y46IW=XY2 r ã ]NB=7Y]N\X98;2¢cmYgue\X?>EX_

B=7Y]N\X98=N\^2sB=VLKM46ILKRN\Y46IW=XY2`IWD625:[N\ýIEg6DJILKM2]N\guN8=e\XY7ý25^ <Y625IWB;>255IWB[N.-<YILK#7\r

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2 R T 7/hXY<?>E52JXY<?>x¤#8;7Y:;F#KB=7Y]N\X98;2

R T < ì 1rÖ ÚGr D6B[NfKMg6<Y25\b!XY<?>ýg6]Nxg6ILK|IW54C>EXO_QB=7Y]N\X98;2HB=VKM46ILK#N\Y46IW=XY2

R2S4uNx<Y625IWB=<Y7

XB=7Y]N\X98;7

R ∪ S 2 R ∩ S :[eB=7Y]N\X98=N\^2JB[VLKM46ILK#N\Y46IW=XY2!K Xrcz\7Y=52JFON\Acb6F[I

IWD625:[N\`25XO_<Y625IWB;>v255IWB[N\<YILK#7\rÖ ñ r3¨MIW<?K#N\Y^>B=7Y]N\X98;7

R2S4uN<Y625IWB=<Y7

A = x ∈ N : x ≤ 10 <Yg67?êu4625IK#N\4674uN\:;F[mYD6Sa8=e\XYIËb

xRy ⇔ 5|(x2 − y2), xSy ⇔ xy 6= 8.

/ NO1x3<fN\:[N\g64625\buY7R8;7Y:;FfbcN

S4627#8;7Y:;F3B=7Y]N\X98=eB=VKR46IK#N\Y46IW=XY2 r

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r/hX*1vkQ>E<Y4uN\XY<?>E A/R

r/hgB1,|<?>

R ∩ S 8;7Y:;F3B=7Y]N\X98=eB=VKM46ILK#N\Y46IW=XY2m3

9Ëòl6 7 ³d¯w³ µ ª5³ ñpø Å

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a ≤ bF[I

min(a, b) = a1rR¨MIW<YDuNaF[B=<Y^>B=7Y]N\X98;7ýB=VLKM46ILKRN\Y46IW=XY2

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〈n,m〉R〈k, l〉 ⇔ min(n,m) = min(k, l); 〈n,m〉S〈k, l〉 ⇔ n = k.

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[〈2, 2〉]Rr

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/Rr

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X2Yru13257YX_

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〈an : n ∈ N〉T 〈bn : n ∈ N〉 ⇔ a0 = b0.

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NN

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¤ ø r ã 7?êu4625Sa8;7Y^>B=7Y]N\X98;m=∗ 4uN P(N)

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A2B:[e:=A\IW6XY<YIW4672

<Y625VWBA8;7Y:;F#K@PhN\=XY2KR>E^æDJIEg6<Y625IWB=7Y^ <Y625IWB=S

BbEF[IôNIEgv46257YZWI^4625798 7Y57Y^7?4C-

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ñCñdù þ ó\² µ ±nó]ª_³ µ ±y×Y7D6B=<Y7Yg6:;FONfKM25XY257Y52IWcI\8;ZdNxDJPiXY2w8;7Y:;FTFU>E57:[N\Î6bXY<?>E52 b25464d>E^2:?PiILKR>Wb!Y7<Y625IWB;>A\IW6257?F2|^mYYXY<?>E<Y4ôN8=e,FON\Aae:[N\ôep525XY<YJmo7Y57Y^7Y4dFOVLKTrN\SdK#N\Y^>\bY7vg6IýF[798A\IW46AC5S6<98;2!g6IW:=<Y525=^>vJ7Y<T525XY<Y7Y462]NGrn,7?F[IEguNPhe\XY<Y7Y462]N`KDuN\B;>@8;7Y:;F|<fNaF[7Y^ g6IW6B[e^7FOICgueTDJIWB=VLKM4C>CK#N\462]N<Y625IWB=VLK

DcIEgQKM<YZW5mYg67Y^µ25IW=XY27Y57Y^7Y4dF[VKTrkQ>EA\IWB=<?>E:=FON\^>F[7YB[N\<F[I,:=DJIW:;F[B=<Y7YY7Y4u257oD6B=<?>g67?êu4625ILK#N\4625SDJIa8;mYXY2]N`B=VLKM46IW525XY<Y46IW=XY26<Y625IWB=VLKÁ/hXY<?>E526DJIW:=2]N\guN\462]NsF[798:[N\^798255IW=XY27Y57Y^7Y4dF[VLKo1 KM257Yg6<fe\X\bdY7sPhe\XY<Y7Y4625SKDuNaB=>7Y57Y^7Y4CF[VLK<Y625IWB=VK

A2BICg6DJILKM2]N\guN

<Y4uN8;g6ILKRN\46257`62 8;7YA\X98;2!DJIW^25mYg6<?>oFU>E^2<Y625IWB[N\^2 rãx»dè Ò Á Ðî?À äÈÊT+W ,[='

A+BR#*ST',XZ#%"Ð'Ë&)(*,+ïéËê½ì`p®r/+~ ìW¿ P &;\^l++_(RV+W& P &v;CRn)OU P #

f[wS.&;)*QSTi#.ºUQ# P a!UQ#oS*ÊT+Wd,[AXÆS*# P &"ÐRV+W& P &Q%R STR#!UQSTRV+W&2R#ÔS*ÊT+Wd,[

BÎEc+_QS.&"Ð'¯X¢&Q%'

A ∼ B¿c#î v;CRn)OU P +

f"d,X¢+m"Ð'*¿È¸.&ÍOi#%^#[Qd,X¢R ,^l+WUQSTR *\*]S*ÊT+W ,[Qd,X

A+BÎ

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<?KM2]e\<YACS<`FU>E^jKMD6B=ILK#N\g6<fN\^>oF[7YT:;>E^JIW |A| /hguNfKM4625798#S6?>CK#N\46I:;>E^cIW5S A= 14uNIW<Y4uN\XY<Y7Y46257`255IW=XY2w7Y57Y^7Y4dF[VLK <Y625IWB=SAbGXY<?>E52" *U's<Y625IWB=S

Aru13257@JmYg6<Y257Y^>

8;7Yg64uN\A<fN\:;FON\4uNfKM2]N\:=25m4uN\gýF9>E^,bXY<?>E^j8;7Y:;F255I\[7Y57Y^7Y4dF[VKÊ<Y625IWB=S46257Y:=AWIWC-XY<YIW467YZWIY/hg6]NT<Y625IWB=VLK:=AWIW6XY<YIW4d>EX_8;7Y:;F|IW4uN52XY<YueT4uNaF[S6B[N\54ueO1bd:[N\^7YZWI<fN\ DJI,-8;mYXY2]N3ÎEX?><Y625IWB=SJmYg6<Y257Y^>S6?>CKRN\52CF9>E5A\I@g6]N3DJIWB=VLKM4C>CK#N\462]Ns255IW=XY2C7Y57Y^7Y4dF[VKgGK|VEXO_,<Y625IWB=VKTr6 <fNaF[7Y^

A ∼ B ⇔ |A| = |B| ⇔ (∃f) f : A −→1−1na B.

¨MVLKM46IW525XY<Y46IW=@<Y625IWB=VLKôNDJ7?KM467`DJICg6:=FONfK|ILK#7@K@PhN\:=46IW=XY2 bGAdF[VWB;>EXO_d>E=^>ICg,4625798#ICXY<Y7YAC2KRN\52 r ¿oÁh»C¹a¼ ºd» Ò Áh»ï Í ^#! ,Xc ,^lRV'OU;eS*ÊT+W ,[Qd,X

A¿B+C"$#%"Ð'

• |A| = |A| Ù• |A| = |B| ⇒ |B| = |A| Ù• (|A| = |B| ∧ |B| = |C|)⇒ |A| = |C| Î

ãx¾E¿ ¼ k>E:;FON\B=XY<?>l:=AWIWB=<?>E:;FON\v<y N\AdF[VLKTb¢?7yhS646AWX98=N,25g67Y4dF9>EXY<Y46IW=XY25ILK#N8;7Y:;F62 8;7YA\X98=eGbGyhS646AWX98=NICgGKMB=I\F[4uNg6I62 8;7YA\X98;2G8;7Y:;F362 8;7YA\X98=eIWB[N\<T<PiIW?7Y46257T62 8;7YA\X98;2G8;7Y:;F62 8;7YA\X98=eGr

¢IKR>EY:=<Y7F9KM257?B[g6<?7Y46257ÎWZEPiIWd>:=S6ZW7YB=IK#N\\bwY7B=VLKM46IW525XY<Y46IW=<Y625IWB=VLK8;7Y:;FB=7Y]N\X98=epB=VKM46ILK#N\Y46IW=XY2 r HN\^25m?FON8;^>x8;7Yg64uN\AcbY7xg6ILK#IW54uNB=7Y]N\X98=N8;7Y:;F<fNfKM:=<Y7B=7Y]N\X98=eoDcIW^25mYg6<?>p7Y57Y^7Y4dFON\^2 DJ7?KM467YZWIS6:;FON\5IW467YZWIv<Y625IWB=S

X/hXY<?>E52c8;7Y:;F`DJIEgC-

<Y625IWB=7Y^X2 1rd¡ >E^XY<fN\:=7Y^ B=VKM46IW25XY<Y46IW=3g6I\FU>EXY<?>KM:=<?>E:;F[AC25X_v<Y625IWB=VLKTbENT<Y625VWBKM:=<?>E:;F[AC25X_<?625IWB=VKTb!8=N\AKM257Y^>Wb 46257p25:;F[4625798;7\r#NaF[7Y^ KIWZWVW54d>E^tD6B=<?>EDuN\g6ACS

²Vò½ñ 74X ª5±E¬õý¬dó\² µ ±nó]ª_³ µ ® ñCñ/546257sÎWY7Y^>K IWZWVW73^VLKM253IB=VLKM46IW525XY<Y46IW=XY2J<Y625IWB=VLK8=N\A\IIB=7Y]N\X98;2cK=XY25:P]>E^:=7Y46:=257RF[7YZWIT:PiILK#NGrCz\7Y=52\8;7Yg64uNaAIWZWB[N\4625XY<?>E^>:=25mRg6ITB=IW<YDuNaF[B;>CK#N\462]N@DJIEg6<Y625IWB=VLKDJ7?KM467YZWIoS6:;FON\5IW467YZWI<Y625IWB=S

XbuF[I¡ KM257YB=g6<Y7Y46257åGr Ö :;FON\46ILKM2 bcY7B=VLKM46IW525XY<Y46IW=

<Y625IWB=VLK-8;7Y:;F3B=7Y]N\X98=eB=VLKM46ILKRN\Y46IW=XY2 4uN P(X)r

¨MIW<?KRN\Y^>F[7YB[N\<,DJ7?KM4uel255IW=xD6B=<?>EA6PhN\g6VK<Y625IWB=VLK%B[VLKM46IW525XY<Y4d>EXO_b AdF[VWB=7D6B=<?>EDJIW^254uN8=eF[7Y\bd8=N\ADJIWD6B[NfKM46257A\IW46:;F[B=S6ILK#N\yhS646AWX98;7S6:;FON\]N8=e\XY7B=VLKM46IW525XY<=-46IW=/h2G8=N\Ao:=D6B[NLKMg6<fN\DcIWD6B[NLKM46IW=`FU>EX_,A\IW46:;F[B=S6AWX98;2m1r¸¹ºWá!ÓÈÇ Àu¼!áÖ r ã ]N<Y625I\B[VLK%:=A\IW6XY<YIW4d>EX_0g67êu4625X98=NQB=VLKM46IW525XY<Y46IW=XY2R<Y625IWB=VLK%<YZdN\g6<fNQ:=25m<v4uNaF[S6B[N\54ue,254dF[S625X98=eGr¢1sND6B=<?>EA6PhN\gb 1, 2, 3 ∼ √2, 3

√2, π bN,yhS646A\X98;m

f1 : 1, 2, 3 → √

2, 3√

2, π S6:;FON\]Na8=e\XfeB=VLKM46IW525XY<Y46IW=TÎWY7Y^>xIWACB=7Y=525K#N\B=S646AaN\^2f1(1) =

√2, f1(2) = 3

√22f1(3) = π

r¤ErN ∼ N+ rwg67?êu4625Sa8;^>xyhS646A\X98;m

f2 : N → N+ KM<YIWB=7Y^ f2(n) = n + 12¢:=D6B[NfKMgË÷Y^>Wb

Y7`S6:;FON\]NIW4uNB=VLKM46IW525XY<Y46IW=\r h N\AdFU>EXY<Y46257\ba8;7Y=52 n 6= mbGF[I

n+ 1 6= m+ 1<fNaF[7Y^f28;7Y:;F`B=VWY46ILKRN\B;F[IW=XY25ILK#NGr¢IW4uN\gGF[I8;7Y=52

k ∈ N+ bcF[I k − 1 ∈ N 2f2(k − 1) = k − 1 + 1 = k

bu<fNaF[7Y^f2

8;7Y:;FRU4uNOrÚGrN ∼ n ∈ N : 2|n r13257?F[B=S6g646I:=D6B[NfKMg6<Y25\b3Y7yhS646A\X98=N

f3 : N → n ∈ N : 2|n <fN\guN\4uNKM<YIWB=7Y^f3(n) = 2n

S6:;FON\]NB=VLKM46IW525XY<Y46IW=\r|NaF[7Y^ 525XY<Y4uNaF[S6B[N\54d>EX_2525XY<Yx4uNaF[S6B[N\54C>EXO_,DuN\B=<?>E:;FU>EXO_8;7Y:;FMFU>E57`:[N\Î6r

ñ rN ∼ Z r¡¢SGFON8 <fN\guN\462578;7Y:;F|46257YXYIF[B=S6g64625798;:=<Y7\rEU4dF[S625X?>L8;462573^S6:=25^>UDJIEPiILK|m3525XY<Y4uNaF[S6B[N\54d>EXO_ /h46Dr6525XY<Yd>DuN\B=<?>E:;F[7*1#D6B=<Y7YAC:=<?FONCPiXY25`4uNKM:=<?>E:;F[AC257T525XY<Yd>4uN.-F[S6B[N\5467\bfNRg6B=S6Zde UDJIEPiILK#m4uNR525XY<Yd>sXfNCPiAWILKM2F[7Sa8;7Y^467\r h IWB=ôNa54627¢yhS646AWX98;mf4 : N→ Z

S6:;FON\]Na8=e\XfeB=VLKM46IW525XY<Y46IW=TÎWY7Y^>v<fN\D625:[N\@KM<YIWB=7Y^%FON\ABb

f4(n) =

n2

8;7Y=522|n

−n−12

8;7Y=52 ¬2|nN\SdK#N\Y^>WbaY7¢8;7Y:;FF[IsyhS646A\X98=N@g6IW6B=<Y7¯/Ø8;7Yg646IW<Y4uN\XY<Y46257*1H<?g67?êu4625IK#N\4uNGr\sd>DJIWAN\<fN\\b#Y7

f4

8;7Y:;FB=VWY46ILK#N\B;F[IW=XY25ILKRNGb|^S6:=25^>B=IW<YDuNaF[B=<?>EýgGKRND6B=<?>O-DuN\g6AC2 r|z\7Y=52

f4(n) = f4(m) ≥ 0F[I

n2m^S6:=<feQd>EpDuN\B=<?>E:;F[7\b|XY<?>E52

n2

= m2

b3N<fNaF[7Y^n = m

r@z\7Y=52T<fN\f4(n) = f4(m) < 0

F[In2m^S6:=<fepd>Ev46257YDuN\B=<?>E:;F[7\bH:;FOe\g −n−1

2= −m−1

2

2 FON\ACY7n = m

rHHIW<YIW:;FON8;78;7Y:=<YXY<Y73DcIWAaN\<fN\\bWY7

f48;7Y:;F :=S6Bh8;7YAWXU8=eGrWk F9>E^ÊXY7Y5SK|7T÷Y^>g6ILK|IW5467

z ∈ Z rz\7Y=52z ≥ 0

bF[I2z8;7Y:;FDuN\B=<?>E:;FOe525XY<Yuev4uNaF[S6B[N\54uev2

f4(2z) = 2z2

= zr

A\IW57Y2HZWgG>z < 0

bJF[I −2z − 18;7Y:;F`46257YDuN\B=<?>E:;FOex525XY<Yue4uNaF[S6B[N\54ueo2HôN\^>

f4(−2z − 1) = −(−2z−1)−12

= zb6XYIA\IW6XY<?>g6IK|VEgr

ñCñ%6 þ ó\² µ ±nó]ª_³ µ ±y×Ý r

(0, 1) ∼ (a, b)/hZWg6<Y257

a < b1r

h S646AWX98=e f5 : (0, 1) → (a, b)S6:;FON\]N8=e\XfepB=VKM46IW525XY<Y46I\[8;7Y:;FKæFU>E^ KR>O-

DuN\g6ACSyhS646A\X98=N`525462ILK#Ns<fN\guN\4uN3KM<YIWB=7Y^f5(x) = (b−a)x+a

rW©3IWB=<?>E:;FON8=e\X<¡ KM257YB=g6<Y7Y462]NåGr Ö KM4625IW:=ACSa8;7Y^>:;FOe\gb!Y7g6ILK#IW5467gGKRNxIEg6XY2546AC2I\FUK#N\B;F[7:[eB=VKM46IW52XY<Y467\r

åGr(0,+∞) ∼ (−∞, 0) rkäIEXY<?>CKM25:;FU>:=DJIW:=VWyhS646AWX98=N

f6 : (0,+∞) → (−∞, 0), f6(x) = −xS6:;FON\]NB=VKM46IW52XY<Y46IW=\rì r

(0, 1) ∼ (1,+∞)r

k FU>E^KR>EDuN\g6ACSyhS646A\X98;mf7 : (0, 1)→ (1,+∞)

S6:;FON\]Na8=e\XfeRB=VLKM46IW525XY<Y46IW=IWACB=7Y=]N\^>KM<YIWB=7Y^

f7(x) = 1x

r¢©3IWB=<?>E:;FON8=e\X<DJI\D6B[<?7Yg64625X_lD6B=<?>EA6PhN\g6VLK2DJIW46ILKM46257<¡!KM257YB=g6<Y7Y462]NpåGr Ö g6IW:;FONa8;7Y^>Wb!Y7og6ILK|IW54d>IEg6XY2467YAýI\F9K#N\B;FU>8;7Y:;FB=VLKM46IW525XY<Y4d><ýg6ILK#IW54ueDJVEPiD6B=IW:;FOeI\FUKRN\B;FOeIWB[N\<ýY7ýg6IK|IW5467ýgGKR257DcVEPiD6B=IW:;F[7TI\FUKRN\B;F[7`:[e<Y7T:=IWueB=VLKM46IW525XY<Y467\r

×Gr(−π

2, π

2) ∼ R r¡ >E^ B[N\<Y7Y^ KR>EA\IWB=<?>E:=FON\^>g6IW6B=<Y7p<Y4uN\4ueQyhS646AWX98;m

f8 : (−π2, π

2) → R

bf8(x) =

F[Z(x)r z\7Y:;FoIW4uNB=IW:=4ue\XYNGb Ný<fNaF[7Y^ B=VWY46ILKRNaB=F[IW=X?25IK#NGr ¢IW4uN\gGF[I

limx→−π

2

F[Z(x) = −∞ IWB[N\<

limx→π

2

F[Z(x) = +∞ 2 A\IWB=<?>E:;FONa8=e\Xo<K@PhN\:-

46IW=XY2 ã N\B=JIWSvI\F[B=<?>E^Sa8;7Y^>\bGY7syhS646AWXU8=NFONM8;7Y:;FR:=S6Bh8;7YA\X98=eGru¢IEg6IW646257 8=N\ADcIWD6B=<Y7Yg64625IÎWY7Y^>:=FOe\gvKR>CKM462IW:=AWILK#N\\buY7@g6IK|IW54d>oIEg6X?25467YAoI\FUK#N\B;F9>2<Y625VWB#525XY<YB=<Y7YXY<?>CKM25:;F9>EXO_:[eB[VLKM46IW525XY<Y467\r

N\SdK#N\Y^>WbCY73KR>EJVWB yhS646AWX98;2JS6:;FON\]N8=e\XY798B=VKM46IW25XY<Y46IW=3462578;7Y:;F 8;7Yg646IW<Y4uN\XY<Y4C>WbF[<Y4rs25:;F[4625798;7g6S6YI0B=VWY4C>EXO_62 8;7YAWX98;2DJIW^25mYg6<?> <Y625IWB[N\^2B=VLKM46IW525XY<Y4d>E^2 rs1sND6B=<?>EA6PhN\gX_6Xfe\X,IWACB=7Y=525xyhS646AWX98;m

f : R → (−1, 1)KM<fN8;7Y^462578;7Yg646IW<Y4uN\XY<?4ue

ÎWY7Y^>S6?>E@<fN\B=VLKM46IKM<YIWB=Sf(x) = 2

π

NaB[XFOZ(x)bL8=N\A2cKM<YIWB=S

f(x) = x1+|x|

rG1sNIWZWVEPuKR>E6257?BONa^>F[mR62 8;7YA\X98;m\bWAdF[VWB[N|8;7Y:;F g6]N@4uN\:KguN\4C>E^ÊÎW^7Y46XY257#4uN89KR>EZWIEgC-4625798;:=<fNGr1sNR<fN\AWIW6XY<?7Y46257 F[7YZWIMDJIEg6B=IW<Yg6<Y2]NCPiSTDcIW<Y4uN\^>@FUKM257YB=g6<Y7Y46257\bLAdF[VWB=7 cmYg6<Y257 4uN\^

:=25m`XY<YmY:;F[IoD6B=<?>EguNfK#N\TKD6B=<?>E:=<PiIW=XY2 r

¿oÁh»C¹a¼ ºd» Ò Áh»ï 5Þ ^#! ,Xc ,^lRV'OU;eS*ÊT+W ,[Qd,XA,B,C

+D"$#%"Ð'/Ô

k#mfÏV&;\^l+A ∼ B

+C ∼ D

¿Z A× C ∼ B ×D Î

kÊjmfÏV&;\^l+A ∼ B

+C ∼ D

,[w#*SA∩C = ∅ + B ∩D = ∅ ¿¢ A∪C ∼ B ∪D Î

kzUmfÏV&;\^l+A ∼ B

¿Z P(A) ∼ P(B)Î

²Vòù¶74X ª5±E¬õ µ ª5®a¬dó\² µ ±nó]ª_³ µ ® ñCñ9ãx¾E¿ ¼ 3g6IK|IEg64625^>FU>E5A\IQDJIEg6D6S646AdFI/hB1bDJIW<YIW:;FONCPi7,DcIW<YIW:;FONLKM2]N8=e\X8=N\A\I?KM25XY<Y7Y46257\rwý<fNCPiIWY7Y462]NKM257Y^>WbuY725:;F[4625798=eyhS646AWX98;7

f : A −→1−1na B

2g : C −→1−1

na Dr

wg67?êu4625Sa8;7Y^>oF[7YB[N\<`yhS646AWX98;mh : A ∪ C → B ∪D 4uN\:;F[mYD6Sa8=e\X?>E^%KM<YIWB=7Y^ b

h(x) =

f(x)

8;7Y=52x ∈ A

g(x)8;7Y=52

x ∈ C .

N\SdKRN\Y^>WbJY7DJIW46257?K#N\<?625IWB;>A2C:[eoB=IW<Phe\XY<Y467\bJF[IoyhS646AWX98=N

h8;7Y:;F`g6IW6B=<Y7

/Ø8;7Yg646IW<Y4uN\XY<Y46257*13<Yg67?êu4625ILK#N\4uNGr D6B[NfKMg6<Y25^>F[7YB[N\<\bJY738;7Y:;F`254a8;7YA\X98=eGrckl7T÷Y^>,KFU>E^%XY7Y5S,g6ILK#I\5467

x, y ∈ A ∪ C b6FON\AC257`Y7 x 6= yr eF[B=<?>xÎWY52~K#IW=XY2Wb

• x, y ∈ A rGk0F[7YgG>xAWIWB=<?>E:;FON8=e\X<`B=VWY46ILKRN\B;F[IW=XY25ILK|IW=XY2yhS646AWX98;2 f I\F[B=<?>E^SC-8;7Y^>h(x) = f(x) 6= f(y) = h(y)

r

• x, y ∈ C rws4uN\5IWZW25XY<Y46257R8=N\A,DJIWD6B=<Y7Yg64625I6bJFU>E^ B[N\<Y7Y^KR>EAWIWB=<?>E:;F[Sa8=e\XB=VW=-46ILK#N\B;F[IW=XY25IK|IW=`yhS646A\X98;2gr

• x ∈ AI\BONa<

y ∈ Crk0F[7YgG>

h(x) = f(x) ∈ B2h(y) = g(y) ∈ D

r!s57B ∩D = ∅ bu<fNaF[7Y^ h(x) 6= h(y)

r¢IW<YIW:;FONa8;7QKR>EAaN\<fN\\b@Y7lyhS646A\X98=N

h8;7Y:;F,U4uNOr33:;FON\5^>0K.F9>E^ XY7Y5Sg6IK|IW5467

z ∈ B ∪D r3z\7Y=52 z ∈ BbRF[IDcIW46257?KRN\ýyhS646AWX98=N

f8;7Y:;FU4uNôN\^>

x ∈ Ab

FON\AC257Y7f(x) = z

rws57KRF[7YgG>x ∈ A ∪ C 2 h(x) = f(x) = z

r!z\7Y=52z ∈ D bcF[IB=IW<YS6^Sa8;7Y^>vDcIEg6IW646257\b6AWIWB=<?>E:;FON8=e\X<@F[7YZWI6buY7`yhS646A\X98=N

g8;7Y:;F3:=S6Bh8;7YAWX98=eGr

Ù < ¢'ÝTÆâ x'?"#~ö3$%YÝÍÞY'| Rx"k257Y^>8;S6\bcXYIF[Io<Y4uN\XY<?>WbJY7gGK#N<Y625IWB;>:[eB=VLKM46IW525XY<Y467\bu<Y4uN\^>xF[7YD6B=<?>O-

A6PhN\gG>pB=VWY4d>EX_DuNaB@46257Y:=A\IW6XY<YIW4d>EX_<Y625IWB=VKB=VLKM46IW525XY<Y4d>EXO_r1sN\guN\H46257KM257Y^>8;7Yg64uN\Acb XY<?>ýKM:=<?>E:;F[AE257<Y625IWB;>ý46257Y:=A\IW6XY<YIW467o^Na8=evFOex:[N\ôexÎEX\r¢¥sAaN\<YSa8;7:=25m\bY7RFON\A46257H8;7Y:;FfrE|ITKM25mYXY798?bdÎWY4uN`DcIEguN\MD6B=<?>EA6PhN\g46257Y:=AWIW6XY<YIW46798B[ICg6<Y254d><Y625I,-B=VLK%46257Y:=A\IW6XY<YIW4d>EX_bFON\AC25798Y7AaN\YgG>8;7987Y57Y^7Y4dFôNý25464ueÎEX\r z\7Y:;F<fNaF[7Y^46257Y:=A\IW6XY<Y7Y46257TKM257Y57`B=VWY4d>EXO_p46257Y:=A\IW6XY<YIW46IW=XY2^N\XY<Y462 8;^>8;7Yg64uN\AICg<fN\:;FON\46ILKM257Y462]Nv:=25m\bXYIxF[Ix<Y4uN\XY<?>Wb ?7gGK#Nx<Y625IWB;>

A2

BRV+W&:[eB=VLKM46IW525XY<Y467\rExg67?êu4625X98;2\8;7Y:;F F[ITB=VKM46ILK#N\Y467346257Y25:;F[46257Y4625SyhS646A\X98;2uKM<fN.-

8;7Y^46257 8;7Yg646IW<Y4uN\XY<Y46798RDJIW^25mYg6<?>o<Y625IWB[N\^2A2BrEkQ>E4625AaN:;FOe\gb6Y7@KR>EAaN\<fN\46257\b

25gGKRN0<Y625IWB;> 46257:[e0B=VLKM46IW525XY<Y467p8;7Y:;Fg6S6YI0F[B=S6g64625798;:=<Y7-IEgDcIWAaN\<fN\462]NGbsY7:[eB=VLKM46IW525XY<Y467\r ¥2557vcIKM257Y^ K%XY7Y5S-S6g6ILK#ICg646257Y462]NýB=VLKM46IW525XY<Y46IW=XY2|KR>E:;FON\B-XY<?>DJIWAaN\<fN\\bY7I+_(RV+W& P &62 8;7YA\X98=NDJIW^25mYg6<?>lguN\4d>E^2|<Y625IWB[N\^2 b XY<?>E52|KM:=AN\<fN\A\IW46ACB=7?F[4C>oIW6257YAdFfbdF[IC>DJIWAN\<fN\\bEY7@462573:[eIW467sB=VKR46IW525XY<Y467MF[B=<Y7YuNS6<fN\:[N\g64625\bY7I¸*#O%R#yhS646A\X98=NlDcIW^25mYg6<?>-FU>E^2M<Y625IWB[N\^2R462578;7Y:;Fv62 8;7YA\X98=eGb XY<?>E52MKR>EA\IW4uN\IWZWVW5467#B=IW<YS6ÎLKRN\46257\rdzWN\AWITD6B=<?>EA6PhN\gFON\AC257YZWITB=IW<YS6ÎLK#N\462]N@DJIWAN\Y7Y^>WbdY7MD6B=<Y7=-g6<Y2]NCP¢I\FUK#N\B;F9>v46257|8;7Y:;F3B[VLKM46IW525XY<Y4d>v<Y7T<Y625IWB=7Y^%525XY<Yx4uNaF[S6B[N\54C>EXO_g6IEguNaF[4625XO_r

ñCñp² þ ó\² µ ±nó]ª_³ µ ±y×¸¹ºWá!ÓÈÇ Àu¼

(0, 1) 6∼ N+ rB=<?>ED6S6=Y^>JILKM257Y^46257 KMD6B=IW:;Ffb\Y7#25:;F[4625798;7 yhS646A\X98=Nf : N+ −→1−1

na (0, 1)rW¥yhS646Ap-

X98;2fÎWY7Y^>,^>E=57YM8=N\AWIvIvXY2]eCZWS525XY<YB=<Y7YXY<?>CKM25:;FU>EX_r¢NCPiIW?>E525=^>xKM25mYX\bwY7

AN\YguNM525XY<YuNR< ICg6XY2546AaN(0, 1)

KR>E:;F[mYD6Sa8;7 KlF9>E^XY2]eCZWSrYk>ED625:=<Y^>ÀC255AN#KR>EB[N\<YVLKF[7YZWIXY2]eCZWS@b

f(1) = 0, a11a

12a

13a

14 . . . a

1n . . .

f(2) = 0, a21a

22a

23a

24 . . . a

2n . . .

f(3) = 0, a31a

32a

33a

34 . . . a

3n . . .2IWZWVW546257

f(n) = 0, an1an2a

n3a

n4 . . . a

nn . . . ,ZWg6<Y257

akm8;7Y:;F

m- FOeX?>CyhB[eDJI0D6B=<Y7YXY2546ACS K B=IW<?KM254625mYXY25Sg6<Y257Y:=25m?F[4d>E^

k- F[7YZWI

KR>EB[N\<YS4uN\:=<Y7YZWIlXY2]eCZWSå/hg6]NýS64625AC4625mYXY2]Ný4625798;7Yg646IW<Y4uN\XY<Y46IW=XY23B=IW<?KRN\YN\^>F9>E5A\IB=IW<?KM254625mYXY2]NTg6<Y257Y:=25m?F[467s<Y7s:=A\IW6XY<YIW4ue255IW=XY2]eTX?>CyhB ó 1rGwg67?êu4625Sa8;7Y^>F[7YB[N\<sDJ7?KM4ue525XY<Ycm

b ∈ (0, 1)K4uN\:;F[mYD6Sa8=e\X?>:=DcIW:=VW@b

b = 0, b1b2b3b4 . . . bn . . . ,

ZWg6<Y257bmb¢XY<?>E52

m- FOe,X?>CyhB=mvDJIpD6B=<Y7YXY2546ACSK B=IW<?KM254625mYXY25Slg6<Y257Y:=25m?F[4d>E^µ525XY<YC>

bKR>E6257YB[N\^>`KFON\AC2E:=DJIW:=VWbad>bm 6= amm

IWB[N\<bm 6= 0

2bm 6= 9

rWn,IWY7Y^>TF[IsIEXY<?>O-KM25=XY257<YB=IW625¯4,<fNfKM:=<Y7ôN\^>,g6IxgG>E:=DJIW<?>EX98;2w8;7Y:=<YXY<Y7:=257Yg67Y^ X?>CyhBYr N\SdK#N\Y^>WbY7TIW:;FONaF[46257`gGK#NK#N\B=S646AC2 <fN\DJ7?KM4625Na8=eGb6Y7

b 6= 02b 6= 1

r¢IW46257?KRNayhS646A\X98=N

f8;7Y:;FTU4uNObF[I,25:;F[4625798;7

n ∈ N+ FON\AC257\b!Y7 f(n) = br!k

:=<YXY<Y7YZWVW546IW=XY2HôN\^>WbuY7ann = bn

bJXYIvDcIW<YIW:;FONa8;7K:=D6B=<Y7YXY<Y46IW[XY2<^7?F[ICgueAWIW4C-:;F[B=S6ILK#N\462]N525XY<Yd>

br¢NaF[7Y^4625725:;F[4625798;7yhS646A\X98=N

f : N+ −→1−1na (0, 1)

b!XYI,AWIW6XY<?>g6ILK|VEgrn,7?F[IEguNsKR>EA\IWB=<?>E:=FON\4uN@KDJILKR>EY:=<Y>E^ÆD6B=<?>EA6PhN\g6<Y257R46IW:=2G4uN\<?K#mE"& %'c[wS.&Ñ

)pa%(RV+W ,Xc& P ruz\7Y=52!cIKM257?^%DJIW^>E=525^>oIX?>CyhB[N\XO_akm8=N\A\II7Y57Y^7Y4CFON\XO_46257Y:=AWIWC-

XY<YIW46798RFON\6525X?>Wb6F[Io525XY<?JmbA\IW46:;F[B=S6IK#N\525=^>oU25gue\XTDcIoD6B=<Y7YAaeaF[46798MF[798RFON\6525X?>v2

4uN,AaN\YgG>E^ ACB=IWACSICg6B=VWY462]Na8=e\X:=25mIEgA\IW5798;467YZWI,KR>EB[N\<YSXY2]eCZWSfrn,7?F[IEguN,FON

ôNKM257Y57`<fN\:;F[IW:=ILKRN\b\8;7Yg64d>E^<T4625XO_8;7Y:;F34uN\:;F[mYD6Sa8=e\XY7AC5N\:;>EXY<Y467TF9KM257YB=g6<Y7Y46257\r ¿oÁh»C¹a¼ ºd» Ò Áh»ï 5Ì ÏxÀ ÒJÇ ¾c¹/Ð ^#! ,Xc ,^lR&i`! ÍS*ÊT+W ,[=

A"$#%"Ð' |A| 6= |P(A)| Î

ãx¾E¿ ¼ NCPiV\Y^>o46257@KMD6B=IW:;FfbcY7T25:;F[4625798;7@yhS646A\X98=Nf : A −→1−1

na P(A).

z\7Y:;FHF[Is<fNaF[7Y^ yhS646A\X98=NGbaAdF[VWB[N3AN\Yg67Y^S7Y57Y^7Y4CF[ILKM2C<Y625IWB=SAD6B=<?>EDJIWB=<fe\g6A\ILKMSa8;7

DcIEg6<Y625VWBR<Y625IWB=SArJn,IWY7Y^>vKM25mYXT<Yg67?êu4625ILK#N\4uNa:=F[mYD6Sa8=eaX?><Y625VWB

Z = x ∈ A : x 6∈ f(x).

²Vò65 «#±C¬dóW² µ õu²s³ µ ª5®°±V³*õý X ª5±E¬dó\² ñCñ Å

¢IW46257?KRNaZ8;7Y:;F`DJIEg6<Y625IWB=7Y^<Y625IWB=S

Ab!NoyhS646A\X98=N

f8;7Y:;F@U4uNObJ25:;F[4625798;7<fNaF[7Y^

a ∈ A FON\AC2 b6Y7 f(a) = ZrJN\:;FON\46VLKM^>v:=25m\buXY<?>

a ∈ Z r egGKM257TÎWY52~K#IW=XY2Wb

• a ∈ Z rGk0F[7YgG><Tg67?êu4625X98;2<Y625IWB=S Z ôN\^> a 6∈ f(a) = Zr D6B[<?7YXY<Y46IW[\r

• a 6∈ Z ruk0F[7YgG>,<g67?êu4625X98;2¢<Y625IWB=S Z ôN\^> ¬a 6∈ f(a) = ZbcXY<?>E52

a ∈ Z r¢IW46ILKM46257T:=D6B=<Y7YXY<Y46IW=Wr

¥sF[B=<?>EôN\467T:=D6B=<Y7YXY<Y46IW=XY2¢<Y4uN\XY<feGbuY7T46257T25:;F[4625798;7syhS646AWX98=Nf : A −→1−1

na P(A)r

,DcIKR>EY:=<Y7YZWIF9KM257YB=g6<Y7Y462]NTÎWY7Y^>KR>CKM4625IW:=A\ILK#N\\bGY7325:;F[4625798;7346257?:[A\IW6XY<Y7=-46257RKM257?57MB=VWY4d>EXO_v46257Y:=A\IW6XY<YIW46IW=XY2 bCcIg6]N`g6IK|IW5467YZWIT<Y625IWB=SA/h46257Y:=A\IW6XY<YIW467YZWI!18;7YZWIp<Y625VWBDJI\F[mYZWILKR>QôN,IEgl46257YZWI,KM25mYXY7987Y57Y^7Y4dF[VK 2 DcIKRFON\B=<fN8=e\XoIWDc7YB[N\X98;m6B[N\462]N<Y625IWB=SDJI\F[mYZWILK|7YZWIg6IW:;FON8;7Y^>-<Y625IWB;>UXYIWB[N\<,uN\B=g6<Y2579846257Y:=AWIW6XY<YIW467Orz\7Y:;F F[ITB=IW<YS6ÎK#N\46257#DJIWD6B[NfKM467\bCY7Yd>@8;7Yg64uN\Ag6IW6B=<Y78;7M:;yhIWB=ôN\52<YILK#N\R^S6:=25^>4uN8;D6257YB;KDcIKM257Yg6<Y257?\buXYIF[I<Y4uN\XY<?>vU^257YsKM25mYXY798R7Y57Y^7Y4dF[VK#Or

Ù; ô ÝT~ö3$%Ðâ$Æx'?" Ý©ZâÜ \¢,'ÝT~öM$k257Y^>`8;S6\bGXYIF[I<Y4uN\XY<?>\buY7@gGKRN<Y625IWB;>ôNa8=eTFOe:[N\ôeÎEX\bCK|XY2]e\ 8;7Yg64uN\A

46257DJI\F[B[Naêu^>,DJIWB=VLKM4d>CKRN\ÎEX?>,<Y625IWB=VKTbJXY<?>E52H:;FUKM257YB=g6<fN\\bwAdF[VWB;>p<4625X_ôNKM25mYXY798o7Y57Y^7Y4dF[VKTr klB=VEY^>4uNXO_dKM255mg6ID6B=<?>CF[IEXY<YIW467YZWI-4uNDJICXY<feaF[ACS0F[7YZWIB=IW<Yg6<Y2]NCPiS,D6B=<?>EA6PhN\g6S254dF[S625X?>L8;467YZWIDJIWB=VLKM4d>CKRN\462]N<Y625IWB=VK525XY<Y,4uNaF[S6B[N\54d>EXO_p2XfNCPiA\IKM2~F9>EXO_r\HIW46257?K#N\

N8;7Y:;FD/iK@PhN\=X?2KR>E^1 DJICg6<Y625IWB=7Y^

ZbKM25mYX KR>EguNfK#NCPiI`:=25m\b

Y7ôNo^4625798s7Y57Y^7Y4dF[VLKTr z\S6KM257Y^>WbwY7FON\Ap4625738;7Y:;Ffb N\57FONv254CF[S625X98=No46257C>uPhNXfNCPiAC257Y^ JPimYg64uNGr6NfK#N\B;FON`K4625798d>uPhNJILKM257Y^ D6B[NLKMg6<Y2K#N254GyhIWB=ôN\X98=NGbWY7s<Y625V\BNôNÎCXîRV+W&X¢+W;)*QSaIEgQÎEX?><Y625IWB=S

Zr Ugue\XFU>E^F[B=IWDJ7Y^KMD6B=IK#N\g6<fN\^>

4uN\:;F[mYD6Sa8=e\Xfeg67?êu4625X98;m\rãx»dè Ò Á Ðî?À £Ô+W&;U;e

A+BÊ=QOa! ,Xc ,^lRV',"Ð+«S*ÊT+W ,[w#%"Ð+(Î d,X¢+m"Ð'*¿¸.&oS*ÊT+Wd,[

A"$#

" *U¼+Èì`0 *Eì`Ko.z)r!í9 I" *U'îS*ÊT+W ,[=B¿ P &;\^l+ P &;Ô[Qd,X¢R ,^l+WUQSTRV'YSÀV&X¢RV',"JB %Ñ

S*ÊT+W ,[Q&" S*ÊT+W ,[=¼$oÎbAB',"$Ê= ,^l+WUQSTRV+W&c+_QS.&"Ð'|A| ≤ |B| ⇔ ∃C(C ⊆ B ∧ A ∼ C).

kQ>E462AaN:;FOe\gIEXY<?>CKM25=XY257IEgB[N\<YSb|Y78;7Y=52A ⊆ B

b F[I |A| ≤ |B| r sd>^VEX:=ACSGF[7YXY<f46257 DJIWB=VLKM4C>CK#N\|ÎCXY7 <Y625IWB=VLKDJI\F[B=<Y7Y6467|4uN\^JmYg6<Y257 4uN\:;F[mYD6Sa8=e\XY7FUKM257YB=g6<Y7Y46257\bGDJIEguN8=e\XY7sK#N\B=S646AC2B=VLKM46ILKRN\Y467sF[7Y^SbEY7@<Y625VWB

AôN462573KM25mYAC:=<fe

ÎEX`4625`<Y625VWBBr

ñCñp× þ ó\² µ ±nó]ª_³ µ ±y× ¿oÁh»C¹a¼ ºd» Ò Áh»ï é ^#$! ,Xc ,^lRV'OU;eRV+W&O('OU;eÒS*ÊT+W ,[Qd,X

A+BR#. P a!U;&oXZ#%Ñ

[=CRn),+Æa[Qd,X¢R ,XZ#*¸TR&Ôk#m |A| ≤ |B| ÙkÊjm+_(RV+W& P &1v;CRn)OU P #

f : A −→1−1 BÙ

kzUm+_(RV+W& P &1v;CRn)OU P #g : B −→na A Î

ãx¾E¿ ¼ B=<?>EDJIW^462 8;^>WbJY7g6<Y25mYAC2HD6B=<Y7YX_6ICg64625IW=XY225^D62AaN\X98;2!KR>E:;FON\B=XY<?>ýDJI,-AN\<fN\\bcY7/ NO1 ⇒ /hB1 ⇒ /hX*1 ⇒ / NO1r/ NO1 ⇒ /hB1rG¢IW46257?K#N\ |A| ≤ |B| bCKM25mYXs<sg67?êu4625X98;2J25:;F[4625798;7 C ⊆ B

FON\AC2 bCY7A ∼ C

r¥@<Y4uN\XY<fN,F[I6b Y7o25:;F[4625798;7yhS646AWX98=N

f : A −→1−1na C

rHn,IWY7Y^>x8=e,DcI\F[B[N\AdF[IK#N\`8=N\AWIyhS646A\X98;mID6B=<Y7YXY2KMg6<Y257Yg6<Y2546257

B2wKRF[7YgG>

f : A −→1−1 Br

/hB1 ⇒ /hX*1r!Q<fNCPiIWY7Y462]NKM257Y^>WbcY725:;F[4625798;7 f : A −→1−1 Br¥@XY<?>CKM25=XY257yhS646AWX98=NoIEgC-

KMB=I\F[4uNf−1 :

B=46Z(f)→ A

8;7Y:;FT:=S6Bh8;7YA\X98=e / Nv4uNfK#7?F`62 8;7YAWX98=eO1bJKR>E:;FON\B=XY<?>o8=eo<fN.-F[7Y^ B=IW<?:[<?7YB=<?>Glg6I-yhS646AWX98;2

gIWACB=7Y=5IW467984uN-XfNCP]>E^ <Y625IWB=<Y7

BrRköF9>E^ X?7Y5S

KR>E6257YB=<Y^>,g6ILK#IW467a0 ∈ A

/Ø8;7Y:;FsF[IoÎWY52K|7\bccIv<Y625VWBA8;7Y:;F@46257YD6S6:;FU>Ë132 g6]N

g6ILK|IW5467YZWIb ∈ B IWACB=7Y=5^> g(b) 4uN\:;F[mYD6Sa8=e\X?>E^%KM<YIWB=7Y^ b

g(b) =

f−1(b)

8;7Y=52b ∈ B=46Z (f)

a0

8;7Y=52b 6∈ B=46Z (f)

.

B=IW:;FU>E^?KM25XY<Y7Y46257Y^%DJIW<YIW:;FON8;7T:=D6B[NfKMg6<Y7Y46257\bcY7g : B −→na A r/hX*1 ⇒ / NO1rdNCPiVWY^>WbY7 g : B −→na A r <YSGAaN\^>F[7YB[N\< FON\AC257YZWI@DJICg6<Y625IWB=S C <Y625IWB=S

BbWC>yhS646A\X98=NTIWJXY25m?FON

g CC>uPhN`62 8;7YAWX98=eGrdHI\46257?KRN\RyhS646A\X98=N

g8;7Y:;FU4uNObW<fNaF[7Y^

g6]NAaN\Yg67YZWIa ∈ A <Y625VWB g−1[a]

/Ø8;7Y:;F#F[I<Y625VWB FU>EXO_7Y57Y^7Y4dF[VLK<Y625IWB=SBbGAdF[VWB=7

D6B=<Y7YXO_6IEg6<fe34uNa1E8;7Y:;F¢46257YD6S6:;F9>Wr\z\7Y=52WKM25mYXK|7T÷Y^257Y^>@FON\Ad2C<Y625VWB

C ⊆ BbLAdF[VWB;>T<

AN\YgG>E^<Y625IWB=7Y^g−1[a]

ôNRg6IWA6PhN\g6462578;7Yg67Y4D6S646AdF¢KM:=DJVW54d>/iFON\AC2C<Y625VWB!4uN\<?>CK#N:=25m:=7Y57YAdF[IWB=7Y^µB=IEg6<Y254d> g−1[a] : a ∈ A 1b!F[I,y N\AdFU>EXY<Y46257 g C : C −→1−1

na A/Ø8;7Y:;F

F[I<Y46VLK0D6B=IW:;F[73?KM25XY<Y7Y462574F[I6bdY73g6I<Y625IWB=SCKR>E6257YB[N\^>7Y57Y^7Y4CFU><3AaN\Yg67YZWI

<Y625IWB=Sg−1[a]

DJILK#ICg6Sa8;7\bCY78;7Y:;F F[I4uN\guN\6yhS646AWX98=N@U4uNObCN`KR>E6257YB[N\46273g6IWA6PhN\g6462578;7Yg6467YZWI`7Y57Y^7Y4dF[S<fN\DJ7?KM46264uN\^B=VWY46IK#N\B;F[IW=XY25ILK|IW=*1rCvg67?êu4625X98;2GôN\^><fNaF[7Y^|A| ≤ |B| buXYIA\IW6XY<?>,g6ILK|VEgr

B=<?>L8;m?F[7IW<Y4uN\XY<Y7?46257:[S6Z\7YB=Sa8;7\b!Y7<YN\57YY46IW= ≤ DJIW^25mYg6<>pÎEXfN\^2¢<Y625IWB=VLK8;7Y:;F3B=7Y]N\X98=eDJIWB=<fe\g6ACSr h IWB=ôNa54627@B=<Y7YXY<625IWB[e\X@FON\Av4627#8;7Y:;FfbuDJIW46257?K#N\/hDcIEg6IWC-462578=N\AB=VKM46IW525XY<?46IW=s<Y625IWB=VLKo1462578;7Y:;F|F[IB=7Y]N\X98=NTK=XY25:P]>E^ :=7Y46:=2573F[7YZWI:PiILK#N4D6B=<Y7YXY257Y@:[N\ÎDcI\8;mYXY2573ÎEX?><Y625IWB=SI/hXY<?>E52J255IW=XY2\8;7YZWI7Y57?^7Y4CF[VLKo146257s<YIW:;FONCPiI:;yhIWB=ôN\52<YILK#N\4672HDJIW<YIW:;FON8;7254CF[S625X?>L8;467\rw¡ >E^46257Y^46257983<fN\57YY46IW=FONvôNoK@PhN\:-46IW=XY2 b6AdF[VWB;>EXO_pIEXY<Y7YAC2K#N\525d>E=^>xICg,B=7Y]N\X98;2DJI\B[<Ye\g6ACSr

²Vò65 «#±C¬dóW² µ õu²s³ µ ª5®°±V³*õý X ª5±E¬dó\² ñCñ ö

¿oÁh»C¹a¼ ºd» Ò Áh»ï Õ ^#! ,Xc ,^lRV'OU;e$S*ÊT+W ,[Qd,XA, B

+C"$#%"Ð'

• |A| ≤ |A| Ù• (|A| ≤ |B| ∧ |B| ≤ |C|)⇒ |A| ≤ |C| Î

ãx¾E¿ ¼ kQ>E:;FON\B=XY<?>:=A\IWB=<?>E:;FON\ý<pXO_uN\B[N\AdF[7YB;>E<fN\X98;2@DcIEguN\46798K¡ KM257YB=g6<Y7Y4625SåGr ñ /hB1IWB[N\<y N\AdF[VLKTbY7 yhS646A\X98=N325g67Y4dFU>EXY<Y46IW=XY25ILKRN8;7Y:;F¢254a8;7YAWX98=eM2d<PiIWY7Y46257 254a8;7YAWX98;28;7Y:;F3254a8;7YA\X98=eGr

zWN\AKM25guN\\bMg6IDJ7Pi467YZWIDJICg6IW6257Y6:;F9K#N<QB=7Y]N\X98=eDJIWB=<fe\g6ACS 6B[N\ACSa8;7l4uN\^K@PhN\:=46IW=XY2s:PhN\J798oN\4dFU>E:;>E^7?F[B=252 r||_6XY257Y525d>E=^>\b d><,y N\AdF[Sb Y7ÎCXp<Y625IWB=S

A8;7Y:;F`46257KM25mYAC:=<fNx2HB=VLKM46IEXY<Y7Y=4625746257^4625798;:=<fNv4625ÎEX<Y625IWB=SBKR>E4625ANCPiI6bwY7

ÎEXY7|F[7R:[e`B=VLKM467\rC¥@AaN\<YSa8;7R:=25m\bWY7#FON\Ay N\AdFU>EXY<Y46257H8;7Y:;FLb?8;7Yg64uN\A46257¢8;7Y:;FF[ITfN\g64ue^2]N\B[ey N\AdFMF[B;>CKM2]N\54C>ù/Ø8=N\AxÎWZEPiIWd>v:=25m`KR>EguNLK#N\*1rz\7YZWIg6ILK#VCg8;7Y:;F 8;7Yg64d>E^<4uN89F[B=S6g64625798;:=<?>EX_ g6ILK#ICg6VK<fNLK#N\B;FU>EXO_ K F9>E^ :=ACB;>EDcXY257l2@K.<?KM25e\<YACS <FU>E^<YIW:;FON\46257`<fN\D6B=7Y<Y7Y4CF[ILK#N\4d>K ¨MIW<Yg6<Y2]N\57 ó r ¿oÁh»C¹a¼ ºd» Ò Áh»ï ï ÏxÀ ÒJÇ ¾c¹x»C¹ Ò¢Å?Ç »EÁ Ò Ð ^#À! ,Xc ,^lRV'OU;e¯S*ÊT+W ,[Qd,X

A+B¿ P &;\^l+

|A| ≤ |B| + |B| ≤ |A| ¿Z |A| = |B| Î

¡ KM257YB=g6<Y7Y46257#N\4dF[IWB[N.-9q|7YB=46:;F[7Y254uN /hXY<YmY:;F[I0K DcIEPhe\XY<Y7Y4625S<¡ KM257YB=g6<Y7Y46257Y^åGr ñ 1 DJIW<?K#N\]N4uN\^æ<Y4uN\XY<Y46257@S6D6B=IW=XY25@g6IK|IEgG>WbEK AdF[VWB;>EXO_ôN\^>KR>EAN\<fN\TB=VLK2-46IW525XY<Y46IW=sDJ7?KM4d>EX_,gGK|VEXO_<Y625I\B[VLKTr h N\AdFU>EXY<Y46257\bGK XY7Y5SxDcIWAaN\<fN\462]NGbEY7 A ∼ B<fN\^2]N\:;FAWI\46:=F[B=S6ILK#N\462]Nl62 8;7YAWX98;2

f : A −→1−1na B

KR>E:;FON\B=XY<?>KM:=AaN\<fN\46Dr gGKM257254a8;7YA\X98;7

f1 : A −→1−1 B2f2 : B −→1−1 A

bWXYI`KKM257Y5SKR>EDuN\g6AaN\XO_T8;7Y:;F g6S6YITD6B=IW:;F[:=<Y7\r¸¹ºWá!ÓÈÇ Àu¼

(0, 1) ∼ [0, 1]r

kQ>E:;FON\B=XY<?><fN\SdK#N\?>E\bY7(0, 1) ⊆ [0, 1] ⊆ (−1, 2)

rw¢IW4uN\gGF[IoKD6B=<?>EA6PhN\g6<Y2574uN:;F[B=IW46257 ÖWÖ ñ S6<fN\:[N\g64625525=^>WbY7#g6ILK|IW5467#gGK#N@D6B=<Y7Yg6<Y2]NdP]>I\F9K#N\B;F[7|:[e@B=VLKM46IW525XY<Y467\rk257Y^>v:;FOe\gbuY7

|(0, 1)| ≤ |[0, 1]| ≤ |(−1, 2)| = |(0, 1)|.N\:;F[IW:=ILKRN\46257`¡ KM257?B[g6<?7Y462]No#N\4dF[IWB[N.-9q|7YB=46:;F[7Y254uNA\IW6XY<?>g6ILK|VEgr¥@XY<?>CKM25=XY257\bJÎWY4uNJ7Y<YDcIW=B=7Yg64u25Iv<Yg67?êu4625ILK#N\62 8;7YAWX98;m

f : (0, 1) → [0, 1]bW8;7YgC-

4uN\A@8;7Y:;F#F[I<Yg67YX?>Eg6ILK#N\46257ùN\B=g6<Y25798 S6X?2]e\Y52K#734625MKR>EA\IW4uN\46257@DJILKR>EY:=<Y7fZWIB=IW<YSC-ÎLKRN\4625NGr¨MIW<Y:[e\g64C>E^tKR>EguN8;7ý:=25mpB=VLKM46257YpIEXY<Y7YAC2K#N\46257\b|d><fN\57YY46IW= ≤ DJIW^25mYg6<?>ÎEXfN\^2C<Y625IWB=VLK-^2]NCPhNRF[7Y|K@PhN\:=46IW=R:=DJVa8;46IW=XY2 baF[<Y4r\d>g6]Nsg6ILK#IW4C>EXO_<Y625IWB=VLK

A2BN\5JI

A^2]NCPH46257KM25mYXY798s7Y57Y^7Y4dF[VLK4625

BbwN\5JI

B46257KM25mYXY798

ArJ¡HN\ACY7

K0FU>E^ KR>EDuN\g6ACSxIWAN\<YSa8;7@:=25m\bCY7@DcIW:;F[S6]NaF F[7Y48;7Y:;F#:=DJ7Pi4625IW4d>WbL8;7Yg64uN\AFON\A:ONaÎ6b

ñdù!ø þ ó\² µ ±nó]ª_³ µ ±y×8=N\AKD6B=<?>EDuN\g6ACS¡!KM257YB=g6<Y7Y462]N-#N\4dF[IWB[N.-9q|7YB=46:;F[7Y254uNl254dF[S625X98=NGb Y78;7Y:;FvF[Ily N\AdFICXY<?>CKM25:;F9>2|uN\4uN\54d>,8;7Y:;FZEPimYJI\AWIýJPimYg64uNGr ¢IEg6IW646278=N\AlDcIWD6B=<Y7Yg64625I6bg6ILK|VEg<YIW:;FON\46257`DJICguN\4C>vK ¨MIW<Yg6<Y2]N\57 ó r ¿oÁh»C¹a¼ ºd» Ò Áh»ï ð ^#! ,Xc ,^lRV'OU;eS*ÊT+W ,[Qd,X

A+B"$#%"Ð'

|A| ≤ |B| ∨ |B| ≤ |A|.

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ÎCX?>x<Y625IWB=SBb6ÎWY7Y^>vDcIKM257Yg6<Y257Y\b6XYIF[I<Y4uN\XY<?>WbcY7TôNÎEX`^462798;:=<feGr

ãx»dè Ò Á Ðî?À £Ô+W&;U;eA+BÊ=QOa! ,Xc ,^lRV',"Ð+«S*ÊT+W ,[w#%"Ð+(Î d,X¢+m"Ð'*¿È¸.&ÔS*ÊT+Wd,[

A"$#

" *U\ø+Èì`0q)z)r!íî E" *U'DS*ÊT+W ,[=B¿ P &;\^ +" *UïS*ÊT+W ,[=

AP &;RV+W&2X¢+W;)*QS*#À ¯" *U'

S*ÊT+W ,[=B+ÆS*ÊT+W ,[='$&ÍRV+W&Ôa$[Qd%X¢R ,^l+WUQSR&Î4AB',"$Ê ,^l+WUQSTRË+W&2+_QS.&"Ð'

|A| < |B| ⇔ |A| ≤ |B| ∧ |A| 6= |B|.N\SdK#N\Y^>\bJY7254GyhIWB=ôN\X98=NGbuY7<Y625VWB

AôN^4625798;:=<feÎEX4625<Y625VWB

B8;7Y:;F

g6S6YI:=2554625798;:=<fNI/h2H4uNxIWZWVEPg6S6YIxF[B=S6g64625798;:=<fNxg6IS6<?>E:=AN\462]NO1s4625254GyhIWB=ôNaX98=NGbJY7<Y625VWB

AôNÎEX46257KM25mYAC:=<fel4625,<Y625VWB

Br|¥ 2557JILKM257Y^ g6IDcIWAaN\<fN\462]NGb Y7

|A| ≤ |B| KR>E:;FON\B=XY<?>,46Dr6<Y4uN\57Y<Y257Y46257$/Ø8;7Yg646798?buA\IW46ACB=7?F[46798Q1|yhS646AWX98;2 f : A −→1−1 Bb

F[IC>vDJIWAaN\<fN\\buY7 |A| < |B| F[B=<Y7YuN@8;7Y:=<YXY<Y7S6g6ILK|IEg64625\b6Y7T46257`25:;F[4625798;7syhS646A\X98=Ng : B −→1−1 A

buXYI6b\8=N\AT8;S6T^VLKM2552=^>WbEKR>EôN\ZdNIWZWVW5467YZWIB=IW<YS6ÎLK#N\462]NGr¸¹ºWá!ÓÈÇ Àu¼!áÖ r |N| < |R| r¢IW46257?K#N\

N ⊆ R b?KM25mYX |N| ≤ |R| rL¢IW4uN\gGF[IM<D6B=<?>EA6PhN\g6VLK4uNR:;F[B=IW4uN\XO_ ÖWÖ Ú2 ÖWÖ ñ KM257Y^>WbcY7N ∼ N+ 2 (0, 1) ∼ R bJN<D6B=<?>EA6PhN\g6Sý4uN:;F[B=IW46257 ÖWÖ åGbuY7

|N+| 6= |(0, 1)| r6k>E4625AaN:;FOe\gbuY7 |N| 6= |R| buXYIAWIW6XY<?>g6ILK|VEgr¤Er ã ]Ng6IK|IW5467YZWI<Y625IWB=S

AôN\^> |A| < |P(A)| rÊ¡ KM257YB=g6<Y7Y462]N0#N\4CF[IWB[NKM257Y^>WbsY7 |A| 6= |P(A)| r`ÊAWIW7Y246257?F[B=S6g646I:=D6B[NfKMg6<Y25\b Y7yhS646A\X98=N

f : A → P(A)<fN\guN\4uNxKM<YIWB=7Y^

f(a) = a 8;7Y:;F254a8;7YA\X98=eGbEXY<?>E52w<@¡ KM257YB=g6<Y7Y462]NåGr ñ g6IW:;FON8;7Y^>WbGY7 |A| ≤ |P(A)| bEXYIA\IW6XY<?>g6ILK|VEgr

q|7Y<YDJIW=B=7Yg64625I<`¡ KM257YB=g6<Y7Y462]NåGr ì KM4625IW:=ACSa8;7Y^>WbGY7|8;7Y=52w<Y625IWB;>A2B46257@:[e

B=VLKM46IW525XY<Y467\bcF[IvN\5JI |A| < |B| bN\5cI |B| < |A| rQAWIW57Y2¢A\IWB=<?>E:;FON8=e\X<¡ KM257YB-g6<Y7Y462]N,#N\4dF[IWB[N.-9q|7YB=46:;F[7Y254uNvÎWY4uNxDJI\AaN\<fN\\b!Y7UB=7Y]NaX98=N<8;7?:=FTD6B=<Y7YXO_6IEg6462]NGb

XY<?>E52(|A| < |B| ∧ |B| < |C|)⇒ |A| < |C|,

²Vòl6 7 ³d¯w³ µ ª5³ ñdùËñ

XYIDJIW<YIW:;FONfKM2]N\^>8=N\AWID6B=IW:;F[7?KM25XY<Y7Y46257\r6kQ>E4625AaN:;FOe\gK:=<YXY<Y7YZWVW546IW=XY2 buY7g6]Ng6ILK#IW5467?ZWI<Y625IWB=S

AôN\^>

|A| < |P(A)| < |P(P(A))| < |P(P(P(A)))| < . . .

|<?>E52|B=IW<YDJICXY<?>E4uNa8=e\XvIEgg6ILK|IW5467YZWI<Y625IWB=S46257Y:=A\IW6XY<YIW467YZWIÎWY7Y^>QS6<?>E:=AaN\XY2]eCZs46257Y:=A\IW6XY<YIW4d>EX_<Y625IWB=VLK-ôN8=e\X?>EXO_XYIWB[N\<|KM25mYX?798H7Y57Y^7Y4dF[VK`r\1325ZWgG>R8;7Yg64uN\A46257pS6<?>E:=AN\^>KF[7Y40:=DcIW:=VWKM:=<?>E:;FOAC25XO_ÎWY52~KR>EXO_U46257Y:=A\IW6XY<YIW46IW=XY2~Ob ZWgG>E625IWB[e\X:[S6^mKR>EB[N\<YVLK FON\AC257YZWIpXY2]edZWSlI\F[B=<?>EôN\^>46ILKR>Q<Y625VWBYb ôNa8=e\X?>ýKR25mYXY7987Y57Y^7Y4dF[VLK4625AN\YgG>p7Y57Y^7Y4dFsXY2]eCZWSrw¥@g,F[7YZWIv46ILK|7YZWIv<Y625IWB=SpÎWY7Y^><fN\XY<fe\A\IW5798;4C>@XY2]edZR2\FON\AsguN\5798?r?k25guN\:;FOe\gbfY746257w8;7Y:;F[7Y=^>@Kl:;FON\46257IWD625:[N\HKM:=<?>E:=F[AC25XO_ÎWY52K#>EX_vU46257Y:=A\IW6XY<YIW46IW=XY2~Ô4T8;7Y:;F325XO_<YC>CF346257Y:=A\IW6XY<Y7Y46257TKM257Y57\rYrYr

Ùy < Íàx'YÖ r ã ]NDJIEguN\4d>EXO_<Y625IWB=VK

A2B<Y4uN\57T÷YyhS646A\X98;mS6:;FON\]Na8=e\XfeB=VLKM46IW525XY<Y46IW=

^25mYg6<?>v4625^2wIWB[N\<T:=D6B[NfKMg6<Y25\buY7#8;7Y:;F3IW4uNy N\AdFU>EXY<Y46257T62 8;7YA\X98=eGr/ NO1

A = 3n+ 1 : n ∈ N, B = Nk

/hB1A = N, B = n ∈ N : (∃k ∈ N)n = 5k k/ X1A = n ∈ N : 2|n, B = n ∈ Z : ¬2|n k/hgB1A = 〈x, y〉 ∈ R2 : x+ y = 1, B = R

k/ 71

A = [0, 1], B = [0, 1)r

¤Er33g6ILK|IEg64625T¡ KM257YB=g6<Y7Y46257åGrØ¤Ë/ NO1|2ï/hX*1rÚGr F[IW:=Sa8=e\XT^7?F[ICg6mTD6B=<Y7YAeaF[4625IK#eS6g6ILK|IEg64625\buY7

N 6∼ NNr

ñ rs¥<Y625IWB[N\XO_A2BKM257Y^>\buY7 |A ∩B| = |A ∪B| ruHIWAN\<fN\\bcY7 A ∼ B

rÝ rswuN\guN\\bcXY<?>g6]Ng6IK|IW54d>EXO_46257Y:=A\IW6XY<YIW4d>EX_<Y625IWB=VK

A,B,C2DD6B[NLK2-

g6<Y2K|7T:[eDJI\4625Y:=<Y7<YguN\462]NGr/ NO1zW7?[2!25:;FO462798;7`yhS646A\X98=NB=VWY46IK#N\B;F[IW=XY25ILK#N

f : A −→1−1 BbuAdF[VWB[N46257|8;7Y:;F

U4uNObGF[I |A| < |B| r/hB1zW7?[2 |A| < |C| 2 |B| < |D| F[I |A ∪B| < |C ∪D| r/ X1,zW7?[2 |A| < |C| 2 |B| ≤ |D| F[I |A ∪B| < |C ∪D| r/hgB1zW7?[2 |A| < |C| 2 |B| < |D| F[I |A×B| < |C ×D| r/ 71,zW7?[2 |A| < |C| 2 |B| ≤ |D| F[I |A×B| < |C ×D| r/iy;1zW7?[2 |A| < |B| 2 C 6= ∅ bGF[I |A× C| < |B × C| r

ñdùEù þ ó\² µ ±nó]ª_³ µ ±y×åGr ã ]Ng6IK|IW54d>EX_,<?625IWB=VK

A,B2CDJIWAN\<fN\\bcY7

(|A| < |B| ∧ |B| < |C|)⇒ |A| < |C|.

ì r313257YXO_ 〈An : n ∈ N〉 JmYg6<Y257TXY2]eCZW257Y^j<Y625IWB=VLK FON\AC25^,b6Y7g6]NAN\Yg6798M525XY<YC>4uNaF[S6B[N\546798nôN\^> |An| < |An+1|

rw3g6ILK|IEg64625\bJY7:=S6ôNK#>EBONa<YVKF[7YZWIXY2]edZWSôNÎEX@KM25mYAC:=<feICg,AN\Yg67YZWI`8;7YZWI7Y57Y^7?4CF[Sb6XY<?>E52

(∀m ∈ N) |Am| < |⋃

n∈N

An|.

Ê++!

TVUµÊ)x .),+-@?!!·+M?K K..),+-@?!!·+M?K

DJIW=B=VEg<Y625IWB=VLKTbL< AdF[VWB;>E^2d:=25m :=FU>EAN\^>Wb4uN8;6525Y:=<Y7 :[e34uN\^ <Y625IWB;>`525XY<YT4uN.-F[S6B[N\54d>EX_

N2w525XY<YvB=<Y7YXY<?>CKM25:;F9>EXO_

Rr61325Xs<fNaF[7Y^jg6<Y2KM467YZWI6bGY7@FON\ACY7sK AdK|7Y:;F[252

255IW=XY27Y57Y^7Y4dF[VLK :[e,IW467g6]N4uN\:D6S646AdF[7Y^ ICg646257Y:=257Y462]NGr¢npN8=e\X<fNaF[7Y^µDc7?KM257Y446257Y:=A\IW6XY<YIW4d>l<Y625VWB

AÎWY7Y^>ý<fN\Dd>CFON\\b¢XY<?>x8;7Y:;FIW4lB=VLKM46IW525XY<Y4d>ý<Y7o<Y625IWB=7Y^

525XY<YQ4uNaF[S6B[N\54d>EXO_b XY<?>E52 XY<?>v8;7Y:;F¯[wS.&^l+WUQS*#%^lRV'ar¢z\7Y=5246257\b!F[I,d>EoÎWY7`8;7Y:;FF[798:[N\^798¢ÎEX?>WbaXYI`<Y625V\BH525XY<YB=<Y7YXY<?>CKM25:;FU>EX_bdXY<>E52" *U'ÀU; ,RV(+mRVCC" g rd¥@XY<?>CKM25=XY257\b8=N\ApKM257Y^>ý<DJIWD6B=<Y7Yg646257YZWIpB=IW<Yg6<Y2]NCPiSb!fNag64uN,<FU>EXO_ÎWY52~K#IW=XY2ÎWY746257<fN.-XO_6IEg6<Y25\bcN\57@KX?IEg6<Y257Y4646IW=XY2 ôNaF[7YôNaFU>EXY<Y46798¯/h46Dr6Ks4uN\525<Y257TnpNaF[7YôNaFU>EXY<Y46798Q1:[eF[I:;>CF[SuN\X98;74uN8;XY<YmY=XY25798M:=DcI\F9>EAN\467\rzWN\AWIM:=<YXY<Y7YZWVW546257K#N\Y467\bÎEXY7FU>EX_<Y625IWB=VLKôN8=e#:;K|I\8;7 IW:=IW6467 IW<Y4uN\XY<Y7Y4625NGr

¥<Y625IWB=<Y7T525XY<Yx4uNaF[S6B[N\54C>EXO_,^VLKM25^>WbGY7#8;7Y:;F3ÎEX?>ù#%^_&DvS.&[; jM2!D625:=<Y7Y^>|N| = ℵ0.1sNaF[IW^2]N\:;FsIx<Y625IWB=<Y7525XY<YB=<Y7YXY<?>CKM25:;F9>EXO_l^VLKM25^>WbJY7s8;7Y:;FTÎEX?>U; ,RV(+mRVCC" 2

D625:=<Y7Y^>|R| = c.

q¢IW46257?KRNa\b?8=N\AM8;S6#DJILKM257Yg6<Y257Y525=^>Wba46257#g67?êu4625Sa8;7Y^>D6B=7YX?>E<?>8;46257abdXY<?>E^ 8;7Y:;F ÎCX<Y625IWB=SbEFON\ACY7@F[7`<fN\D625:;>oF[B[N\AdF[Sa8;7Y^>8=N\A\I:=ACB=V\F9>Wb6XY<?>E52g6]Ng6IK|IW5467YZWI<Y625IWB=S

A<fN\D625: |A| = ℵ0IW<Y4uN\XY<fNRKR>uPhe\XY<Y46257 |A| = |N| bLNM<fN\D625: |A| = c

IW<Y4uN\XY<fN |A| = |R| rs4uN\5IWZW25XY<Y46257@B=<Y7YXY<TôN:=25m@K KR>EDuN\g6ACS4627YB=VKM46IW=XY2 rk FU>E^ B=IW<?g6<Y2]N\57o<fNa8;^257Y^>:=25mvD6B=<Y7Yg67xKM:=<?>E:;F[AC25^ K@PhN\:=46IW=XY2]N\^2 <Y625IWB=VLK

D6B=<Y7Y525XY<fN\54d>EXO_rCN\XY<Y46257Y^>38;7?g64uN\AIEg:=D6B=7YX?>E<YIK#N\462]NTDcI\8;mY\bWAdF[VWB;>E^26cmYg6<Y257Y^>:=25m`DJIW:PiS6ZW2K#N\\rs¥xzTw ª £Wy £..|z= ja.e§.y z©W xwZ y Wz©i2= apWú..©i=¡ (ª y zþN W2 y Wz©i2¥§.y =£,Z§p2þ= *¥ ª !

ñdùO6 74X ª5±E¬õ > ¬YL®pó]ª_³³!ó µ ®oª µ ª5® > ¬Y®%ó]ª_³³!ó µ ®ãx»dè Ò Á Ðî?À £Ô+W&;U;e

AÊ=Q*ST+W&À! ,Xc ,^lRV'," S*ÊT+W ,[Q&"$Î d%X¢+m"Ð'*¿È¸.&oS*ÊT+Wd,[

AP &;

¥ Î zhoBéÒ1p®rCé+~n¿DÒ%' P &;À ,RfS*ÊT+W ,[Q&" O('," ^lËÊî+_(RV+W& P &Y^l+WUQS*Ê;#IR#%(C[w#%^lR#n ∈ N ¿Zi#.)p#Ð¸.& A ∼ 1, 2, . . . , n Î

« Î +Èì`0/zhoBéÒ1p®rCé+~n¿ÈÒ%'$RV+W& P &;¢;)O ®ÓUQS. ,RV'pÎ° ÎP~&hry0pê½ì`p®r!çVê>+~n¿Ò%' P &;ï[Qd,XïR %^ +WUQSTRV'ÀS.&ÔS*ÊT+W ,[Q&"^l+WUQS*ÊÍR#%(C[Q#%^lRV'OU;eËÎB Î'p.é +¢ç)q)*,.-y0qx~&hry0pê½ì`p®r!çVê>+~n¿Ò%' P &;[wS.&^l+WUQS*#%^lRV'Ð^lËÊ¯;)O ®ÓUQS. ,RV'pÎC Î +Èì`0d~&hry0pê½ì`p®r!çVê>+~n¿ÈÒ%'$RV+W& P &;U; $R# P X¢'¸.& P [wS.&^l+WUQS*#%^lRV'pÎ

k FU>E^^25798;:=XYS,F[B=<Y7YuNo<YB=IW62525:;F[I\F[4ueSdKRN\ZWmF[7YB=^2546IW5I\ZW2XY<Y4ueGrw¥sF[VWK52F[7YB[N.-F[S6B=<Y7sôNFO7?ôNaF9>EXY<Y46798 <Y625IWB=7Y^ D6B=<Y7Y525XY<fN\54d>E^ 4uN\<?>CK#N:=25m3XY<YmY:;F[Iî/h254uN\XY<Y798 4625MKDcIKR>EY:=<Y798g67?êu4625X98;2m1s<Y625VWBYbAdF[VWB;>o8;7Y:;F:=A\IW6XY<YIW4d>Q5S6B=VLKM46IW525XY<Y4C><Y7<Y625IWB=7Y^525XY<Y4uNaF[S6B[N\54d>EXO_5/hXY<?>E52#<Y625VWBYb¢AdF[VWB;>^>l4uN\<?>CK#N\^>XYIý4uN89KR>EY798D6B=<Y7Y52XY<fN\¨-4d>E^1rHA\IW4dF[7YAC:;F[Sp8;7Yg64uN\AÎWY4uN,4uN,IWZWVEPMPhNaFUK#I,KR>CKM462IW:=AWILK#N\\bHAdF[VWB[N,AWIW4C-K|7Y46X98=N@8;7Y:;F3:;F[IW:=IK#N\4uNGr¢IKR>EY:=<feg67?êu4625X98;mTÎWY4uN4uN\:;F[mYD6Sa8=e\XYID6B=<Y7?yhIWB=^SJPiIK#N\\r

¿oÁh»C¹a¼ ºd» Ò Áh»ýð Í £Ô+W&;U;eAÊ=Q*ST+W&À! ,Xc ,^lRV'," S*ÊT+W ,[Q&"$ÎÔY&Q%'

• A P &;¢;)O ®ÓUQS. ,RV' ⇔ |A| < ℵ0

Î• A P &;ZRV+W&;;)O ®ÓUQS. ,RV' ⇔ |A| ≥ ℵ0

Î• A P &;[wS.&^l+WUQS*#%^lRV' ⇔ |A| = ℵ0

Î• A P &;U; $R# P X¢'¸.& P [wS.&^l+WUQS*#%^lRV' ⇔ |A| ≤ ℵ0

Î• A P &;ZRV+W&[wS.&^l+WUQS*#%^lRV' ⇔ |A| > ℵ0

Îãx¾E¿ ¼ k25mYAC:=<YI\[KRN\B=S646A\VLKKR>E4625ANKMD6B=IW:;F@<g67?êu4625X98;2 2¢¡!KM257YB=g6<Y7Y462]NvåGr ì r3<fN\:[N\g646257Y462]NTKR>EôN\ZdNFON\A4uN\D6B[NfKMg6msF9>E5A\Iî/h254dF[S625X?>L8;462573IEXY<?>CKM25:;F[7*1|:;FUKM257YB=g6<Y7=-46257\b\Y7M<Y625VWBJ8;7Y:;F :=AWIW6XY<YIW4d>KRF[7YgG>2GFU>E5A\I`KRF[7YgG>WbdZWgG>ôN@^4625798H7Y57Y^7Y4dF[VLK4625<Y625VWB KM:=<?>E:;F[AE25XO_,525XY<Yv4uNaF[S6B[N\54d>EX_rG¡¢Is8;7Y:;F8;7Yg64uN\Ao<fN\guN\46257sKR>EôN\ZdN8=e\XY7@S6?>O-XY2]Ns2546g6S6AWX98;26^NaF[7YôNaF9>EXY<Y46798?baKMB=VEXY25^><fNaF[7Y^Êg6I`46257YZWI@K4uN\:;F[mYD64d>E^ B=IW<Yg6<Y2]N\57ar

¢IW46257?KRNa-DJIa8;mYXY257:=A\IW6XY<YIW46IW=Xf2@8;7Y:;F4uN0IWZWVEP<YB=IW<YS6^2]NCPi7\b@46257JmYg6<Y257Y^>:=25mv<fN\ZEPimY62]N\oKjB=IW<?K#N\fN\462]N4uNpFO7?4FO7?ôNaF2#:=A\IW46XY7Y4dF[B=Sa8;7Y^>:=25mv4uNý<Y625IWB[N\XO_46257Y:=A\IW6XY<YIW4d>EX_r3B=<Y7YgGF[7Y^ DcIEguN\^>FU>E5AWI5/hJ7Y<g6ILK|IEg6SB1F9KM257YB=g6<Y7Y46257\b#AdF[VWB=7DcIW:PiS6?>uPiIK£@ £ KM257YACSx46257Y^257YXAC27Y^SoôNaF[7YôNaFU>EA\IKM2w¨Tr ã 7Yg67YAC2546g6ILKM2!g6I254C-467YZWI<Yg67?êu4625ILKRN\462]NDcI\8;mYXY2]N<Y625IWB=S:=A\IW6XY<YIW467YZWI6r

Å ò½ñ 74X ª5±E¬õ > ¬Y®%ó]ª_³³!ó µ ® ñdùy9 ¿oÁh»C¹a¼ ºd» Ò Áh»ýð 5Þ äÈÊT+Wd,[È;)O ®ÓUQS. ,RV'RV+W& P &;[Qd,X¢R ,^l+WUQSTRV'cS¸*#O%RV',"åXc ,+m"ÃV %ÑS*ÊT+W ,[Q&"¹XÈº#.\*U+mX¢',"$ÎDäÈÊT+Wd,[îRV+W&;;)O ®ÓUQS. ,RV' P &;À[Qd,X¢R ,^l+WUQSTRV'ðS$V&X¢RV'," Xc ,+m"V *S*ÊT+W ,[Q&"XÈº#.\*U+mX¢',"$Î

ÙTØ < ¢'ÝTÆâxH #"Þ?'| R¯Þfx"w625IWB;>`D6B=<Y7Y525XY<fN\5467 :[e34uNa8;^462798;:=<?>E^2d<Y625IWB[N\^2d46257Y:=A\IW6XY<YIW4d>E^2 r\g67?êu4625X98;2

<Y625IWB=7Y^jD6B=<Y7Y525XY<fN\54d>E^8;7Y:;F3<Y625VWBR525XY<Y4uNaF[S6B[N\54d>EXO_rJýD6B=<?>EA6PhN\g6VLK 4uN:;F[B=IW46257ÖWÖ ÚKM257?^>ýF[7Y\b¢Y7oD6B=<Y7Y525XY<fN\5467:[ep<Y625IWB;>525XY<YQ4uNaF[S6B[N\54d>EX_lg6IEguNFO462X_b 525XY<Y4uNaF[S6B[N\54d>EX_-DuN\B=<?>E:;FU>EX_2R525XY<Y-XfNCPiA\ILKM2FU>EX_rkëFU>E^ B=IW<Yg6<Y2]N\57x<fN\:;FON\46ILKM25^>:=25m\b\8=N\AC27|8;7Y:=<YXY<Y7<Y625IWB;>v:[eD6B=<Y7Y525XY<fN\5467\rN\XY<Y46257Y^>IEgpD6B[I\:=F[7YZ\Iv:=DJIW:;F[B=<Y7YY7Y462]NGbJAdF[VWB=7g6IW6B=<Y7IEg6guN8;7254dF[S625X98;mD6B=<Y7=-

525XY<fN\546IW=XY2 rH¥sF[VWoy N\AdFfbHY7A8;7Y:;F<Y625IWB=7Y^ D6B=<Y7Y525XY<fN\54d>E^ IW<Y4uN\XY<fNGbY7x25:;F[4625798;7

62 8;7YA\X98=Nf : N −→1−1

na ArU464d>E^2M:PiILKR>Wb yhS646A\X98=N

f4CS6^7YB=Sa8;77Y57Y^7Y4CFU><Y625IWB=S

A525XY<YuN\^2 4uNaF[S6B[N\54d>E^2 b¢N\5JI6b ^VKM2]e\X`8;7Y:=<YXY<Y7254uN\XY<Y798?b US6:;FONfKM2]N8;7oKæXY2]eCZaOr

n,IWY7Y^><fNaF[7Y^J7Y<:=Dc7YX98=N\5467YZWIx4uN\g6S6?>EXY2]NxD62:ONaA = a0, a1, a2, . . . , an, . . ./hZWg6<Y257

an = f(n)1r

B=<Y798;gË÷Y^>vF[7YB[N\<Tg6ID6257YB;KM:=<Y7YZWIFUKM257YB=g6<Y7Y462]NGr ¿oÁh»C¹a¼ ºd» Ò Áh»ýð 5Ì AVC"$#Y%Xcd*U;eS*ÊT+W ,[Qd,X [wS.&^l+WUQS*#%^lRV'OU;e P &;[wS.&^l+WUQS*#%^lR#OÎãx¾E¿ ¼ 13257YXO_

A2BcmYgueg6ILK|IW54C>E^23<Y625IWB[N\^23D6B=<Y7Y525XY<fN\54d>E^2 r U:;F[4625798=e

<fNaF[7Y^Æ62 8;7YA\X98;7f : N −→1−1

na A2g : N −→1−1

na BrCwg67?êu4625Sa8;^>yhS646A\X98;m

h : N→ A∪BK4uN\:;F[mYD6Sa8=e\X?>:=DJIW:=VW@b

h(n) =

f(k)

8;7Y=52n = 2k

g6]NDc7?KM467YZWIk ∈ N

g(k)8;7Y=52

n = 2k + 1g6]NDc7?KM467YZWI

k ∈ N.N\SdKRN\Y^>WbY7 yhS646A\X98=N

h8;7Y:;F:=S6Bh8;7YA\X98=eGra9:;F[I\F[46257\b46257YXO_

y ∈ A∪B rLkQ>E4625AaN3:;FOe\gbY7y ∈ A 5S6 y ∈ B raz\7Y=52 y ∈ A bfF[I3DJIW4627?KRN\yhS646A\X98=N f : N→ A

8;7Y:;F¢:=S6Bh8;7YAWX98=eGbKM25mYX 8;7Y:;F

m ∈ N bFON\AC257 Y7 f(m) = yras57 KRF[7YgG>

h(2m) = f(m) = yraHICg6IW646257\b

8;7Y=52y ∈ B

b|F[IA\IWB=<?>E:;FONa8=e\X<F[7YZWI6b#Y7yhS646AWX98=Ng : N → B

8;7Y:;F,:=S6Bh8;7YA\X98=e<Y4uN8;g6Sa8;7Y^>

m ∈ N b!FON\AC257Y7 g(m) = y2KRF[7YgG>

h(2m + 1) = g(m) = yr

¡ KM257YB=g6<Y7Y462]NåEr ñ KR>E4625AN<fNaF[7Y^,b6Y7 |A ∪B| ≤ |N| r¥@XY<?>CKM25=XY257\b6KM257Y^>F[7Y\buY7A ⊆ A ∪B b6XY<?>E52 |A| ≤ |A ∪B| ru©3IWB=<?>E:;FON8=e\X`<<fNCPiIWY7Y462]Ng6IW:;FON8;7Y^>

|N| = |A| ≤ |A ∪B| ≤ |N|. F[IW:=Sa8=e\X¡ KM257YB=g6<Y7Y46257Q#N\4dF[IWB[N.-9q|7YB=46:;F[7Y254uNI\F[B=<?>E^Sa8;7Y^> |A ∪ B| = |N| b|XYIA\IW6XY<?>g6IK|VEgr

¥@gGK#IEPiSa8=e\XM:=25m3g6I254CF[S625X98;2EUS6:;FONfKM2]N\462]N@K0XY2]eCZaObWDcIKR>EY:=<?>og6ILK|VEgo^IWY4uNTd>IWD625:[N\o4uN\:;F[mYD6Sa8=e\XYI6r w625IWB;>

A2BS6:;FONLKM2]N\^>KæXY2]eCZW2Wb

A = a0, a1, a2, . . .b

ñdù!² 74X ª5±E¬õ > ¬YL®pó]ª_³³!ó µ ®oª µ ª5® > ¬Y®%ó]ª_³³!ó µ ®

B = b0, b1, b2, . . ./hIEg6DcIKM2]N\guN8=e<fNF[IyhS646AWX98;7

f2g1bwN4uN\:;F[mYD646257TUD6B=<Y7YD6]N.-

FON\^>d|8;7`<Y7T:=IWueð/iFON\Avg6<Y2]NCPhNyhS646A\X98=Nh1|g6IW:;FON8=e\X

A ∪B = a0, b0, a1, b1, a2, b2, . . ..k FU>E^æXY2]eCZWSvDc7?KM467`KR>EB[N\<?>vÎWZde:=25m@DcIKRFON\B=<fN\Ò/hcI<Y625IWB;>

A2B46257s^S6:=<fe

d>E B=IW<Phe\XY<?467*1baN\574uNRDJ7?KM46I8;7Y:;F¢K4625^ 46257Y:=A\IW6XY<Y7Y46257 KM257Y57B=VWY4d>EXO_KR>EB[N\<YVLKTbXYIAWIW6XY<?>g6ILK|VEgrÃÊÒ Áh¾ Å »CÓð é ABC"$#Ð%Xcd*U;eÍS*ÊT+W ,[Qd,XÃU; ¯R# P X¢'¸.& P [wS.&^l+WUQS*#%^lRV'OU;e P &;¢U; ¯R# P X¢',Ñ¸.& P [wS.&^l+WUQS*#%^lR#OÎãx¾E¿ ¼ N\SdK#N\Y^>Wb Y7ý4uNlÎEX?>¡ KM257YB=g6<Y7Y4625N-åEr ñ <Y625VWBT8;7Y:;FxXYI4uN89KR>EY798D6B=<Y7Y525XY<fN\54d>WbJZWgG>p25:;F[4625798;7:=S6Bh8;7YA\X98=Nx<Y7<Y625IWB=S525XY<Y4uNaF[S6B[N\54C>EXO_ý4uNoF[7Y4ý<Y625VWBYr¢IW46257?K#N\@K g6ILK|IEg6<Y257`¡ KM27YB=g6<Y7Y462]N ì r Ú46257@AWIWB=<?>E:;FON\525=^>o<@F[7YZWI6bGY7@yhS646A\X98;7

f2g:[eB=VWY46ILK#N\B;F[IW=XY25ILK#7\bG<fNaF[7Y^ KR>E:;FON\B=XY<?>xIWD6S6=XY25@IW:;FNFO462wN\AaN\D62F|K g6ILK#ICg6<Y257

F[7YZWIFUKM257YB=g6<Y7Y462]NGbud>xI\F[B=<?>EôN\`g6IK|VEgxKM4625IW:=ACSr

DJIW:;F[B=<Y7YY7Y46257TS6XY<?>E4625IW467TD6B=<Y7YgX_dKM25]eK g6ILK|IEg6<Y257@kl4625IW:=ACS ì r ñ JmYg6<Y257Y^>8;7Y:=<YXY<Y7ýKR>EA\IWB=<?>E:=FU>CKRN\ýKF9>E^ B=IW<Yg6<Y2]N\57\b|d><ýS6g6ILK|IEg64625IW4d>EXO_0F9KM257YB=g6<Y7YI<Y625IWB[N\XO_ýD6B=<Y7?525XY<fN\54C>EXO_KR>EXY2]edZdN\KM4625IW:=AC2Iv<Y625IWB[N\XO_ýXYIx4uN89KR>EY798`D6B=<Y7Y52XY<fN\¨-4d>EXO_rW|<?>CF[7Y54625AWILKM2CDJIW<YIW:;FONfKM2]N\^>TDcIWAaN\<fN\46257 25464d>EXO_K#7YB=:h8;2CDJILKR>EY:=<f7YZWIsKM4625IW:-ACSb 46Dr¢:=S6ôN<Y625IWB=SlD6B=<Y7Y525XY<fN\5467YZWI2 <Y625IWB=SlXYIp4uN89KR>EY798D6B[<?7Y525XY<fN\5467YZWI8;7Y:;FD6B=<Y7Y525XY<fN\54uNEr3g6ILK#ICg64625^>F[7YB[N\<A\IW5798;4C>ý<fN\:=AN\ACSa8=e\X?>y N\AdFfb!Y7DuN\B/hS6DcIWB=<fe\g6AWILK#N\4d>EX_B1

525XY<Y`4uNaF[S6B[N\54d>EXO_#8;7Y:;F FU>E57:[N\ÎMX?I3525XY<Y`4uNaF[S6B[N\54d>EXO_r FOe\gR8;S6|PhNaFUK|IRKR>CKM4625IW:-ACSa8;7Y^>WbuY7`525XY<YxKR>E^257YB=4d>EXO_F[7Y#8;7Y:;F3D6B=<Y7Y52XY<fN\546257sKM257Y57\ruN\4625^æD6B=<?>E:;FedD625^>g6IyhIWB=ôN\5467YZWIxg6IK|IEg6Sb <fN\D625:=<Y^>DuN\B;>ý525XY<YQ4uNaF[S6B[N\54d>EX_K /h46257Y:=A\IW6XY<YIW46798Q1FON\6525X?>o2!:=D6B=VW6Sa8;^>xS6:;FONfKM25#8;7@KXY2]eCZ6r

〈0, 0〉 → 〈0, 1〉〈0, 2〉 → 〈0, 3〉 . . .↓ ↑ ↓

〈1, 0〉 ← 〈1, 1〉〈1, 2〉〈1, 3〉 . . .↓ ↑ ↓〈2, 0〉 → 〈2, 1〉 → 〈2, 2〉〈2, 3〉 . . .

↓〈3, 0〉 ← 〈3, 1〉 ← 〈3, 2〉 ← 〈3, 3〉 . . .

↓rrr rrr rrr

13257#8;7Y:;F3F[B=S6g646Io<fN\SdKRN\?>E\bJY725gue\XTKF[7Y4p:=DcIW:=VW,UICgGKM257Yg6<Y25^>dÀN\YgG>,7Y57=-^7Y4dF FON\6525X?>`2dKg6ICguNaF[ACS<YB=IW625^>@F[IMFU>E5AWI3B[N\<\bXY<?>E52CDcIKR>EY:=<fNs^7?F[IEguNMIWD625:=Sa8;7Dc7?KM4ue62 8;7YA\X98;m`DJIW^25mYg6<?>o<Y625IWB[N\^2

N2N×N rc¥@XY<?>CKM25=XY257\bu46257 8;7Y:;FRF[Is8;7YgG>E4uN

Å ò½ñ 74X ª5±E¬õ > ¬Y®%ó]ª_³³!ó µ ® ñdù Å

ÎWY52K#N,62 8;7YA\X98=NGb¢Kjg6ICguNaF[ACSF[B=S6g646Iýd>uPiIWC>,8=ep<fN\D625:[N\oKM<YIWB=7Y^,r ã ]NaF[7YZWIpKDJIW4625Y:=<?>E^g6ILK#ICg6<Y257S6?>L8;7Y^>p62 8;7YA\X98;2 bcAdFOV\BONoICg6DJILKM2]N\guNS6:=FONfKM2]N\4625SpDJIvA\IW57Y265IWA\VK FU>EXO_pDuN\BYbGKAdF[VWB;>EX_KM:=DJVEPiB=<YmYg6467guNa8=eFOe:[N\ôe:=S6^m\b6F[<Y4r

N× N = 〈0, 0〉, 〈0, 1〉, 〈1, 0〉, 〈0, 2〉, 〈1, 1〉, 〈2, 0〉, 〈0, 3〉, . . .. ¿oÁh»C¹a¼ ºd» Ò Áh»ýð Õ |N× N| = |N| Îãx¾E¿ ¼ ¨MIW<?K#N\Y^>vyhS646A\X98;m

f : N× N→ N<fN\guN\4ueKM<YIWB=7Y^

f(x, y) =(x+ y)(x+ y + 1)

2+ x.

¢IWAaN\Y7Y^>Wb!Y7s8;7Y:;FIW4uNoKM<fN8;7Y^46257s8;7Yg646IW<Y4uN\XY<Y4uNGrkQ>EA\IWB=<?>E:;FON\^>KFU>E^XY7Y5SDJIWÎCXY4625XY<fe`yhS646A\X98;m

ϕ : N→ N/hXY<?>E52cXY2]edZ525XY<Y4uNaF[S6B[N\54d>EX_B1<fN\guN\4ue`KM<YIWB=7Y^

ϕ(n) = n(n+1)2

rEN\SdKRN\Y^>WbdY78;7Y:;F|IW4uN/h=XY25=57*1B=IW:=4ue\XfNT2uB=IW<Y6257YY4uNTg6I46257Y:=A\IWC-XY<YIW46IW=XY2 IWB[N\<`Y7

f(x, y) = ϕ(x+ y) + xr

kl7T÷Y^>F[7YB[N\<gGKM257B=VWY467DuN\B;>p525XY<Yý4uNaF[S6B[N\54d>EXO_ 〈x, y〉 2 〈x′, y′〉 r exgGKM257ÎWY52K|IW=XY2 rÖ rx+ y = x′ + y′

rGkQ>E4625AN:;FOe\gbGY7x 6= x′

/hcIZWgG>Ed>x = x′

bGF[Iy = y′

b6XYI8;7Y:;F346257YÎWY52~K|7\b6ZWgG>E 〈x, y〉 6= 〈x′, y′〉 1rus57@KRF[7YgG>

f(x, y) = ϕ(x+ y) + x 6= ϕ(x+ y) + x′ = ϕ(x′ + y′) + x′ = f(x′, y′).

¤Erx + y 6= x′ + y′

rq|7Y<0<Y^4625798;:=<Y7Y462]NIWZWVW546IW=XY2oÎWY7Y^>Æ<fNCPiIW>E\bY7x+ y < x′ + y′

b XY<?>E52x + y + 1 ≤ x′ + y′

r|©sI\B[<>E:=FON8=e\Xp<pF[7YZ\I6b Y7ϕ8;7Y:;F3B=IW:=4ue\XfNI\F[B=<?>E^Sa8;7Y^>

f(x, y) < ϕ(x+ y) + x+ y + 1 = ϕ(x+ y + 1) ≤ ϕ(x′ + y′) ≤ f(x′, y′).

|<?>E52!2wK F9>E^jKR>EDuN\g6ACSpôN\^>f(x, y) 6= f(x′, y′)

r¢IWAaN\<fN\525=^>v<fNaF[7Y^buY7`yhS646A\X98=N

f8;7Y:;F3B=VWY46ILK#N\B;F[IW=XY25ILKRNGr

13257YX_F[7YB[N\<m ∈ N JmYg6<Y257xg6ILK|IW54ue525XY<?ue4uNaF[S6B[N\54ueGrH¢IW46257?K#N\vyhS646A\X98=N

ϕ8;7Y:;F,B=IW:=4ue\XfN2`B=IW<Y627YY4uN-g6I46257Y:=AWIW6XY<YIW46IW=XY2 b#KM25mYXx8;7Y:;F

k ∈ N FON\AC257\bRY7ϕ(k) ≤ m < ϕ(k + 1)

rB=<?>L8;^2 8;^>x = m − ϕ(k)

2y = k − x

r¢IW46257?KRN\ϕ(k + 1)− ϕ(k) = k + 1

bGF[I0 ≤ x ≤ k

buXY<?>E52x, y ∈ N rJnpN\^>oKRF[7YgG>

f(x, y) = ϕ(x+ y) + x = ϕ(k) + (m− ϕ(k)) = m.

|<?>E52yhS646A\X98=Nf8;7Y:;FMU4uNLOb6XYIA\IW6XY<?>g6ILK#VCgr

ñdù!× 74X ª5±E¬õ > ¬YL®pó]ª_³³!ó µ ®oª µ ª5® > ¬Y®%ó]ª_³³!ó µ ®ã ILK#VCgDcIW4625Y:=<Y7YZWIpD6B=<?>EguNaF[467YZWIpKM4625IW:=ACSlDJIW<YIW:;FONfKM2]N\^>o8=N\A\IpD6B=IW:;F[7?KM2¨-

XY<Y7Y46257\rÃÊÒ Áh¾ Å »CÓð ï=^_ *UQST',RI)p#%[=&QS P #KÓn;),+c%Xcd*U;e k(;)O ®ÓUQS.&RV+W&ÍX¢+W&^l)mÒS*ÊT+W ,[Qd,X¼[wS.&^l+mÑUQS*#%^lRV'OU;e P &;ÈS*ÊT+W ,[Q&"Â[wS.&^l+WUQS*#%^lRV',"$Î

npN\^>l8;S6ýF[7YB[N\<KM:=<?>E:=F[AC257QDJIaFOB=<Y7?6467Q4uN\B=<YmYg6<Y2]N-g6I-DJIWAN\<fN\462]NGb#Y7Q25XY<YKR>E^257YB=4d>EXO_8;7Y:;FD6B=<Y7Y525XY<fN\546257,KM257Y57\b|XYI-ÎWY7ýD6B=<Y7YXY<?>EQ254dF[S625X98;2 b Y7o8;7Y:;F,25XO_Ug6S6YIaKDJIWB=VLKM4uN\4625Sg6Il525XY<Y4uNaF[S6B[N\54d>EXO_r#N\SdK#N\Y^>ý8;7Yg64uN\Acb Y7p254dF[S625X98=NFONx6257YB=<Y7:=25m<K@PhN\:=46IW=XY2 DcIWB=<fe\g6AWILKR>EXO_ 4p<Y625VWB`525XY<YýKR>E^257YB=4d>EX_x8;7Y:;FZWmY:;F9>K <Y625IWB=<Y7525XY<YQB=<Y7YXY<?>CKM25:;F9>EXO_5/hXY<?>E52 DcIW^25mYg6<?>g6ILK|IW54C>E^2B=VWY4C>E^2525XY<YuN\^2B=<Y7YXY<?>CKM25:;FU>E^2E<Y4uNa8;g6<Y257Y^>T525XY<YJmKR>E^257YB=4ueO1bLXYI 8;7Y:;F¢KIWDJIW<?>EX98;2Eg6IMUB=<fN\g6A\IW=XY2~<Y625IWB=S525XY<Y4uNaF[S6B[N\54d>EX_rDJIW4625Y:=<Y7YZWIF9KR257YB=g6<Y7Y462]NpKR>E4625AaN<fNaF[7Y^,bHY7v<Y625VWBUg6S6?>d`K8;7?g64C>E^:=7Y46:=257TÎWY7`d>EÙôNCP]>dsK25464d>E^,r ¿oÁh»C¹a¼ ºd» Ò Áh»ýð ð |Q| = ℵ0

Îãx¾E¿ ¼ N\XY<Y462 8;^>oICg<Yg67?êu4625ILK#N\462]NyhS646A\X98;2

f : N×N+ → q ∈ Q : q ≥ 0KM<YIWB=7Y^f(k, l) = k

l

rHI\46257?KRN\vAaNaYguNý525XY<YuN,KR>E^257YB=4uN8;7Y:;FSJPhN\ÂC257Y^,b!KM25mYXyhS646A\X98=N

f8;7Y:;FRU4uNOrcý¡ KM257YB=g6<Y7Y462]NåGr ñ ôN\^>o<fNaF[7Y^

|q ∈ Q : q ≥ 0| ≤ |N× N+|.ã N\5798?b g6<Y25mYAC2 ¡ KM257YB=g6<Y7Y4625SåGrØ¤Ë/ NO1`ôN\^> |N × N+| = |N × N| /hJIpKM257Y^>Wb Y7N ∼ N+ 1ru¢IW4uN\gGF[IIEXY<?>CKM25=XY257 N ⊆ q ∈ Q : q ≥ 0 r6©3IWB=<?>E:;FONa8=e\X`<@¡ KM257YB=g6<Y7=-462]N ì r Ý I\F[B=<?>E^Sa8;7Y^>

|N| = |N× N| = |N× N+| ≥ |q ∈ Q : q ≥ 0| ≥ |N|.¡ KM257YB=g6<Y7Y462]N#N\4dF[IWB[N.-9q|7YB=46:;F[7Y254uNKR>E4625AaN<fNaF[7Y^,bGY7T<Y62VWBR52XY<YvKR>E^257YB=4d>EX_46257YSa8;7Y^4d>EXO_8;7Y:;FD6B=<Y7Y525XY<fN\54C>\rMk XY7Y5S<fN\AWIW6XY<Y7Y462]Ng6IK|IEg6S<fN\SCK#N\Y^>WbsY7q ∈ Q : q ≥ 0 ∼ q ∈ Q : q ≤ 0 rc¢IW46257?KRN\

Q = q ∈ Q : q ≥ 0 ∪ q ∈ Q : q ≤ 0,KM25mYX`4uNÎEX?>v¡ KM257YB=g6<Y7Y462]N ì r Ú<Y625VWB#525XY<YxKR>E^257YB=4d>EXO_8;7Y:;F3D6B=<Y7Y525XY<fN\54d>Wr

DJILKR>EY:=<Y7fZWI-FUKM257YB=g6<Y7Y462]NlKR>E4625AaNICgB[N\<YS254dF[7YB=7Y:=Sa8=e\X?>KM4625I\:[7?AcbRAdF[VWB;>^VLKM2w4uN\^,bEY7`525XY<Yx46257?KR>E^257YB=4d>EXO_8;7Y:;FRKM25mYXY798|4625@525XY<YvKR>E^27YB=4C>EXO_b6XYIB=VLK2-46257Y@ÎWY7@46257@d>E`254dFOS62X?>L8;46257sIEXY<?>CKM25:;F[7\b\8=N\A\IY7`<@525XY<YuN\^2cKR>E^257YB=4d>E^2w:=DJI,-FU>EAaN\^>l:=25moXY<YmY=XY25798?r¢k guN\5:=<Y798XY<YmY=XY2#F[7YZWIB=IW<Yg6<Y2]NCPiSDcIWAaN\Y7Y^>8;7Y:=<YXY<Y72546467KR>E4625AC2!D6B=<Y7YXY<fe\XY7TF[7YZWIFU>ED6S,254CF[S625X98;I\^,r

Å ò½ñ 74X ª5±E¬õ > ¬Y®%ó]ª_³³!ó µ ® ñdù ö

ÃÆÒ Áh¾ Å »CÓð õÂäÈÊT+Wd,[E^l+WUQS*ÊÍRV+W&X¢',"Ð+W&[=RV'OU;e P &;ZRV+W&[wS.&^l+WUQS*#%^lRV'pÎãx¾E¿ ¼ NCPiVWY^>p46257KMD6B=IW:;Ffb!Y7<Y625VWB@525XY<Yý46257?KR>E^257YB=4d>EX_

R \ Q 4625738;7Y:;F46257YD6B=<Y7Y525XY<fN\54d>Wb XY<?>E52!8;7Y:;FXYI4uN89KR>EY798D6B=<Y7Y525XY<fN\54d>Wr¢HIW46257?K#N\KM257Y^>x8;S6\b¢Y7<Y625VWB@525XY<YKR>E^257YB=4d>EX_

Q8;7Y:;FTD6B=<Y7Y525XY<fN\54d>Wbw<fNaF[7Y^<YZWIEg646257<kQ4625IW:=AC257Y^ ì r ñ

<Y625VWB(R \Q) ∪Q = R

8;7Y:;FoD6B=<Y7Y52XY<fN\54C>Wr¡ Ix8;7Yg64uN\A462578;7Y:;FÎWY52~K#7ù/hDuNaF[B=<D6B=<?>EA6PhN\gp4uN:;F[B=IW46257 Ö ¤ ø 1ru3<?>E:=AaN\4uN:=D6B=<Y7YXY<Y46IW=A\IW6XY<?>g6IK|VEgr

¨MIW<?KRN\Y^>F[7YB[N\<4uN\:;F[mYD6Sa8=e\XY7ogGK#N25:;F[I\F[467D6B=<?>EA6PhN\gG>Wb¢AdF[VWB=7DJIWAN\fe,4uN\^,b8=N\AvÎ\Y4uNS6<fN\:[N\g6462]N\TD6B=<Y7Y525XY<fN\546IW=`Dc7?KM4C>EXO_<Y625IWB=VLKTr¸¹ºWá!ÓÈÇ Àu¼!áÖ rsw625VWB S KM:=<?>E:;F[AC25X_D6B=<Y7?g6<Y2]NCPiVK5/hI\F9K#N\B;FU>EXO_B1 IsIW6SA\IW6XfN\XO_KR>E^27YB=4C>EXO_8;7Y:;F3D6B=<Y7Y525XY<fN\54d>Wr¨MIW<YDuNaF[B=<Y^>-yhS646A\X98;m

f : S → Q2 <fN\guNa4uelKM<?IWB=7Y^ f((a, b)) = 〈a, b〉 rzWN\APhNaFUK|I<fN\SCK#N\?>Ev8;7Y:;FxF[I254a8;7YA\X98=NGb#XY<?>E52s4uNÎCX?>¡ KM257YB=g6<Y7Y462]NåGr ñôN\^> |S| ≤ |Q2| rJk257Y^>F[7Yg6<Y25mYAC2H¡ KM257YB=g6<Y7Y4625S ì r ì 2 kl4625IW:=AWILKM2 ì r åGbY7 |Q2| = ℵ0

rxg6B=S6ZW25798:;F[B=IW4d>IWACB=7Y=5^>yhS646A\X98;m

g : N→ S KM<YIWB=7Y^ g(n) = (n, n+ 1)r

3KÆF9>E^KR>EDuN\g6ACSPhNaFUK#I,DcIWAaN\<fN\\b Y7yhS646AWX98=Ng8;7Y:;F254a8;7YAWX98=eGb!XY<?>E52FON\A

8=N\ADJIWD6B=<Y7Yg64625IôN\^> |S| ≥ |N| = ℵ0r#¡¢7?BONa<ýKR>E:;FON\B=XY<?>0<fN\:;F[IW:=ILKRN\

¡ KM257YB=g6<Y7Y46257#N\4dF[IWB[N.-9q|7YB=46:;F[7Y254uNGb6C>xI\F[B=<?>EôN\ |S| = ℵ0r

¤Er ã ILK|IW54C><Y625VWBDuN\B[N\^2GB=IW<Phe\XY<Y4d>EXO_D6B=<Y7Yg6<Y2]NCPiVLKÁ/hI\F9K#N\B;FU>EXO_B1J8;7Y:;F XYI`4uNa8-KR>EY798MD6B=<Y7Y525XY<fN\54d>Wr13257YXO_ T cmYg6<Y257 <Y625IWB=7Y^ DuN\B[N\^2WB=IW<Phe\XY<Y4d>EXO_D6B=<Y7Yg6<Y2]NCPiVLKI\FUKRN\B;FU>EXO_r ã ]NAaNaYg67YZWIsD6B=<Y7Yg6<Y25NCPiS

I ∈ T KR>E6257YB=<Y^>`DJ7?KM4ueMA\IW46ACB=7?F[4ue3525XY<YcmKR>E^257YB=4ue<#F[7YZWITD6B=<Y7Yg6<Y2]NCPiSð/iKM27Y^>Wb\Y7RFON\A\ILK#N`25:;F[4625798;7\bdN@4uNfK#7?FfbWY78;7Y:;F 25XO_uN\B=g6<YIg6S6YI6b|N\57,^>KR>E6257YB[N\^>-F9>E5A\I8;7Yg64ueO1234uN\<?KM2 8;^>ý8=e

qIr#wg67?êu4625Sa8;^>

F[7YB[N\<,yhS646A\X98;mh : T → Q

KM<YIWB=7Y^h(I) = qI

r#N\SCK#N\Y^>Wb Y78;7Y:;FoF[IyhS646A\X98=NB=VWY46IK#N\B;F[IW=XY25ILK#NGr|U:;F[I\F[46257\b!8;7Y=52sôN\^>

I1, I2 ∈ T , I1 6= I2b

F[I<<fNCPiIWY7Y462]NlKM257Y^>Wb Y7I1 ∩ I2 = ∅ r#HIW4627?KRN\ h(I1) = qI

1∈ I1

2h(I2) = qI

2∈ I2

bJF[Ix^S6:=2d>Eh(I1) 6= h(I2)

r!NaF[7Y^<¡ KM257YB=g6<Y7Y462]NxåGr ñKR>E4625AN |T | ≤ |Q| = ℵ0

b6XYIA\IW6XY<?>g6ILK#VCgr

¥@AaN\<YSa8;7o:=25mab Y7o¡ KM257YB=g6<Y7Y46257 ì r Ú,ÎWY4uNS6IWZWVW546254uN,:=S6^mKM25mYAC:=<Y798255IW=XY2<Y625IWB=VLKTr ¿oÁh»C¹a¼ ºd» Ò Áh»ýð 5â AVC"$#fU; IR# P X¢'¸.& P [wS.&^l+WUQS*#%^lRV+W&X¢+W&^l S*ÊT+W ,[Qd,X [wS.&^l+WUQS*#%^lÑRV'OU;e P &;@[wS.&^l+WUQS*#%^lR#OÎ

ñ/5Cø 74X ª5±E¬õ > ¬YL®pó]ª_³³!ó µ ®oª µ ª5® > ¬Y®%ó]ª_³³!ó µ ®ãx¾E¿ ¼ ¨MIW<?K#N\Y^>B=IEg6<Y2546mx<Y62IWB=VK An : n ∈ N FON\AC25X_bHY7 |An| = ℵ0g6]NoAaN\Yg6798@525XY<Yd>4uNaF[S6B[N\546798

nrwk <?KM2]e\<YACS<FU>E^ g6]NvAN\Yg67YZWI

n ∈ N ôN\^>yhS646A\X98;mfn : N −→1−1

na Anrc13257YXO_

A =⋃

n∈N

An.

N\SdK#N\Y^>WbY7KÊF[798@:=S6^257s8;7Y:;FTXYI,4uN89KR>EY798`D6B=<Y7Y525XY<fN\546257KM27Y57<Y625IWB=VLKTr!13257ÎWY4uN:;FUKM257YB=g6<Y25\bcY7M8;7Y:;Fs25XO_pg6IWA6PhN\g646257D6B=<Y7Y525XY<fNa546257`KM257Y57\buJIÎWY7:=25m<YguN.-B=<?>E\bCY73K0B=ICg6<Y2546257 An : n ∈ N 8;7Y:;F FU>E5AWI:=A\IW6XY<Y7Y462573KM257Y57MB=VWY4d>EX_o<Y625IWB=VKTbZWgG>E`F[7Y4,:[N\^%<Y625VWBRÎWY7`:=25m`DJILKRFON\B=<fN\`KM257Y5IWACB=I\F[46257@<TB=VWY4d>E^2!2546g67YAC:[N\^2 rã 7?êu4625Sa8;7Y^>pF[7YB[N\<yhS646A\X98;m

f : N × N → AKM<YIWB=7Y^

f(n, k) = fn(k)/Ø8;7Y=52

D6B=<?>L8;^257Y^>WbfY7yhS646A\X98=Nfn4ES6^7YB=Sa8;7 525XY<YuN\^2\4uNaF[S6B[N\54d>E^2W<Y625VWB

AnbfF[I

f(n, k)8;7Y:;Fk- FU>E^ 7Y57Y^7Y4dF[7Y^ <Y625IWB=S

An1rG¢IWAaN\Y7Y^>WbEY7

f8;7Y:;F#:=S6Bh8;7YA\X98=eGrEk FU>E^æXY7?5S

K|7T÷Y^>g6IK|IW5467x ∈ A ruk0F[7YgG>25:;F[4625798;7`525XY<YuN4uNaF[S6B[N\54uN n0

buFON\ANY7x ∈ An

0

rs57RyhS646AWX98=N

fn0

8;7Y:;F U4uNObd<fNaF[7Y^8;7Y:;Fk0 ∈ N

bdFON\AC2573Y7x = fn

0(k0) = f(n0, k0)

bXYI4uN\57YfNCPiIDJIWAN\<fN\\r©3IWB=<?>E:;FONa8=e\XT<`¡ KM257YB=g6<Y7YåGr ñ 2 ì r Ý IWB[N\<@F[7YZWI6buY7

A0 ⊆ AI\F[B=<?>E^Sa8;7Y^>

|N| = |A0| ≤ |A| ≤ |N× N| = |N|2S6?>EXY257¡!KM257YB=g6<Y7Y462]No#N\4dF[IWB[N.-9q|7YB=46:;F[7Y254uNAWIW6XY<?>g6ILK|VEgr

s4uN\5IWZW25XY<Y46257 8=N\AKD6B=<?>EDuN\g6ACSo¡ KM257YB=g6<Y7?462]N ì r ÚGbWÎWY7Y^>KR>E:=4ES634uN\:;F[mYD6SC-8=e\XY7@KM4625IW:=AC2 buAdF[VWB;>EXO_,S6<fN\:[N\g646257Y46257TDJIW<YIW:;FONfKM2]N\^>8=N\A\ID6B=IW:;F[7T?KM25XY<Y7Y46257\rÃÊÒ Áh¾ Å »CÓð ÍCÂ ABC"$#[wS.&^l+WUQS*#%^lRV+W&X¢+W&^lùS*ÊT+W ,[Qd,X [wS.&^l+WUQS*#%^lRV'OU;e P &;Z[wS.&^l+mÑUQS*#%^lR#OÎ

ÃÊÒ Áh¾ Å »CÓð ÍcÍ ABC"$# U; ¾RB# P X¢'¸.& P [wS.&^l+WUQS*#%^lRV+W&ùX¢+W&^l9S*ÊT+W ,[Qd,X U; ¾R# P X¢'¸.& P[wS.&^l+WUQS*#%^lRV'OU;e P &;U; ÒR# P X¢'¸.& P [wS.&^l+WUQS*#%^lR#OÎ

ÃÊÒ Áh¾ Å »CÓð ÍEÞ ABC"$#Í[wS.&^l+WUQS*#%^lRV+W&ÍX¢+W&^lYS*ÊT+W ,[Qd,XåU; R# P X¢'¸.& P [wS.&^l+WUQS*#%^lRV'OU;eP &;@[wS.&^ +WUQS*#%^lR#OÎ

zWN\A\IoD6B=IW:;F[7<fN\:;F[IW:=ILK#N\46257DcIKR>EY:=<?>EX_pF9KM257YB=g6<Y7YpB=IW<YDuNaF[B=<Y^>x4uN\:;F[mYD6Sa8=e\X?>D6B=<?>EA6PhN\gr¸¹ºWá!ÓÈÇ Àu¼jw625VWB W KM:=<?>E:;F[AE25XO_KM257Y5IW^25N\46VLKIsKM:=DcVEPiXY<?>E4u4625AN\X_XfNCPiA\ILKM2F9>EXO_8;7Y:;F3D6B=<Y7Y525XY<fN\54d>Wr13257YXO_ Wn

IW<Y4uN\XY<fN-<Y625VWBoKM:=<?>E:=F[AC25XO_ KM257Y5IW^25N\46VLK:;F[IWD6462]NnIKM:=DJVEPiXY<?>E4C-

4625AN\X_XfNCPiA\ILKM2FU>EX_rHIW46257?K#N\ AaN\YgG>`KM257Y5IW^25N\4anx

n+an−1xn−1+. . .+a1x+a0

Å òù¶74X ª5±E¬õ°±V³*õI³± µn: ª µV?? ° ñ/5VñÎWY4uNsSGF[IWY:[N\^25 <#XY2]edZW257Y^ 525XY<YXfNCPiA\ILKM2FU>EX_ 〈an, an−1, . . . , a1, a0〉

ba<fNaF[7Y^J7Y<F[B=S6g6S,ÎWY7Y^>vS6g6ILK#ICg64625\buY7 |Wn| = ℵ0

r6s57@KM257Y^>\buY7

W =⋃

n∈N

Wn,

KM25mYXT:;F[IW:=Sa8=e\X`kl4625IW:=7YA ì r ÖLø g6IW:;FON8;7Y^> |W| = ℵ0

rB=<?>EDJIW^462 8;^>WbcY7525XY<YuNB=<Y7YXY<?>CKM25:;FON

a4uN\<?>CK#Nv:=25m#%^ `!&ÊT[Q#%+WUQSTRBaWba8;7Y=52¢25:;Fz-

4625798;7 KM257Y5IW^25N\4/h:;F[IWD6462]N3XYIs4uNa8;^4625798!D6257YB;KM:=<Y7YZWI!1 I3KM:=DJVEPiXY<?>E464625AaN\XO_XfNCPiA\IKM2¨-FU>EX_

W (x)bdFON\AC2cY7

W (a) = 0rE!25XY<YJm3B=<Y7YXY<?>CKM25:;FOeGbEAdF[VWB[N462578;7Y:;FRN\5ZW7Y6B[N\25XY<Y4uN

4uN\<?>CK#N\^>ù[wS.&;Ra\r¢!25XY<YuN\^2 N\5ZW7Y6B[N\25XY<Y4d>E^2 :[ep<fNaF[7Y^µ525XY<Yd>KR>E^257YB=467\bNFON\ACY7TD6257YB;KM2]N\:;F[AC2N\B;>CF[^7?FU>EXY<Y467Tg6ILK#IW5467YZWI:;F[IWD6462]N<@525XY<YvKR>E^257YB=4d>EX_r6U4C-4d>E^2:PiILKR>Wb6KM25mYAC:=<YI\[T525XY<YbG<ÀdF[VWB;>E^2w:=25m@4uNXYIg6<Y257Yx:=DJI\FU>EAaN\^>vF[IK@PhN\=46257525XY<Yd>xN\5ZW7Y6B[N\25XY<Y467\rc¥@AaN\<YSa8;7T:=25m|8;7Yg64uN\AcbcY7T25XO_xF[7Y#8;7Y:;F346257?KM257Y57\r¸¹ºWá!ÓÈÇ Àu¼w625VWB#525XY<YN\5ZW7Y6B[N\25XY<Y4C>EXO_

A8;7Y:;F3D6B=<Y7Y525XY<fN\54d>Wr

N\SdKRN\Y^>Wb6Y7

A = a ∈ R : (∃W ∈ W)W (a) = 0 =⋃

W∈Wa ∈ R : W (a) = 0,

ZWg6<Y257 W 8;7Y:;F<Y625IWB=7Y^.KM:=<?>E:;F[AC25X_KM257Y5IW^2]Na46VKÆIpKM:=DJVEPiXY<?>E464625AaN\XO_-XfNCPiA\IKM2¨-FU>EX_rk DJIWD6B=<Y7Yg64625^ D6B=<?>EA6PhN\g6<Y257DJIWAaN\<fN\525=^>WbwY7<Y625V\B W 8;7Y:;FTD6B=<Y7Y525XY<fN\54d>WbKM257Y^>lDcIW4uN\gGF[I6bHY7xAN\YgG>KR257Y5IW^2]Na4lôNp:=AWIW6XY<Y7Y46257vKM257Y57o^25798;:=Xo<Y7YB=IKR>EXO_rNaF[7Y^j4uNÎEX?>okl4625IW:=ACS ì r Ö ¤g6IW:;FON8;7Y^>WbuY7 |A| = ℵ0

r¨MIW<YS6^Sa8=e\X#DcIEg6IW646257\b?8=N\ATK#7#kl4625IW:=ACS ì r ×@ÎWY7Y^>DcIWAaN\<fN\\bWY7M<Y625VWB525XY<Y

D6B=<Y7Y:;F[mYD64d>GXO_x8;7Y:;F`46257YD6B=<Y7?525XY<fN\54C>Wrwz\7Y:;F`25XO_<fNaF[7Y^KM25mYXY798s4625525XY<YýN\5ZW7Y6B[N\25XY<=-4d>EX_r¡ 7Y4ýy N\AdF`ÎWY7F[7YD6B=<Y7?XY<?>G254dF[S625X98;2 buJIvK<fN\:[N\g6<Y25738;7YgG>E4d>E^2525XY<YuN\^2D6B=<Y7Y:;F[mYD64d>G^2 XY<Ym?[X?25798M:=DJI\FU>EAaN\4d>E^2 :[e

e2πr

Ù < ¢'ÝTÆâ ÝÊZâ cÝÐÜ 'Y$ÛîÛ\ã IF[798|DcIWB;>vDcIW<Y4uN\525=^>AC255AaND6B=<?>EA6PhNag6VK <Y625IWB=VLK 46257YD6B=<Y7Y525XY<fN\54d>EXO_ruN8-

^257Y^>ý:=25mF[7YB[N\<DJ7?KM4uepAC]N\:[e,<?625IWB=VKÆ46257YD6B=<Y7Y525XY<fN\54d>EX_b N,^2]N\46ILKM25XY27<Y625I,-B[N\^26B=VKM46IW525XY<Y4d>E^26<Y73<Y625IWB=7Y^Æ525XY<YB=<Y7YXY<?>CKM25:;FU>EX_

RrG¥FON\AC25XO_<Y625IWB[N\XO_^V,-

KM25^>Wb\Y7R:[e`ÎCX?>XYIW4dF[254CS6S6^ /Ø8;7Y=52A8;7Y:;FFON\AC25^<Y625IWB=7Y^,bF[I`D625:=<Y7Y^> |A| = c

1r1sNa8;D6257YB;K-8;7Yg64uN\AvS6g6ILK#ICg64625^>v57YôNaFfr

ü »E½lÀ Ç ð ÍEÌ ^#! ,Xc ,^lR&i`! ÍS*ÊT+W ,[A"$#%"Ð' P(A) ∼ 0, 1A Î

ñ/5Gù 74X ª5±E¬õ > ¬YL®pó]ª_³³!ó µ ®oª µ ª5® > ¬Y®%ó]ª_³³!ó µ ®ãx¾E¿ ¼ wg67?êu4625Sa8;^>-yhS646A\X98;m

F : P(A) → 0, 1A KM<YIWB=7Y^ F (B) = χB/hD6B=<?>EDcIW^462 8;^>WbWY7χBF[IyhS646A\X98=NXO_uN\B[N\AdF[7YB;>E:;F9>EXY<Y4cN<Y625IWB=S

B1r D6B[NfKMg6<Y7Y46257\b

Y7F8;7Y:;FH62 8;7YA\X98=e8;7Y:;FHD6B=IW:;F9>E^ ?KM25XY<Y7Y46257Y^bAdF[VWB=7 DJIW<YIW:;FONfKM2]N\^>T|<?>CF[7Y54625A\IKM2 r

yhIWB=^SJPiSa8;7Y^>F[7YB[N\<TZEPiVLKM467`FUKM257YB=g6<Y7Y46257`F[7YZWIDJIEg6B=IW<Yg6<Y2]NCPiSr ¿oÁh»C¹a¼ ºd» Ò Áh»ýð Í é |R| = |0, 1N| = |NN| = |P(N)| Îãx¾E¿ ¼ ã ILK|VEg,D6B=<Y7YD6B=ILK#N\g6<Y25^>vKD6S646AdFON\XO_rÖ rs!7YôNaF[S ì r Ö ÚK#>E4625AaNGbuY7 |P(N)| = |0, 1N| r¤Er3¢IW46257?K#N\ÀaN\YgG>xXY2]eCZ<Y7YBM2E8;7YgG>E467YA8;7Y:;F3XY2]edZW257Y^æ525XY<Yx4uNaF[S6B[N\54d>EXO_bGKM25mYX0, 1N ⊆ NN

r¢HIW4uN\gGF[I8;7Y=52f ∈ NN

b F[Ip<g67?êu4625X98;2 yhS646A\X98;2 KR>E4625AaNGb¢Y7f ⊆ N×N rW|<?>E52 NN ⊆ P(N×N)

rW¡ KM257YB=g6<Y7YåErØ¤Ë/hX*1 2 ì r Ý KM4625IW:=ACSa8;7Y^>WbY7 P(N× N) ∼ P(N)

ru <fNaF[7Y^%ôN\^>

|0, 1N| ≤ |NN| ≤ |P(N× N)| = |P(N)|.ã <Y25mYAC2uDJIEg6D6S646AdF[ILKM2cD6257YB;KM:=<Y7Y^So2u¡ KM257YB=g6<Y7Y4625Sv#N\4CF[IWB[N.-9q|7YB=46:;F[7Y254uN`g6I,-:;FON8;7Y^> |0, 1N| = |NN| = |P(N)| r

ÚGr3¢IWAN\Y7Y^>F[7YB[N\<\b Y7 |R| ≤ |P(N)| r HICg6IW6462578=N\AD6B=<Y7YgXO_dKM25]eGbg6<Y25mYAC2KR>EA\IWB=<?>E:;FON\4625S ¡ KM257YB=g6<Y7YåErØ¤Ë/hX*1o2 ì r ì KR>E:;FON\B=XY<?>4uN\^S6g6IK|IEg64625\b#Y7|R| ≤ |P(Q)| rJk FU>E^XY7Y5Sý<Yg67?êu4625Sa8;^>yhS646AWX98;m

f : R → P(Q)KM<YIWB=7Y^

f(x) = q ∈ Q : q < x rJN\SCK#N\Y^>WbcY7 f 8;7Y:;FsB=VWY46ILKRN\B;F[IW=XY25ILK#NGrcU:;F[I\Fz-46257\b46257YXO_x, y ∈ R, x 6= y

r q|7Y<<Y^4625798;:=<Y7Y462]NxIWZWVW546IW=XY2HÎWZWm<fNCPiIW?>E\bY7

x < yr#¢IW46257?K#N\DJIW^25mYg6<?>g6IK|IW54d>E^23gGKM257YôNlB=VWY4d>E^2s525XY<YuN\^2

B=<Y7YXY<?>CKM25:;FU>E^2M<fNfKM:=<Y7,ÎWY7Y^><Y4uN\57T÷Y525XY<YJmvKR>E^257YB=4ueGbHKM25mYXx25:;F[4625798;7q0 ∈ Q

b!FNaAC257oY7x < q0 < y

r s57KRF[7YgG>q0 6∈ f(x)

2q0 ∈ f(y)

b¢XY<?>E52f(x) 6= f(y)

rJNa:=F[IW:=ILK#N\46257`¡ KM257YB=g6<Y7Y462]NåGr ñ guN8;7Tfe\guN\4ue46257YB=VKM46IW=\rñ r ã I`<fN\AWIW6XY<Y7Y4625NTg6ILK|IEg6S6B[N\ACSa8;7R4uN\^8;7Y:=<YXY<Y78;7Yg64679846257YB=VLKM46IW=XY2 bdN@^2]N.-46ILKM25XY257 |0, 1N| ≤ |R| rq >lF[I<YB=IW625\bHAN\Yg67Y^SXY2]eCZWILKM2 <Y7YB2 8;7YgG>E467YAD6B=<?>ED625:=<Y7Y^>525XY<YcmvB=<Y7YXY<?>CKM25:;FOeGbAdF[VWB=798A\IW5798;467vX?>CyhB;>B=IW<?KM254625mYXY2]N,g6<Y257=-:=25m?F[467YZWI-<YZdN\g6<fN8=e:[2m<A\IW5798;4d>E^2sKR>EB[N\<fN\^2`4uN\:=<Y7YZWIXY2]edZWSrRq#N\B=g6<Y25798yhIWB=ôN\4627\bGg67?êu4625Sa8;7Y^>oyhS646AWX98;m

f : 0, 1N → RKM<YIWB=7Y^

f(〈an : n ∈ N〉) = 0, a0a1a2a3 . . . .

z\7Y:;F@F[Ixg6IW6B=<Y7IWACB=7Y=5IW4uNoyhS646A\X98=NxB=VWY46ILK#N\B;F[IW=XY25ILKRNGbJJIxB=VWY467XY2]eCZW2¢<fN.-guN8=eB=VWY467B=IW<?KM254625mYXY2]Ng6<Y257Y:=25m?F[467\bJNF[7<A\IW57Y2!KR>E<Y4uN\XY<fNa8=eoB=VWY467525XY<Yd>

Å òù¶74X ª5±E¬õ°±V³*õI³± µn: ª µV?? ° ñ/5y5B=<Y7YXY<?>CKM25:;F[7¯/hgGK#N3B=VWY467|B=IW<?KM254625mYXY2]NRg6<Y257Y:=25m?F[467 ÎWZdeRKMD6B[NfKMg6<Y257|<fN\guNfK#N\FOe:[N\ôe525XY<YJm@B=<Y7YXY<?>CKM25:;FOeGbcN\573F9>E5A\IZWgG>oK-8;7Yg64C>E^j<`4625X_xIEgxDJ7?KM467YZWI^25798;:=XfN,KR>E:;F[mYD6Sa8;7,X?>CyhB[N ó bHXYIýKæ4uN\:=<?>E^ KR>EDuN\g6ACSIEXY<?>CKM25=XY257x46257vôN^25798;:=XfNO1r6©sIWB=<?>E:;FON8=e\XT<T¡ KM257YB=g6<Y7Y462]NåEr ñ I\F[B=<?>E^Sa8;7Y^>vfe\guN\4C>vKR>E4625Acr

N\:;F[IW:=ILKRN\46257¡!KM257YB=g6<Y7Y462]N#N\4dF[IWB[N.-9q|7YB=46:;F[7Y254uN-g6I-KR>E4625AWVLK.<QDJICg6D6S646AdF[VKF[B=<Y7YXY257YZWI2MXY<?KRN\B;F[7YZWI /hD6B=<?>-KR>EAWIWB=<?>E:;FON\4625SKR>E4625ACS0<pDcIEg6D6S646AdF[S0g6B=S6ZW257YZWI!1A\IW6XY<?>g6IK|VEgr

N\:=AaN\ACSa8=e\X?>E^ KM4625IW:=AC257Y^ <F[7YZWI-FUKM257YB=g6<Y7Y462]Ný8;7Y:;F,y N\AdFfbMY7QD6S646AdF[VLKµ4uND6B=IW:;F[798H8;7Y:;FRF9>E57`:[N\Î6bGXYID6S646AdF[VLK4uNDJPhN\:=<YXY<?>C÷Y4u257/hIW6:=7YB;KRN\X98=NFNKR>CK#IEPhNCPhN:=DJIWB=7ýA\IW4CF[B=ILK|7YB=:h8;7ýKM=B=VEg ôNFO7?ôNaF9>EA\VLK DcIEgA\IW46257YX£@ £.KM257YACSB1r ã I8;7YZWIg6ILK#ICg6S,JmYg6<Y257Y^>xDJI\F[B=<Y7YcIK#N\gGK#N57YôNaFU>Wrü »E½lÀ Ç ð Í Õ ^#! ,Xc ,^lRV'OU;eS*ÊT+W ,[Qd,X

A,B+C¿ P &;\^l+

B ∼ C¿ï

AB ∼ AC .

ãx¾E¿ ¼ 13257YX_ϕ : B −→1−1

na CJmYg6<Y2573yhS646AWX98=eS6:;FON\]Na8=e\XfeB=VLKM46IW525XY<Y46IW=\r6wg67?ên-

4625Sa8;7Y^>@yhS646AWX98;mF : AB → AC

KM<YIWB=7Y^F (f) = f ϕ−1 r ã ]NMg6IK|IW54d>EXO_yhS646A\X98;2

f1, f2 : B → AFON\AC25XO_bcY7

f1 6= f28;7Y:;F

x ∈ B bcg6]NAdF[VWB=7YZWI f1(x) 6= f2(x)rcs57

ôN\^>ϕ(x) = y ∈ C 2guN\5798

f1 ϕ−1(y) = f1(x) 6= f2(x) = f2 ϕ−1(y),

XY<?>E52F (f1) 6= F (f2)

rJNaF[7Y^æyhS646A\X98=NF8;7Y:;F3B=VWY46ILK#N\B;F[IW=XY25ILKRNGr

z\7Y=52HF[7YB[N\<ôN\^>pg6ILK#IW5467g : C → A

bwF[Ig67?êu4625Sa8=e\Xf : B → A

KM<YIWB=7Y^f(x) = g ϕ g6IW:;FONa8;7Y^> F (f) = (g ϕ) ϕ−1 = g

rJ|<?>E52yhS646A\X98=NF8;7Y:;FMU4uNOb

XYIA\IW6XY<?>g6ILK#VCgr

ü »E½lÀ Ç ð Í ï ^#! ,Xc ,^lRV'OU;eS*ÊT+W ,[Qd,XA,B

+C¿ P &;\^l+

B ∩ C = ∅ ¿Z

AB × AC ∼ AB∪C .

ãx¾E¿ ¼ ¨MIW<YDuNaF[B=<Y^>oyhS646AWX98;mF : AB∪C → AB × AC <fN\guN\4ueKM<YIWB=7Y^

F (f) = 〈f B, f C〉.

h S646A\X98=NFON,8;7Y:;Fg6IW6B=<Y7ý<Yg67?êu4625ILK#N\4uNGb#JIg6IW^ (f) = B ∪ C b KM25mYXýÎ\Y7Y^>IWACB=7Y=525TIEg6DcIKM257Yg64627IWcXY25mYXY2]NGrc¢IWAaN\Y7Y^>WbcY7TyhS646A\X98=NF8;7Y:;Fs62 8;7YAWX98=eGr6kl7T÷Y^>

K%FU>E^tXY7Y5SgGKM257xyhS646AWX98;7f1, f2 : B ∪ C → A

bFON\AC257Y7f1 6= f2

r|¥@<Y4uN\XY<fNF[I6b¢Y78;7Y:;F

x ∈ B ∪ C FON\AC2 b¢Y7f1(x) 6= f2(x)

rz\7Y=52 F[7YB[N\<x ∈ B

b¢F[IôN\^>f1 B 6= f2 B

b!N8;7Y=52x ∈ C bwF[IvKRF[7YgG> f1 C 6= f2 C

rwk IW6SýKR>EDuN\g6AaN\XO_ôN\^>

F (f1) 6= F (f2)buXY<>E52yhS646AWX98=N

F8;7Y:;F3254a8;7YA\X98=eGr

ñ/5!6 74X ª5±E¬õ > ¬YL®pó]ª_³³!ó µ ®oª µ ª5® > ¬Y®%ó]ª_³³!ó µ ®3:;FON\5^>F[7YB[N\<g6IK|IW5467 〈g, h〉 ∈ AB × AC

r|npN\^><fNaF[7Y^g : B → A

2h : C → A

rJwg67?êu462Sa8;^>oF[7YB[N\<f : B ∪ C → A

KM<YIWB=7Y^

f(x) =

g(x)

8;7Y=52x ∈ B

h(x)8;7Y=52

x ∈ C .

h N\AdFfb!Y7`8;7Y:;FTFOIg6IW6B=<Y7<Yg67?êu4625IK#N\4uNxyhS646AWX98=NKR>E462AaN,<B=IW<Phe\XY<Y46IW=XY2 <Y625IWB=VLKB2Cr ¢IW4uN\gGF[I,D6B=IW:;FU>E^.B[N\X_CS646AC257Y^.ÎWY7Y^>ý:=D6B[NfKMg6<Y25\b Y7

F (f) = 〈g, h〉 r|<?>E52yhS646A\X98=NF8;7Y:;F3:=S6Bh8;7YA\X98=eGbcXYIA\IW6XY<?>g6ILK#VCgr

¿oÁh»C¹a¼ ºd» Ò Áh»ýð Í ð |R× R| = |R| Îãx¾E¿ ¼ ¥@<Y4uN\XY<Y^>0D6B=<Y7Y<

P<Y625VWBv525XY<Y04uNaF[S6B[N\54d>EXO_DuN\B=<?>E:;F9>EXO_rRk257Y^>

8;S6\bY7 |P | = |N \ P | = |N| r !7YôNaF[S ì r ÖÝ 2#¡ KM257YB=g6<Y7Y462]N ì r Ö ñ ôN\^>l<fN.-F[7Y^ |0, 1P | = |0, 1N\P | = |0, 1N| = |R| rW¡¢7YB[N\<Mg6<Y25mYAC2u¡ KM257YB=g6<Y7Y4625SoåGrØ¤Ë/ NO12!!7YôNaF[IKR2 ì r Ö åg6IW:;FON8;7Y^>

R× R ∼ 0, 1P × 0, 1N\P ∼ 0, 1P∪(N\P ) = 0, 1N ∼ R,XYI4uN\57YfNCPiIg6ILKM257Y=\r

¢IEg6IW64625738=N\AKKR>EDuN\g6ACSýkl4625IW:=ACS ì r åGbJFON\ACY72¢F[7Y4ýDJIW4625Y:=<?>ôNo4uNaFU>EXO_C-^2]N\:;F[ILKR>vg6ILK|VEgrÃÊÒ Áh¾ Å »CÓð Í õ=^_ *UQST',Rð)p#%[=&QS P #KÓn;),+Z%Xcd*U;e¯k(;)O ®ÓUQS.&RV+W&¯X¢+W&^l)mÐS*ÊT+W ,[Qd,X " *U'U; ,RV(+mRVCC" P &;S*ÊT+W ,[Q&"J" *U'U; ,RV(+mRVCC"$Î

k257Y^>M8;S6\bdY7R<Y625VWB525XY<Y46257?KR>E^257YB=4d>EXO_@8;7Y:;F 46257YD6B=<Y7Y525XY<fN\54d>Wrd¥sAaN\<YSa8;7M:=25m\bY7|ÎWY4uN3DJIWAN\<fN\|KM25mYXY798?ba^2]N\46ILKM25XY27\bLY7¢8;7Y:;FIW4ÎCX?>XYIW4dF[254CS6S6^,r ã IK|VEg38;7Y:;F8;7Yg64uN\Av25:;F[I\F[46257sF[B=S6g64625798;:=<?>ICg,g6ILK|IEg6S,46257YD6B=<Y7Y525XY<fN\546IW=XY2wF[7YZWI<Y625IWB=Sr ¿oÁh»C¹a¼ ºd» Ò Áh»ýð ÍEâ äÈÊT+Wd,[E^l+WUQS*ÊÀRV+W&X¢',"Ð+W&[=RV'OU;e P &;Z" *U'U; %RËW+mRËCC"$Îãx¾E¿ ¼ 3:;FON\5^>4uN8;D6257YB;KÊg6IK|IW54uex62 8;7?AWX98;m

f : R → R × R /h25:;F[4625798;7IW4uN4uNÎEX?>¡ KM257YB=g6<Y7Y462]N ì r Öì 1rC13257YX_

A = f [Q]rG¥@XY<?>CKM25=XY257 |A| = ℵ0 < |R|

IWB[N\<|R \Q| = |f [R \Q]| = |(R× R) \ A| r¨MIW<?K#N\Y^>vF[7YB[N\<`yhS646A\X98;mB=<YSGF[ILKRN\462]N4uND6257YB;KM:=<feIW

π : R× R −→na R, π(x, y) = x.

¢IW46257?K#N\ |π[A]| ≤ |A| < |R| buKM25mYX π[A] 6= R2 25:;F[4625798;7

x0 ∈ R \ π[A]rck0F[7YgG>

x0 × R ⊆ (R× R) \ A rc|<?>E52

c = |x0 × R| ≤ |(R× R) \ A| ≤ |R× R| = c.

Å òù¶74X ª5±E¬õ°±V³*õI³± µn: ª µV?? ° ñ/5.9ã <Y25mYAC2H¡ KM257YB=g6<Y7Y4625SQ#N\4dF[IWB[N.-9q|7YB=46:;F[7Y254uNoôN\^> |R \ Q| = |(R × R) \ A| = c

bXYI4uN\57YfNCPiIg6ILKM257Y=\r

ã ]NA\IW4dF[B[N\:;F[SDcIWAaN\Y7Y^>WbJY725464d>uN\B=g6<YIoXY<YmY:;F[Iv:=DJI\FU>EAaN\4d>¼/h<?K@PhN\:=<YXY<fNv4uNKR>EA6PhN\g6<Y257M<Rs4uN\525<?>nNFO7?ôNaF9>EXY<Y46798Q1<Y625VWB46257YD6B=<Y7Y525XY<fN\54d>46257H8;7Y:;F ÎEX?>XYIW4C-F[254CS6S6^,r |_6IEg6<Y23Il<Y625VWB

RRKM:=<?>E:;F[AC25X_yhS646AWX98;23B=<Y7YXY<?>CKM25:;FU>EX_r|U:;F[I\F[46257\b <fN.-

XO_6IEg6<Y2!4uN\:;F[mYD6Sa8=e\XY7TF9KM27YB=g6<Y7Y46257\r ¿oÁh»C¹a¼ ºd» Ò Áh»ýð 5ÞuÂ |RR| = |P(R)| > c

Îãx¾E¿ ¼ 3B[Z\S6^7Y4CF[Sa8=e\Xpg6IWA6PhN\g646257FON\A:[N\ÎGb!8=N\AKg6IK|IEg6<Y257pD6257YB;KM:=<?>EXO_gGK|VEXO_pDcIEg6D6S646AdF[VLK ¡ KM257YB=g6<Y7Y462]N ì r Ö ñ I\F[B=<?>E^Sa8;7Y^>

|P(R)| = |0, 1R| ≤ |RR| ≤ |P(R× R)| = |P(R)|.N\:;F[IW:=ILKRN\46257@¡ KM257YB=g6<Y7Y462]N#N\4CF[IWB[N.-9q|7YB=46:;F[7Y254uNî/h2¡ KM257YB=g6<Y7Y462]No#N\4dF[IWB[NO1|A\IWC-XY<?>g6ILK|VEgr

<fNaF[7Y^ yhS646A\X98;2JB=<Y7YXY<?>CKM25:;F9>EXO_8;7Y:;F|KM25mYXY798 46253525X?<YoB=<Y7?XY<?>CK325:;FU>EX_ruz\7Yg64uN\A2 F[SýXY<?7YAaNx4uNo4uN\:sDJ7?KM4uNv46257Y:=DJICg6<Y2]N\46AaNGrw1sNos4uN\525<Y257npNaF[7YôNaF9>EXY<Y46798suN\B=g6<YIXY<YmY:;F[IQDcI\8=NfKM2]N8=e:=25moyhS646A\X98;7XY2]eCZEPi7\bHXYIýÎWY7x:=S6ZW7YB=ILK#N\\bY7x:[eýIW467vKæKM25mYAp-:=<YIW=XY2|KM=B=VCgKM:=<?>E:;F[AE25XO_yhS646A\X98;2|B=<Y7YXY<?>CKM25:;FU>EX_r ¥@AaN\<YSa8;7v:=25m\b¢Y7o2 FONp254dF[S625X98=N8;7Y:;F@JPimYg64uNÐ4vyhS646A\X98;2¢XY2]eCZEP]>EXO_8;7Y:;FsFU>E5AWIXYIW4CF[254CS6S6^,buXY<?>E52 FON\Ax4uN\D6B[NfKMg6môN.-5SGF[A\IoKDcIWB=VKR4uN\4625Sg6Iv255IW=XY2!KM:=<?>E:=F[AC25XO_ýyhS646A\X98;2HB=<Y7YXY<?>CK325:;FU>EX_r ã ILK#VCgF[7YZWIy N\AdF[S,DcIW<YIW:;FONLKM25^>8=N\A\I<fN\guNa46257Tg6]N<fN\254CF[7YB=7Y:=ILK#N\4d>EX_ý|<?>CF[7Y54625A\VLKTr1sNT<fN\AWIW6XY<Y7Y46257MF[7YZWIB=IW<Yg6<Y2]NCPiSDJIW:;FONfKM^>:=IW62578;7Y:=<YXY<Y7 8;7Yg646IDd>CFON\46257\rGzWN\A

<fN\SdKRN\>E525=^>WbHKM:=<?>E:;F[AC257,DcI\8=NfKM2]N8=e\XY7x:=25mxg6IýF[798DJIWB;><Y625IWB;>46257YD6B=<Y7Y525XY<fN\5467^2]NCP]>ÎEXlXYI04uNa8;^4625798XYIW4CF[254CS6S6^,r@|<?><fNaF[7Y^ÜÎWY4uN<Y4uN\57T÷Y<?625VWB46257=-D6B=<Y7Y525XY<fN\54d>vÎEX?>A"ÒRË+W& P QS.& P 4625`XYIW4dFO24ES6S6^3N\SdK#N\Y^>WbuY7ZWgG>Ed>xFON\AC2!<Y625V\B4625725:;F[462]NCPUbcFOIxAaN\?gG>ýDJIEg6<Y625VWB@<Y625IWB=Sý525XY<YýB=<Y7YXY<?>CKM25:;F9>EXO_ld>uPiC>XYI4uNa89KR>EY798D6B=<Y7Y525XY<fN\54d>xN\5JI^2]NCP!ÎEX`XYIW4dF[254ES6S6^,r¦325DJI\F[7Y<Ym\bRY7ýFON\AK@PhN\=46257v8;7Y:;FfbRXY<?>E52 b#Y7AaN\YgG>046257YD6B=<Y7Y525XY<fN\54C>DJICg6<Y625VWB

<Y625IWB=ST525XY<Y`B=<Y7YXY<?>CKM25:;FU>EX_ôN#ÎEXXYIW4dF[254CS6S6^ :;yhIWB=^SJPiILK#NCPdKl£@ £KM257YACS#N\4C-F[IWB@2H4uNa<?KRNCP8=eîÌÔ+ B ,&QS*aò[Z ,RV(+mRVCC"rB=<Y7Y<KM257Y57]NaF@D6B=VWJILK#N\46I8;798sg6ILKM257Y=N\5JI38=eIWuN\525\bE<Y4uN8;g6Sa8=e\X`46257YD6B=<Y7Y525XY<fN\54d>oDcIEg6<Y625VWB#<Y625IWB=Sx525XY<YxB=<Y7YXY<>CK325:;FU>EX_ÎEX?>Q^4625798;:=<Y7984625oXYIW4dF[254ES6S6^,r!kl:=<?>E:;F[AE257oF[7vD6B=VWd>lIWAaN\<fNCP]>l:=25moJ7Y<YILK|IEXY467\r¢IQ]NaFON\XO_IWAaN\<fNCPiI:=25m\bY7D6B=<?>EXY<?>E4C>F9>EXO_46257YDJILK|IEg6<Y7Y:[euN\B=g6<YIZEPimYJIWAC257\r¢IWAaN\<fN\46IJILKM257Y^,bdY734uNZWB=S646XY257MDJILKM:=<Y7YXO_64u257sD6B=<?>L8;m?F[798yhIWB=ôN\52<fN\X98;26ôNaF[7YôN.-FU>EAC2¢/hIAdFOV\B[7U8 4uNF9>E^ KR>EA6PhN\g6<Y257@462573^VLKM25^>Ë14uNTF[IDd>CFON\46257sDJID6B=IW:;F[Sx4627sguN:=25mIEg6DcIKM257Yg6<Y257YarHn,VLKM2]e\XIW6B[N\<YILK#I6b!:[exFON\AC257U;KM2]NaFU>ýôNaF[7YôNaFU>EXY<Y467Ob!ZWg6<Y257XfNCPhNU<?KR>EA6PhN,ôNaF[7YôNaF9>EANv8;7Y:;FFON\AaNGb8=N\Aex8=e<Y4uN\^>-2RFON\AC257<Y625IWB;>-4625725:;Fz-4625798=eA/hXY<?>E2H¦325DJI\F[7Y<fNv|IW4dF[254CS6S6^Æ8;7Y:;F@D6B[NfKMg6<Y2K#NO1bwN\57:[eF[7Y2546467\bc<4uN\:=<Y7YZWI

ñ/5C² 74X ª5±E¬õ > ¬YL®pó]ª_³³!ó µ ®oª µ ª5® > ¬Y®%ó]ª_³³!ó µ ®D6S646AdF[SvKM2g6<Y7Y462]NB=VLKM46257sg6IW6B=7MU;KM2]NaFU>dObCZWg6<Y257MFON\AC257s<Y625IWB;>:[eY/hXY<?>E52w¦325DJI\F[7Y<fN|IW4dF[254CS6S6^æ8;7Y:;F`y NCPi:=<?>CKRNO1rk <?KM2]e\<YACSQ<FU>E^46257guNx:=25mS6g6<Y257Y525fN\g646798|8;7YgC-46IW<Y4uN\XY<Y46798¢IEg6DcIKM257Yg6<Y2C4uN3Dd>CFON\46257 IsD6B[NLKMg6<?2K#IW=#¦325DcI\F[7Y<?>|IW4dF[254CS6S6^,bAdF[VWB[NIWcIKM2]e\<?>CK#NCPhN\d>vK|7TKM:=<?>E:;F[AC25XO_,U;KM2]NaFON\XO_EOr

Ù; < Íàx'YÖ r33g6ILK|IEg64625\bu?7TyhS646A\X98=N

f : N× N→ N<fN\guN\4uNKM<YIWB=7Y^

f(x, y) = 2x(2y + 1)− 1

8;7Y:;F362 8;7YA\X98=eGr¤Er3¢IWAN\<fN\\bGY7@:=S6ôN<Y625I\B[SoD6B=<Y7Y525XY<fN\5467YZWI2J<Y62IWB=SvXYI4uN89KR>EY798#D6B=<Y7Y52XY<fN\¨-467YZWI@8;7Y:;F3<Y625IWB=7Y^%D6B=<Y7Y525XY<fNa54C>E^,r

ÚGr33g6ILK|IEg64625@kl4625IW:=7YA ì r åGrñ r3¢IWAN\<fN\\bEY73IW6B[N\<3<Y625IWB=SD6B=<Y7Y525XY<fN\5467YZWI`KM<YZW5mYg67Y^ g6ILK|IW546798yhS646AWX98;2W8;7Y:;F<Y625IWB=7Y^XYI,4uN89KR>EY798`D6B=<Y7Y525XY<fN\54d>E^,r!|<?>F[I:[N\ÎxÎWY4uNxDcIKM257Yg6<Y257YID6B=<Y7YXY2K|IW6B[N\<Y257*3

Ý r33g6ILK|IEg64625@kl4625IW:=7YA ì r ÖLø råGr33g6ILK|IEg64625@kl4625IW:=7YA ì r ÖWÖ rì r33g6ILK|IEg64625@kl4625IW:=7YA ì r Ö ¤Er×Gr313257YXO_

P = 2n : n ∈ N r6kl:=AaNa<fN\<Y625VWB X ⊆ N bGFON\AC2!Y7

|P ∩X| = |X \ P | = |N \X| = ℵ0.

ó r3¢IWAN\<fN\\bcY7TDJIW4625Y:=<Y7T<Y625IWB;>v:[eXYI4uN89KR>EY798MD6B=<Y7Y525XY<fN\5467\r/ NO1w625VWB S KM:=<?>E:;F[AE25XO_AWVEP64uNsDJPhN\:=<YXY<?>C÷Y46257#I@D6B=IW^257Y462]N\XO_TKR>E^257YB=4d>EX_2!=B=IEg6AN\XO_,KD6S646AdFON\XO_pIIW6SKM:=DJVEPiB=<YmYg64d>EX_,KR>E^257YB=4d>EX_r/hB1

(Q× Z) \ (Q× N)r

/hX*1,w625VWB#DuN\B[N\^2B=IW<Phe\XY<Y4d>EXO_pAWVEP¢4uNDJPhN\:=<YXY<?>C÷Y46257\r/hgB1,w625VWB#:=AWIW6XY<YI\4C>EX_pXY2]eCZWVLK 525XY<Yx4uNaF[S6B[N\54C>EXO_r/h7*1,w625VWBKM:=<?>E:;F[AE2X_%KR257Y5IW^2]Na46VKIKM:=DJVEPiXY<?>E464625AaN\XO_XfNCPiA\ILKM2F9>EXO_:;F[IWD6462]N Ý r

/iy;1w625VWB#D6S646AdF[VK46257YXY2]eCZEPiIW=XY2wÎW46I\F[IW4625XY<Y46798|yhS646A\X98;2f : R→ R

r

Å ò6587 ³d¯w³ µ ª5³ ñ/5 Å/hZ!1w625VWB#KM:=<?>E:;F[AC25XO_:=A\IW6XY<YIW4d>EX_pDJIEg6<Y625IWB=VLK <Y625IWB=SD6B=<Y7Y525XY<fN\5467YZWI6r/h_B1w625VWBRDuN\B[N\^2wB=IW<Phe\XY<Y4d>EX_p52F[7YB#¡ 4uN\B;>E:[ILK#N\4d>EX_p4uNDJPhN\:=<YXY<>C÷f46257\r©@F[VWB=7`<`FU>EX_,<Y625IWB=VLK :[ND6B=<Y7Y525XY<fN\5467*3©@F[VWB=7`ÎWZded>ED6B=<Y7Y525XY<fN\5467*3

Öfø r3¢IWAaN\<fN\\b6Y7`<Y625VWB|KM:=<?>E:;F[AC25XO_KM257Y5IW^25N\46VLKIKM:=DJVEPiXY<?>E464625AaN\XO_,XfNCPiA\IKM2¨-FU>EX_,S6:;FON\5IW467YZWI:;F[IWD6462]N

n8;7Y:;F3D6B=<Y7Y525XY<fN\54d>Wr

Ö\Ö r313257YXO_f : N → N

rHHIWAN\<fN\\bY7 | B=46Z (f)| = ℵ05S625:;F[4625798;7FON\AaNp525XY<YuN

4uNaF[S6B[N\54uNnbuY7 |f−1[n]| = ℵ0

rÖ ¤ErszWN\Avg6S6Y798RÎEX?>vÎWY7`C>ETB=IEg6<Y254uN A ⊆ P(N)

FON\AaNGbuY7

(∀A,B ∈ A)(A ⊆ B ∨B ⊆ A)?

Ö ÚGrszWN\AaNg6S6Y798ÎEX?>0ÎWY7ýC>EQB=ICg6<Y254uN-DuN\B[N\^2@B=IW<Phe\XY<?4C>EXO_ DJIEg6<Y625IWB=VLK<Y625IWB=S525XY<Yx4uNaF[S6B[N\54C>EXO_B3

Ö ñ rs|<?>25:;F[4625798;7@<Y625VWBAFON\AC2 b6Y7 |P(A)| = ℵ0

3ÖLÝ r33g6ILK|IEg64625!7YôNaF ì r Ö ÚGrÖ åGr33g6ILK|IEg64625`kl4625IW:=7YA ì r Ö ×GrÖLì r33g6ILK|IEg64625\buY7Tg6]Ng6ILK|IW54d>EX_,<Y625IWB=VLK

A,B,C,DôN\^>

/ NO1zW7?[2A ∼ B

2C ∼ D

b6F[IAC ∼ BD r

/hB1(A×B)C ∼ AC ×BC r

/ X1(AB)C ∼ AB×C r

Ö ×Gr3¢IWAaN\<fN\\bY7vg6]Npg6IK|IW54d>EXO_-<Y625IWB=VLKA2B8;7Y=52 |A| = c

2 |B| < c

b¢F[I|A \B| = c

rÖfó r33g6ILK|IEg64625\buY7T4uN\:;F[mYD6Sa8=e\XY7<Y625IWB;>vôN8=eÎCX`XYIW4CF[254CS6S6^,r

/ NO1xHICg6<Y625VWB!<Y625IWB=ST525XY<Y`B=<Y7YXY<?>CKM25:;F9>EXO_<fNfKM257YB[N8=e\X?>`D6B=<Y7Yg6<Y2]NCPEI\FUK#N\B;F9>Wr/hB1

A×B buZWg6<Y257 A ôNÎEX`XYIW4dF[254CS6S6^,buN B 8;7Y:;F3D6B=<Y7Y525XY<fN\54d>Wr/ X1R4 r

/hgB1 P(Z)r

/ 71,w625VWBR525XY<?D6B=<Y7Y:;F[mYD64d>GXO_!r/iy;1x©sIEPiI4uNDJPhN\:=<YXY<?>C÷Y46257I=B=IEg6ACS,K D6S646AWXY257

(0, 0)2!D6B=IW^257Y4625S Ö r

/hZ!1¥@ACB[eCZ4uNDJPhN\:=<YXY<?>C÷Y46257I=B=IEg6ACS,KD6S646A\XY257(0, 0)

2!D6B=IW^257Y4625S Ö r

ñ/5C× 74X ª5±E¬õ > ¬YL®pó]ª_³³!ó µ ®oª µ ª5® > ¬Y®%ó]ª_³³!ó µ ®/h_B1,w625VWB#:=AWIW6XY<YI\4C>EX_pDJIEg6<Y62IWB=VK 525XY<YB=<Y7YXY<?>CKM25:;FU>EX_r/h2m1RN

/hXY<?>E52!<Y625V\B#KM:=<?>E:=F[AC25XO_pXY2]eCZWVLK 525XY<YB=<Y7YXY<?>CKM25:;FU>EX_B1r¤ ø r33g6ILK|IEg64625\bcY7g65Ng6ILK#IW4C>EXO_<Y625IWB=VLK

A,B,C,D8;7Y=52

A ∼ C, B ∼ DIWB[N\<B ⊆ A, D ⊆ C

b6F[IA \B ∼ C \D r

¤ Ö r3¢IWAN\<fN\\bY7 <Y625VWBA = f ∈ RR : f

8;7Y:;F3XY2]edZEPhN ôNÎCX XYIW4dF[254CS6S6^/ ÃÊÅ ÓGÀ6º ¿ÓGÀç©TN\YguNyhS646A\X98=NXY2]eCZEPheý8;7Y:;F8;7Yg646IW<Y4uN\XY<Y46257lKR>E<Y4uN\XY<YIW4uND6B=<Y7Y<T:;K#Ia8;7`KRNaB=F[IW=XY2!4uN<YJIWB=<Y7T525XY<YxKR>E^257YB=4d>EXO_B1r

¤W¤Er33g6ILK|IEg64625\b\Y7#25:;F[4625798;7|B=IEg6<Y254uNs<Y625IWB=VLK-D6B=<Y7Y525XY<fN\54d>EX_ A ⊆ P(N)b\AdF[VWB[N

8;7Y:;FÎEX?>XYI\4CF[254CS6S6^.IWB[N\<FON\ANGb Y7og6]Ng6ILK|IW54C>EXO_A,B ∈ A D6B=<Y7YACB=Va8

A ∩B 8;7Y:;F3:=A\IW6XY<YIW4d>Wr¤\ÚGr313257YXO_ An : n ∈ N cmYg6<Y257g6ILK#IW54ueB=IEg6<Y254ueý<Y625IWB=VLKjD6B=<Y7Y525XY<fN\54d>EXO_r¢IWAN\\bcY725:;F[4625798;7TB=IEg6<?254uN Bn : n ∈ N <Y625IWB=VLKD6B=<Y7Y525XY<fN\54d>EXO_,DuN\B[N\^2B=IW<Phe\XY<?4C>EXO_buFON\AaNY7

(∀n ∈ N)Bn ⊆ Anr

Ê++!

ÐK.LÆ·o ·+ÃK. ÷), )po

U46g6S6AWX98=N-ôNaF[7YôNaFU>EXY<Y4uNp8;7Y:;F,uN\B=g6<YIK#N\Y4ue-2@XY<YmY:;F[I:;F[IW:=ILK#N\4ue-^7?F[IEgueg6ILK#ICg6<Y7Y462]NFUKM257YB=g6<Y7Yýg6I\FU>EXY<fe\X?>EXO_:=<Y7YB=IWA\IxB=IW<YS6^2]N\4d>EX_,K@PhN\:=46IW=XY2H525XY<Y,4uN.-F[S6B[N\54d>EX_b<fN\B=7YACS6B=:h8=Nv8;7Y:;F6525:=AWIý<462]eý<?KM2]e\<fN\4ueý^7?F[IEgueý:PiS6fe\Xfeg6IQg67?ên-4625ILKRN\4625NIW6257YAdF[VLKTr|N\B=VLKM46Il<,2546g6S6AWXU8=eGb!8=N\A2M<,B=7YACS6B=:h8=el:=DJI\FU>EAaN\^>:=25mxKôNaF[7YôNaFU>EXY7TXY<YmY:;F[I6bcX_6IC4625798;7Yg646IWACB=I\F[46257T46257Y;KM2]N\g6IW^257\rEk KM257Y5Sg67?êu4625X98=N\XO_D6B=<Y7YD6B=ILKRN\g6<fNa4C>EXO_QDJIW<YIWB=46257J7Y<S6?>EXY2]Nv2546g6S6A\X98;2c8;7Y:;FTIW4uNT8;7Yg64uN\ApS6ACB;>CFONvDJIEg4uN\D625:[N\^2FU>ED6S,

. . .5S6,:;yhIWB=^SJPiILK#N\462]N\^2wFU>ED6SU2!FON\AxguN\5798UOrc¢ICg6IW646257C>CK#N

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¿oÁh»C¹a¼ ºd» Ò Áh»õ Í Ï ß À Å Àu¼ À Ò ¼ Ë ÓJÐîYÁi À Ç »E½lÀ Ç á ÐWº Ò »Yî)Ð¯£Ô+W&;U;eϕ(x)

Ê=Q*ST+W&v;CRn)OU P aÓS*O#%RV+W ,XZa ÓS*#.),[Q&;+W&oST"Ð+W&RVR& Px ∈ N ÎEY&Q%' P &;\^l+

k#mϕ(0)

,[w#*SkÊjm

(∀n ∈ N)(ϕ(n)⇒ ϕ(n+ 1))¿

(∀n ∈ N)ϕ(n)

Î

h IWB=ôN\54d>xg6IK|VEgDcIW^2 8=N\^>WbuDJIW46257?K#N\KR>EôN\ZdNIW4S6?>EXY2]NyhIWB=ôN\5467U8Mg67?ên-4625X98;2525XY<Y4uNaF[S6B[N\54C>EXO_buAdF[VWB=798R46257`JmYg6<Y257Y^>xKMD6B=ILK#N\g6<fN\\rcU464C>xg6IK|VEgKR>EA\I,-B=<?>E:;F[Sa8;7 y N\AdFfbY7 <Y625VWB¢52XY<Y4uNaF[S6B[N\54C>EXO_Y/h<Y7 <?KR>EA6P]>E^DJIWB=<fe\g6AC257Y^1E8;7Y:;FHg6IW6B=<Y7

ñ%6nø û µ ¯ ? ¶C³[³v°³ : ®\°³ : õB³ µ ³pª¢¬Y®,¶ ? ¬ 8 [³S6DcIWB=<fe\g6AWILK#N\4d> g buAdF[VWB=7YZWIog6ILK|VEg8;7Yg64uN\AvF[7Y:=D6B=ILK#N\g6<fNo:=25mTg6IyhIWB=ôN\546798#g67=-êu4625X98;2J525XY<Yx4uNaF[S6B[N\54d>EXO_rGU4CF[S625X?>L8;46257s4uNaF[IW^2]N\:;F#<fN\:[N\guNFON38;7Y:;FRg6IW=ÌEXY<?>CKM25:;FON48;7Y=52¢XO_6XY7Y^>pDcIWAaN\<fN\\bwY7g6]Ng6ILK#IW5467YZWIoS6:;FON\5IW467YZWI

n0 ∈ ND6B[NfKMg6<Y2K|738;7Y:;F

ϕ(n0)b\F[I`<fN\X?<?>E4uN\^>IEg

ϕ(0)/hAdF[VWB=7¢8;7Y:;F D6B[NLKMg6<Y2K|7*1H2GDJI`<YB=IW6257Y4625S

n0ACB=IWA\VK

/h4uNAN\YgG>EÂCB=IWACSpA\IWB=<?>E:;FON8=e\X<TK#N\B=S646ACSpg6B=S6ZW257YZWI!1|g6IW:;FONa8;7Y^>xD6B[NfKMg6<Y2K#IW=ϕ(n0)

rNaF[7Y^ KR>EAN\<fN\46257\bY7DJ7?KM4uNýK@PhN\:=46IW=

ϕ8;7Y:;FD6B[NLKMg6<Y2K#Nýg6]NpKM:=<?>E:;F[AE25XO_

525XY<Y4uNaF[S6B[N\54d>EXO_DcIW57YZdNs4uN@:=D6B[NLKMg6<Y7Y4625SgGK|VEXO_<fNCPiIWY7YoN\:[N\gG>U46g6S6A\X98;2JnpN.-F[7YôNaFU>EXY<Y46798?r6kN\B=S6467YA /hB1|4uN\<?>CKRN\^>xXY<YmY:;F[Iî),[Q *),+W&"J+mR%O)OU' P RV',"rN\:[N\gG>U46g6S6AWX98;2!npNaF[7YôNaFU>EXY<Y46798|ÎWY4uNTKR>ED6B=ILK#N\g6<Y25@:=<Y7YB=7YZs8;798 K#N\B=2]N\4C-

F[VLK /hXYIDcIKM254646Ipd>Eog6]N4uN\:g6IW=oIEXY<?>CKM25:;F[7\bu8;7Y=52 <YB=IW<YS6^257Y525=^>ýDJILKR>EY:=<fe254dF[S625X?>L8;4ue254dF[7YB=D6B=7?FON\X98;m*1bGFON\AC25X_8=N\AÏV&;\^l+

ϕ(c)+(∀n ≥ c)(ϕ(n)⇒ ϕ(n+ 1))

¿Z (∀n ≥ c)ϕ(n)

rÏV&;\^l+

ϕ(0) ∧ ϕ(1)+(∀n ∈ N)(ϕ(n)⇒ ϕ(n+ 2))

¿ï (∀n ∈ N)ϕ(n)

rÏV&;\^l+

ϕ(0)+(∀n ∈ N)((∀k < n)ϕ(k)⇒ ϕ(n))

¿ï (∀n ∈ N)ϕ(n)

rljwguN\B=<fN8=e:=25m`B=IW<YS6ÎLKRN\462]NIWDuN\B;F[7ÌuN\B=g6<Y25798R:=A\IW^D6525A\ILKRN\467`:=XO_67YôNaFU>Wbu46DrÏV&;\^l+

ψ(0, 0) ,[w#*SIn^#f! ,Xc ,^lRV'OU;e

m,n ∈ N S$[w#%XZ*ST+mXc *\*U+S*O#%RV+(#ψ(m,n)X¢',RV+_)p#À[w#%XZ*ST+mXc *\*]ES!#KÓ

ψ(m + 1, n)+ψ(m,n + 1)

¿2 $S*O#%RV+W&ψ(m,n)

P &;[w#%XZ*ST+mXc&Ín^#$XÈQST'*m),+WU;e

m,n ∈ N rB=<?>L8;B=<?>L8;^>:=25mRF[7YB[N\<RF9>EDcIKR>E^ <fN\:;F[IW:=IK#N\4625IW^ N\:[N\gG>U46g6S6A\X98;2npNaF[7YôN.-

FU>EXY<Y46798?r¸¹ºWá!ÓÈÇ Àu¼!áÖ rÀ/ 13257YB=VLKM46IW=oq|7YB=46IWS65527YZWI!1 ã ]Ng6ILK|IW54d>EX_

n ∈ N IWB[N\< x > −1ôN\^>

(1 + x)n ≥ 1 + nxr

k÷F9>E^jKR>EDuN\g6ACSϕ(n) = (∀x > −1)((1 + x)n ≥ 1 + nx)

r/ NO1x13257YXO_,4 ø rGk0F[7YgG>g6]Ng6ILK|IW5467YZWI

x > −1ôN\^>

(1 + x)0 = 1 ≥ 1 + 0 · x buXY<?>E52 ϕ(0)<fN\XO_6IEg6<?2 r

s ¬8¨ £.T¤gDiÓ£,=xz@=£,D¡m Wz§@p v ®Ey £.y |Z.|Nyi ª £,=§.©iQ«vu¡½ D©W~;£.Q@=z£,@p v*xzy ~»m£,v* ª ¥ ¡myn®¯;Wz|;m*¥xz£.(¡zýCz£À@=©Wy =££,=xz@ y u2¥xw z|À@p v »m£,v* ª ¥ ¡myiË©Wx v ª Q(¡%¬ m|5§,v ª Í£.y jæW©W,v*£.Æ§% ª =xwIg=zzxwlzz£*y ϕ(0)|2z£,«§.,¤W¥yg=i¡m V£.ÆyTi ª O ¥ .Q@=£. Q $v*©W.þy (¡!¥xzuw¤W¥yxwlzz£.y .

×Vò½ñ û µ ¯ ? ¶C³[³x°³ : ®\°³ : õB³ µ ³ ñ%6Bñ

/hB1x©`B=IWA,2546g6S6A\X?>L8;4d>Wr!B=<?>ED6S6=Y^>CqbJY7<fN\XO_6IEg6<Y2ϕ(n)

r|_6XY7Y^>DcIWAaN.-<fN\\b?7 D6B[NfKMgue8;7Y:;F

ϕ(n+1) = (∀x > −1)((1+x)n+1 ≥ 1+(n+1)x)r

9:;F[I\F[46257\b6g6]Ng6ILK|IW5467YZWIx > −1

ôN\^>

(1+x)n+1 = (1+x)n(1+x) ≥ (1+nx)(1+x) = 1+(n+1)x+nx2 ≥≥ 1 + (n+ 1)x

b

D6B[<>-XY<?>E^ D6257YB;KM:=<fN<46257YB=VKM46IW=XY2#F[I<fN\:;F[IW:=IK#N\46257<fNCPiIWY7Y462]Ný254C-g6S6AWX?>L8;467YZWI6r

oN\:[N\gG>U46g6S6A\X98;26npNaF[7YôNaFU>EXY<Y46798 KM4625IW:=ACSa8;7Y^>WbaY7ϕ(n)

8;7Y:;FHD6B[NLKMg6<Y2K|7g6]Ng6IK|IW5467YZWI4uNaF[S6B[N\5467YZWI

nbuXYI4uN\57YfNCPiIg6ILKM257Y=\r

¤Er ã ]Ng6ILK|IW5467YZWI<Y625IWB=S:=A\IW6XY<YIW467YZWIXôN\^> |P(X)| = 2|X| r

N\:;F[IW:=Sa8;7Y^>2546g6S6A\X98;mRôNaF[7YôNaFU>EXY<Y4uesKM<YZW5mYg67Y^Ê255IW=XY2G7Y57Y^7Y4dF[VLK<Y625IWB=SXrc|<?>E52

ϕ(n) = (∀X)(|X| = n⇒ |P(X)| = 2n)r

/ NO1x13257YX_4 ø rs57 |X| = 0IW<Y4uN\XY<fNGb!Y7

X = ∅ bXY<?>E52 P(X) = ∅ 2|P(X)| = 1 = 20 bu<fNaF[7Y^ ϕ(0)

<fN\XO_6IEg6<Y2 r/hB1x©`B=IWA2546g6S6A\X?>L8;4d>Wr6B=<?>ED6S6=Y^>WbGY7sg6]NTg6ILK#IW5467YZWI<Y625IWB=S

n-U7Y57Y^7Y4C-

FOILK|7YZWI3255IW=w8;7YZWI3DJICg6<Y625IWB=VKQKR>E46IW:=22nrfk XY7Y5SDJIWAaN\<fN\462]N

ϕ(n+1)K#7T÷Y^>Qg6ILK#I\54C><Y625VWBXb!FON\AC2 Y7 |X| = n + 1

r¢3:;FON\5^>ýg6ILK#IW5467a ∈ X rc13257YXO_ X ′ = X \ a /hIEXY<?>CKM25=XY27\b |X ′| = n

1rcnpN\^>

P(X) = A ∈ P(X) : a ∈ X ∪ A ∈ P(X) : a 6∈ X = A ∪ a : A ∈ P(X ′) ∪ P(X ′)

r

HIW46257?K#N\3<Y625IWB;> A∪a : A ∈ P(X ′) 2 P(X ′):[eB=IW<Phe\XY<Y467sIWB[N\<

|A ∪ a : A ∈ P(X ′)| = |P(X ′)| bGF[I

|P(X)| = |P(X ′)|+ |P(X ′)| = 2 · 2n = 2n+1,

XYI4uN\57YfNCPiIDcIWAaN\<fN\\r¡HN\A8=N\ADJIWD6B=<Y7Yg64625I6bJDcIK|IEPhN\46257:=25m4uNvN\:[N\g6mU46g6S6AWX98;2npNaF[7YôNaFU>EXY<Y46798A\IW6XY<?>g6IK|VCgr

kQ>EAWI\B[<>E:=FON\^>F[7YB[N\<ý2546g6S6A\X98;mýôNaF[7YôNaFU>EXY<Y4ueg6Ig6IK|IEg6S0FUKM257YB=g6<Y7Y46257<¨MIW<Yg6<Y2]NCPiS Ý b6AdF[VWB;>xIWD6S6=XY25525=^>WrzýnD§.©Wxz§. xw¥xzz£.y c£,=xz=|ZÍw(qw,5lpxTi,;N¦¥o|2z£,.xz£,IgZxw (ª ©W~=È*vO½xwlzz£.y ª ©W ª Dy £,v* ª ¥Q¡m£.zþ Q ±

ñ%66ù û µ ¯ ? ¶C³[³v°³ : ®\°³ : õB³ µ ³pª¢¬Y®,¶ ? ¬ 8 [³ ¿oÁh»C¹a¼ ºd» Ò Áh»õ 5Þ £Ô+W&;U;e 〈X,≤〉 Ê=Q*ST+W&oRV+W&O('," ;)O ®ÓUQS. ,RV',"NS*ÊT+W ,[Q&"JUQS.;\ÑU+W ,Xc Ò*V ,[wS*aO.)O ,XZ#%RV',"$ÎoY&Q%'$X

X+_(RV+W& P &Ð&^_&"&RVï"$#.)*',"$#%^lRV'pÎ

ãx¾E¿ ¼ N\:;F[IW:=Sa8;7Y^>l2546g6S6AWX98;mvôNaF[7YôNaFU>EXY<Y4uepKM<YZW5mYg67Y^ 255IW=XY2 7Y57Y^7Y4dF[VK:=A\IW6XY<YIW467YZWI<Y625IWB=SXY<YmY=XY25IK|IS6DcIWB=<fe\g6AWILK#N\467YZWI6r ã ]NTS6D6B=IW:=<YXY<Y7Y462]NGb

XJmYg6<Y257

<fNfKM:=<Y7IW<Y4uN\XY<fNCPiI<Y625VWBRXY<YmY=XY2IK|IoS6DcIWB=<fe\g6AWILK#N\4d>D6B=<Y7Y<B=7Y]N\X98;m ≤ rGk0F[7YgG>

ϕ(n) = (∀X)(|X| = n⇒ KX25:;F[4625798;7`7?57Y^7Y4CFMôN\AC:;>EôN\54d>

).r

¢IW46257?K#N\X 6= ∅ bF[IQ2546g6S6A\X98;m,<fN\XY<?>E4uN\^>ICg n = 1

rs57g6]Ný<Y625IWB=VLK8;7YgC-46IC7Y57Y^7Y4CF[ILKR>EXO_p25:;F[46257Y46257`7Y57Y^7Y4CF[S,ôN\AC:;>EôN\5467YZWI@8;7Y:;F3F[B;>CKM2]N\5467\ruHIW<YIW:;FON8;7<fNaF[7Y^%ACB=IWAv2546g6S6A\X?>L8;4C>\rN\A6PhN\guN\^>WbLY7

ϕ(k)8;7Y:;FHD6B[NLKMg6<Y2K|7 g6]NRKM:=<?>E:;F[AE25XO_

k < nrLkl7T÷Y^>Tg6IK|IW54d>

DcIWB=<fe\g67YA 〈X,≤〉 b FON\AC2@Y7 |X| = nr#3:;FON\5^>g6ILK#IW467

a ∈ XrRz\7Y=52

a8;7Y:;F

7Y57Y^7Y4dF[7Y^ ôN\AC:;>EôN\54d>E^ KXbF[IQA\IW46257YX\r z\7Y=52R46257\bF[IB=IW<?KRN\Y^>IWJXY25mYXY257

DcIWB=<fe\g6ACS ≤ g6I<Y625IWB=S X ′ = x ∈ X : a < x r ¢IW46257?K#N\ a 6∈ X ′ bwF[IôN\^>|X ′| < |X| = n

2 4uNpÎCX?>Q<fNCPiIWY7Y462]N,2546g6S6A\X?>L8;467YZWIý25:;F[4625798;77Y57Y^7Y4dFôN\AC:;>O-ôN\54d>3K

X ′ rg67?êu4625X98;2d<Y625IWB=S X ′ KR>E4625AaNGbLY78;7Y:;F¢IW4T7Y57Y^7Y4dF[7Y^ôN\AC:;>EôN\54d>E^FON\ACY7`KXr

NaF[7Y^ 4uNÎEX?>N\:[N\gG>U46g6S6A\X98;2s¢IWB=<fe\g6A\IK|798ϕ(n)

8;7Y:;FoD6B[NfKMg6<Y2K|7,g6]Ng6ILK|IW5467YZWI4uNaF[S6B[N\5467YZWI

nr

B=<?>L8;B=<?>E^>:=25m`F[7YB[N\<@F[B[<?7Y^DcIWS6XY<fNa8=e\X?>E^D6B=<?>EA6PhN\g6IW^ ú r¸¹ºWá!ÓÈÇ Àu¼!áÖ r3¢IWAN\Y7Y^>WbuY7Tg6]Ng6ILK|IW546798#525XY<YC>v4uNaF[S6B[N\546798

n<fN\XO_6IEg6<Y2 46257YB=VLKM46IW=

30n < 2n + 110. (∗)B=<Y7YD6B=ILK#N\g6<Y25^>xg6IK|VEg2546g6S6AWX?>L8;4d>Wr/ NO1 ã ]N4 ø :=D6B[NLKMg6<fN\^>xJ7Y<YDJIW=B=7Yg64625IËb

0 < 20 + 110 = 111r

/hB1,N\A6PhN\guN\^>F[7YB[N\<\b Y730n < 2n + 110

r|3g6ILK|IEg64625^>46257YB=VLKM46IW=30(n + 1) < 2n+1 + 110

r F[IW:=Sa8=e\X<fNCPiIWY7Y462572546g6S6AWX?>L8;467I\F[B=<?>E^SC-8;7Y^>

30(n+ 1) = 30n+ 30 < 2n + 110 + 30 = 2n+1 + 110 + 30− 2n << 2n+1 + 110

b

.©W|= £.y ϕ(n) = (∀X)(|X| = n⇒ (∃a ∈ X)(∀x ∈ X)¬(a < x))

üjiBy z©W xzx¢£.y ¥Ô§p*¥..v*xzyOxzx.y ©W~; v*©whN,.¬À©W~I. z (ª y zþ,

×Vò½ñ û µ ¯ ? ¶C³[³x°³ : ®\°³ : õB³ µ ³ ñ%6.5D6B[<>-XY<?>E^tIW:;FONaF[462]Ný46257YB=VLKM46IW=<fN\X_6ICg6<Y2Mg6]N

n ≥ 5r NaF[7Y^ 46257=-

B[VLKM46IW=(∗) <YIW:;FONCPhNS6g6ILK|IEg64625IW4uNg6]N n ≥ 5

rEHIW<YIW:;FON8;7s:=D6B[NLKMg6<Y25\bY7g6]N

n = 1ôN\^>

30 < 21 + 110 = 112b

g6]Nn = 2

ôN\^>60 < 22 + 110 = 114

bg6]N

n = 3ôN\^>

90 < 23 + 110 = 118b

g6]Nn = 4

ôN\^>120 < 24 + 110 = 126

r¡ >E^:[N\^>E^46257YB=VLKM46IW=

(∗) <YIW:;FONCPhNS6g6ILK#ICg64625IW4uNg6]NKM:=<?>E:;F[AC25X_525X?<Y4uNaF[S6B[N\54d>EXO_nr6k :=<YX?<Y7YZWVW546IW=XY2 g6]N

n = 6IaFOB=<?>E^Sa8;7Y^>

180 < 174r

¤Er3¢IWAaN\Y7Y^>WbuY7AN\YguN525XY<YuN4uNaF[S6BONa54uNDcIW:;FON\XY22n+ 1

8;7Y:;F3DuN\B=<?>E:;FONGrNCPiVWY^>WbuY7

2n + 18;7Y:;FsDuN\B=<?>E:;FO7abwXY<?>E52¢25:;F[4625798;7

k ∈ N bcY7 2n + 1 = 2kr

¢IWAaN\Y7Y^>WbuY72(n+ 1) + 1

8;7Y:;F3DuN\B=<?>E:;F[7\rcU:;F[I\F[46257\b

2(n+ 1) + 1 = 2n+ 1 + 2 = 2k + 2 = 2(k + 1),

XYIAWIW6XY<?>g6ILK|VEg,2546g6S6A\X?>L8;4C>\rÚGr33<fN\:[N\g64625^>WbCY73KM:=<?>E:;F[AC257@A\I\FU>4uN;KM257YXY2573:[eF[7YZWI:[N\^7YZWIA\IW5IWB=SËûfrE1sNa8-D6257YB;K :;yhIWB=ôN\52<YSa8;7Y^>D6B=IW657Y^,rG13257YXO_

KIW<Y4uN\XY<fN<Y625VWB|AWI\F[VLKTr ã ]NAaN\=-

g67YZWIAWI\FONx ∈ K 46257YXO_

k(x)IW<Y4uN\XY<fN@8;7YZWIA\IW5IWBYr6B=<?>L8;^2 8;^>

ϕ(n) = (∀X ⊆ K)(|X| = n⇒ (∀x, y ∈ X) k(x) = k(y)),

XY<?>E52wyhIWB=^SJPhNϕ(n)

:;FUKM257YB=g6<fNGbuY7#8;7Y=52wôN\^>g6IK|IW54d>v<Y625VWBnA\I\F[VLKTb6F[I

:[eIW467F[7YZWI:ONa^7YZWI,A\IW5IWB=SrkQ>E:;FON\B=XY<?>Q<fNaF[7Y^2546g6S6AWX?>L8;46257DJIWAN\<fN\\b Y7(∀n ∈ N+)ϕ(n)

r/ NO1

ϕ(1)8;7Y:;F3:=DJ7Pi4625IW467`F[B;>CKM2]N\546257\r

/hB1x©`B=IWA2546g6S6A\X?>L8;4C>WrCNCPiVWY^>WbaY7#<fN\XO_6IEg6<Y2ϕ(n)

rW13257YXO_X ⊆ K

JmYg6<Y257g6IK|IW54d>E^Æ<Y625IWB=7Y^ÆAWI\F[VLKTbWFON\AC25^ÆY7 |X| = n+1

rC3:;FON\5^>g6ILK#IW5467a ∈ X 2¢46257YX_

X ′ = X \ a rck0F[7YgG> |X ′| = nbJ<fNaF[7Y^<<fNCPiIWY7Y462]N

2546g6S6AWX?>L8;467YZWIKM:=<?>E:=F[AC257TAWI\FU>oK<Y625IWB=<Y7X ′ :[eF[7YZWI:[N\^7YZWIAWIW5I\B[SrHIW<YIW:;FON8;7R:=D6B[NfKMg6<Y25RA\IW5IWBHA\I\FON

araktFU>E^ÊXY7Y5SK|7T÷Y^>

b ∈ X b b 6= a2 46257YXO_X ′′ = X \ b rc¢IEg6IW646257#8=N\AvDJIWD6B=<Y7Yg64625IKM:=<?>E:;F[AC257A\I\FU>xK<Y625IWB=<Y7X ′′ :[eF[7YZWIp:[N\^7YZWI,AWIW5I\B[SQIWB[N\< a ∈ X ′′ b <fNaF[7Y^.KM25g6<Y25^>WbY7A\I\FaôN,FON\AC2#:[N\^ AWIW5IWBM8=N\AlDJIW<YIW:;FONCPi7xA\I\FU>Kj<Y625IWB=<Y7

XbXYI

4uN\57YfNCPiIDJIWAN\<fN\\r¢ILK#IEPhN\46257`:=25m`4uNoN\:[N\g6mÙ46g6S6AWX98;2HnpNaF[7Y^NaF9>EXY<Y46798RA\IW6XY<?>,g6ILK|VEgr

ÿN C©W~;£.y zv;~*vOzzB xz ª y ª = W ¥xw=©W£. ª W~©W¡m n§%.v*I.£Tw= B£.y y v*z£ *¥xz£

ñ%6C6 û µ ¯ ? ¶C³[³v°³ : ®\°³ : õB³ µ ³pª¢¬Y®,¶ ? ¬ 8 [³kl:=<?>E:;F[AC257#DJILKR>EY:=<Y7 UB=IW<YS6ÎK#N\462]NR2546g6S6A\X?>L8;467|ôNa8=e8;7Yg64ueMKM:=DJVW54ue3XY7YXO_6m

4o:[e46257YD6B[NfKMg6<Y2K#7\ruU5S6:;F[B=Sa8=eIW467`FU>EDJILK#7JPimYgG>\buKR>E4625AaN8=e\XY7<T46257Y<YB=IW<YS6^257Y462]N25:;F[I\FU>v2546g6S6A\X98;2!ôNaF[7YôNaF9>EX?<Y46798?rkëD6257YB;KM:=<?>E^ D6B=<?>EA6PhN\g6<Y257:=D6B[NfKMg6<Y25525=^>Wb¢Y746257YB=VLKM46IW=8;7Y:;FD6B[NfKMg6<Y2K#N

g6]Nn ≤ 4

IWB[N\<\baY7#ACB=IWAT2546g6S6AWX?>L8;4d>guNs:=25m KR>EA\IW4uN\Rg6]Nn ≥ 5

r\13257?:=F[7?FU>WbF[7#gGK#ND6B[NfKMg6<Y2K|7Ry N\AdF9>46257R:=A6PhN\guNa8=e`:=25m#KQ8;7Yg64ueTXfNCPiIW=\rC13257RguN`:=25mRJILKM257Y^5KR>EAWIW4uN\ACB=IWACSE<

4g6I

54ýg6I,F[7YZWIp46257Y<YJmYg64uN8;7Y:;FD6B[NfKMg6<Y2K|IW=oACB=IWACS2546g6S6AWX?>L8;467YZWI

g6]Nn ≥ 4

rk g6B=S6ZW25^D6B=<?>EA6PhN\g6<Y257IEXY<?>CKM25=XY25746257:=D6B[NfKMg6<YIW46I6bJXY<?>,F[7Y<fN<fN\XO_6IEg6<Y2Hg6]N

4 ø r68=N\AKM2]N\g6IWÎ6bG46257`<fN\XO_6IEg6<Y2 rk D6B=<Y7?KMB=I\F[4C>E^ D6B=<?>EA6PhN\g6<Y2573F[B=<Y7YXY25^ /hg6IW6B=<Y7sIWDJILKM257Yg6<Y2]N\4d>:;FON\46ILKM2cF[B=S6gC-

4ue<fN\ZdN\g6A\m*1o46257guN:=25mpKR>EA\IW4uN\ACB=IWACS2546g6S6AWX?>L8;467YZWI-g6]Nn = 2

r|U:;FOIaFO4627\bKRF[7YgG><Y625IWB;>

X ′ 2 X ′′ :[eTB=IW<Phe\XY<Y467\bCXY<?>E52cA\IW6XYILKR>KM4625IW:=7YA462578;7Y:;F DJIWD6B[NfKM4C>Wr

Ù F "R&Ûv/¥ 2557l2546g6S6A\X98=NôNaF[7YôNaFU>EXY<Y4uNQ8;7Y:;F^7?F[ICgueg6ILK|IEg6<Y7Y462]NFUKM257YB=g6<Y7YbsFON\A

B=7YACS6B=:h8=Nl8;7Y:;Fý^7?F[IEgueg67?êu4625ILKRN\462]NIW6257YAdF[VLKTr@1sN8;XY<YmY=XY25798JmYgueF[IyhS646A\X98;7\bAdF[VWB;>EXO_g6<Y257Yg6<Y254ue8;7Y:;Fý<Y625VWBp525XY<Y4uNaF[S6B[N\54d>EX_bsXY<>E52XY25eCZW2 r ã 7?êu4625X98=NFON\AaN/h<?K#N\4uNg67?êu4625X98=eB=7YACS6B=7Y46X?>L8;4ue 2 1vDJIW57YZdN-4uN-DJIEguN\4625S DcIEXY<feaF[A\IK|7YZWIKR>EB[N\<YS/h5S6TAC255ACSDcIEXY<feaF[A\IKR>EXO_KR>EB[N\<YVLKo1 XY2]edZWSIWB[N\< IWACB=7Y=57Y4625SbfKp8=N\AC2d:=DJIW:=VWI\F[B=<?>O-^>CK#N\`KR>EB[N\<?>DJVp÷Y4625798;:=<Y7T<Y4uN8=e\X`KR>EB[N\<?>xK|XY<Y7Y=4u25798;:=<Y7\r¢IEg6:;FONfK#ILKR>:=XO_67YôNaF`g67?êu4625IK#N\462]NoB=7YACS6B=7Y46X>8;467YZWIg6]NoXY2]edZWS

anKR>EZW]e\guN

<fNaF[7Y^%4uN\:;F[mYD6Sa8=e\XYI6r• ¥sACB[7?[5N\^>oK#N\B;F[IW= a0

r

• NaA6PhN\guNa8=e\X\b!Y7o<Y4uN\^>v8;S6K#N\B;F[IW=XY2 a0, . . . , anIWACB=7Y=]N\^>^7?F[IEg6mKR>E<=-

4uN\XY<Y7?462]NK#N\B;F[IW=XY2an+1

r13257QJmYg6<Y257Y^> =XY25=5798xyhIWB=ôN\2<YIK#N\DJIa8;mYXY2]Ng67?êu4625X98;2B=7YACS6B=7Y46X?>L8;46798?bsFU>E^

uN\B=g6<Y25798?bJY7DcIEg6IW646257M8=N\AxKD6B=<?>EDuN\g6ACSlN\:[N\gG>,U46g6S6AWX98;2npNaF[7YôNaFU>EXY<Y46798@ôNIW4uNpB=VWY467KRN\B=2]N\4dFU>WrN\^2]N\:;FTF[7YZWID6B=<?>L8;B=<?>E^>l:=25moAC255ACSD6B=<?>EA6PhN\g6IW^,b¢ôNa8=e\X;KM2]N\g6IWÎW=p25:;F[46257Y462]NýF9KM257YB=g6<Y7Y4625NGb|AdF[VWB=7^VLKM2 b Y7pF[7Y40:=DJIW:=VW0g67?êu4625ILK#N\462]N8;7Y:;F3DcIWD6B[NLKM4d>x2!I\F[B=<?>EôN\4d>xXY2]eCZs8;7Y:;FMKR>E<Y4uN\XY<YIW4d>8;7Yg646IW<Y4uN\XY<Y46257\r¸¹ºWá!ÓÈÇ Àu¼!áÖ r h S646AWX98;mÓ+m^lRV+(#g67?êu4625Sa8;7Y^>x4uN\:;F[mYD6Sa8=e\XYIËb

a0 = 1

an+1 = (n+ 1) · an .

3 !hEv*jc,£.y ¥ ¡ y £,v* ª ¥ ¡m£

×Vòùvþ ®,¶ ? ¬ 8 [³ ñ%69k0F[7YgG>

an = n!r

¤Er3©`]N\:;>EXY<Y4d>E^jD6B=<?>EA6PhN\g67Y^%IW6257YAdF[Svg67?êu4625IK#N\467YZWIB=7YACS6B=7Y46X?>L8;46257|8;7Y:;FÓU+(a*`ï+(Ê= ,R#!U;U+W&i`! abEIWACB=7Y=5IW4d>vKRN\B=S646AN\^2

a0 = 0a1 = 1

an+2 = an+1 + an

.

ÚGrsnpN8=e\XvguN\467v<Y625IWB;>A0, . . . , Am

ÎWY7Y^>Q<Yg67?êu4625ILKRN\v25XO_:=S6^mvA\IWB=<?>E:-FON8=e\X`<T4uN\:;F[mYD6Sa8=e\XY7YZWIo:=XO_67YôNaF[S¼/hIEXY<>CKM25=XY257A\IW46:;F[B=S6A\X98=NôN:=7Y46:MFU>E5A\Ig6]N

n < m1r

a0 = A0

an+1 = an ∪ An+1.

k0F[7YgG>am = A0∪ . . .∪Am

rWk N\4uNa5IWZW25XY<Y4d>:=DJIW:=VWoÎWY7Y^><Yg67?êu4625ILK#N\D6B=<Y7YACB=Va8M:=A\IW6XY<YIW46798M255IW=XY2w<Y625IWB=VKTrn,IWY4uNv:=25m<fN\Dd>CFON\\bwDJIvXYIvFON\AS6g6<Y2KM462]N\\r¥sF[VWK FO7?4ý:=DJIW:=VWýACB=IWD6AC2¢K4uN\D625:=257

A0 ∪ . . . ∪ AmD6B=<Y7Y:;FONa8=eýd>EoUôN\X_uNa46257Y^ B=mYAN\^2~ObHN:;FONa8=e:=25m

DJIWB=<fe\g64d>E^.d>CF[7Y^.ôNaF[7YôNaF9>EXY<?4C>E^,r ¥@XY<?>CKM25=XY257\b!KÆD6B[N\AdF9>EXY7ôNaF[7YôN.-FU>EXY<Y46798UACB=IWD6AC2 #DJI\8=NfKM2]N8=e@:=25m#XY<YmY:;F[I6bf8;7Yg64uN\A<fNLKM:=<Y7abCZWgG>:=25m#25X_S6?>CKRNEbDJILKM254646I:=25mS6^257Y4uN\D625:[N\ICg6DJILKM257Yg6462]eUDJIWB=<fe\g64ueg67?êu4625X98;mB=7YACS6B=7Y4C-X?>L8;4ueGr

¥@XY<?>CKM25=XY257\b62546g6S6A\X98=NôNaF[7YôNaFU>EXY<Y4uN2wB=7YACS6B=:h8=NôN8=e<Y7@:=IWueuN\B=g6<YIg6S6YIKM:=DJVW5467YZWI6r¥@627YAdF9>,<Yg67?êu4625ILK#N\467B=7YACS6B=7Y46X?>L8;46257:[eT8=N\ACC>xU:;FUK#IWB=<YIW467g6Iog6I,-K|IEg6VLK2546g6S6AWX?>L8;4d>EXO_rwB=<?>L8;B=<?>L8;^>:=25m`DJIW4625Y:=<Y7Y^S,D6B=<?>EA6PhN\g6ILKM2 r¸¹ºWá!ÓÈÇ Àu¼ 13257YXO_ 〈an〉n∈N

cmYg6<Y257ýXY2]eCZ\257Y^ h 25JIW4uN\XYXY257YZWI6r HIWAN\Y7Y^>2546g6S6Ap-X?>L8;46257\buY7Tg6]NAN\Yg67YZWIn ∈ N ôNa^>

an =

√5

2

[(1 +√

5

2

)n

−(

1−√

5

2

)n]

.

Ú NaFUK#I:=D6B[NfKMg6<Y25\buY7Tg6]Nn = 0

2n = 1

DJILKR>EY:=<LNB=VKM46IW=T<fN\XO_6IEg6<Y2 rcNCPiVWY^>F[7YB[N\<\buY7T<fN\XO_6IEg6<Y2 IW4uNg6]N

n2n+ 1

ruHIWAN\Y7Y^>WbcY7T<fN\XO_6IEg6<Y2!FON\ACY7Tg6]Nn+ 2

rN\SCK#N\Y^>04uNa8;D6257YB;KTbR?7Q525XY<YC>

a = 1+√

52

2b = 1−

√5

2

:[e-gGK|IWôN-D6257YB-KM2]N\:;F[AaN\^2wB=VKM4uN\462]N

x2 = x+ 1rJýg67?êu4625X98;2!ôN\^>

an+2 =√

52

(an − bn) +√

52

(an+1 − bn+1) =√

52

(an+1 + an)−√

52

(bn+1 + bn) =

=√

52

[an(a+ 1)− bn(b+ 1)] =√

52

[an · a2 − bn · b2] =√

52

(an+2 − bn+2)b

ñ%6n² û µ ¯ ? ¶C³[³v°³ : ®\°³ : õB³ µ ³pª¢¬Y®,¶ ? ¬ 8 [³XYIýA\IW6XY<?>g6IK|VEgACB=IWACS2546g6S6A\X?>L8;467YZWI6rHkQ>E:;FON\B=XY<?>p8;S6vFU>E5AWIýDcIK|IEPhN\v:=25mx4uNN\:[N\g6mÙ46g6S6AWX98;2HnpNaF[7YôNF9>EXY<Y46798?r1sNM<fN\A\IW6XY<Y7Y46257 D6B=<Y7Yg6:;FONLKM25^>`g6IK|VEgTFUKM257YB=g6<Y7Y462]NM< ¨MIW<Yg6<Y2]NCPiS ì bfKAdF[VWB;>E^

S6?>L8;7Y^><fN\B=VLKM46I 2546g6S6A\X98;2 b|8=N\A2B=7YACS6B=:h8;2 r ã ILK#VCgF[7Y4ÎWY7KR>EguNfK#N\:=25mg6<Y2KM4d>WrB=<?>uN\B=g6<Y25798g6IWA6PhN\g64C>E^ DcIEg6798;=XY25Sg6IýôNaF[7YôNaFU>EAC2!8;7Y:;FIW4p8;7Yg64uN\A46257Y<YcmYg64C>ý2ÎWY7d>ED6B=<?>EA6PhN\g67Y^.4uNvF[I6bY7S6g6ILK|IEg646257Y46257y N\AdF[SQ254dF[S625X?>L8;46257ICXY<?>CKM25:;F[7YZWIoÎWY7`d>Eg6IW=T<fNfKM25A6PhN\467T2!46257FOB;>CKM25N\5467\r

¿oÁh»C¹a¼ ºd» Ò Áh»õ 5Ì äÈÊT+Wd,[XP &;¢;)O ®ÓUQS. ,RV' ⇔ |X| < |N| = ℵ0

Îãx¾E¿ ¼ klD6B=ILK#N\gË÷Y^>I\<Y4uN\XY<Y7Y46257

An = 1, 2, . . . , n r|3g6ILK|IEg64625^>-gGK#NKR>E4625AN\462]NGr/⇒ 1#NdPiVWY^>WbCY7@<Y625VWB X 8;7Y:;FR:=A\IW6XY<YIW4d>WrJz\7Y=52 X = ∅ bCF[IF[7Y<fN38;7Y:;FRIEXY<>CKM25:;FONGrn,IWY7Y^>Q<fNaF[7Y^ D6B=<?>L8=e\\b¢Y7

X 6= ∅ b¢XY<?>E52 25:;F[4625798;7 n ∈ N b FON\AC257Y7 X ∼ An

rN\SdK#N\Y^>ý4uN8;D6257YB;KTbY7KR>E:;FON\B=XY<?>DJIWAN\<fN\\b!Y7g6]Nvg6ILK|IW5467YZWI

n ∈ N ôN\^>|An| < |N|

r¢IW46257?K#N\An ⊆ N

bwXY<?>E52 |An| ≤ |N| bJF[I /hg6<Y25mYAC2H¡ KM257YB=g6<Y7Y4625SQåGr ñ 1KR>E:;FON\B=XY<?>ýDJIWAN\<fN\\b!Y7g6]NoAaN\Yg67YZWIn ∈ N 4625725:;F[4625798;7yhS646A\X98=N f : N −→1−1 An

r¡¢I<fN\MDJIWAaN\Y7Y^>x2546g6S6A\X?>L8;46257=Lrã ]N

n = 1F[7Y<fN8;7Y:;FT:=DJ7Pi4625IW4uNoF[B;>CKM2]N\546257Y/hJI

NôNoKM25mYXY798@462538;7Yg67Y47Y57=-

^7Y4dF=1r|NCPiVWY^>F[7YB[N\<\b Y7p46257,25:;F[4625798;7xyhS646AWX98=Nf : N −→1−1 An

2MD6B=<?>ED6S6=f^>46257KMD6B=IW:;FLb Y725:;F[4625798;7

g : N −→1−1 An+1r# <fNCPiIWY7Y462]NQ2546g6S6AWX>L8;467YZWIKR>E4625AaNGb

Y7n + 1 ∈ B=46Z (g) r13257YX_ a ∈ N JmYg6<Y257 /Ø8;7YgG>E4ueO1525XY<Yue4uNaF[S6B[N\54ueGb FON\AaeY7

g(a) = n+ 1rc¨MIW<?K#N\Y^>vyhS646A\X98;m

h : N→ N \ a <fN\guN\4ueKM<YI\B[7?^

h(n) =

n

g6]Nn < a

n+ 1g6]N

n ≥ a.

Ú NaFUK|I:=D6B[NfKMg6<Y25\buY7#8;7Y:;FMFOI62 8;7YA\X98=NGruHIW46257?K#N\`yhS646A\X98=Ng′ = g (N \ a) 8;7Y:;F254a8;7YA\X98=eY/Ø8=N\A\IIWcXY25mYXY257s254a8;7YA\X98;2m1bdF[IyhS646A\X98=N

g′ h : N→ An

8;7Y:;F#254a8;7YAWX98=eY/Ø8=N\A\I<PiIWY7Y462573254a8;7YAWX98;2m1bdXYI38;7Y:;F#:=D6B=<Y7YXY<Y467`<3<fNCPiIWY7Y46257Y^ 2546g6S6A\X?>L8;4C>E^,rG¢ILK#IEPhN\462573:=25m4uNN\:[N\g6mTU46g6S6A\X98;2HnpNaF[7YôNaFU>EXY<Y46798MA\IW6XY<?>g6ILK#VCgr/⇐ 1sNCPiVWY^>WbcY7 |X| < |N| rwq|7Y<<Y^462798;:=<Y7Y462]NIWZWVW546IW=XY2 ÎWY7Y^>,D6B=<?>L8=e\\bJY7X ⊆ N bJ<fNaF[7Y^<fN\A6PhN\guN\^>WbwY74625725:;F[4625798;7254a8;7YA\X98=N f : N −→1−1 X

rwB=<?>ED6S6=f^>462573KMD6B=IW:;FfbGY7

X8;7Y:;FR46257YD6S6:;FU>v2Jg6]NAN\Yg67YZWI

n ∈ N ôN\^> X 6∼ An

r6wg67?êu4625SC-8;7Y^>vB=7YACS6B=7Y46X?>8;46257TyhS646A\X98;m

g : N→ Xr

Ö r33:;FON\5^>og6IK|IW5467a ∈ X /hÎWY4uNGb6JI

X 6= ∅ 1#2DJIEPiVWY^> g(0) = a0r

?Qxz yϕ(n) = ¬(∃f)(f : N→ An ∧ f

¡m 1− 1)

×Vò6587 ³d¯w³ µ ª5³ ñ%6 Å

¤Er3B=<?>ED6S6=Y^>WbY7n > 0

2 K#N\B;F[IW=XY2g(0), . . . , g(n − 1)

<?IW:;FNdP]>x8;S6o<Yg67?ên-4625ILK#N\467@IWB[N\<s:[eIW467@B=VWY467\r6¢IW46257?KRN\ g(0), . . . , g(n− 1) ∼ An

bE<fNaF[7Y^X 6= g(0), . . . , g(n−1) rdM:=FON\^>g6IK|IW5467 b ∈ X \g(0), . . . , g(n−1)2DJIEPiVWY^>

g(n) = br

¥sF[B=<?>EôN\4uNK F[7Y4Ê:=DcIW:=VWyhS646A\X98=N8;7Y:;FlB=VWY46ILK#N\B;F[IW=XY25ILKRNGb@XYI DJIW<YIW:;FON8;7K:=D6B=<Y7YXY<Y46IW=Xf2 <T<fNCPiIWY7Y46257Y^,r

Ù; < Íàx'YÖ r33g6ILK|IEg64625TD6B=<Y7Y<T2546g6S6AWX98;mT4uN\:;F[mYD6Sa8=e\XY7TFUKM257YB=g6<Y7Y462]NGr

/ NO11 + 5 + 9 + . . .+ (4n− 3) = n(2n− 1)

k/hB1

13 + 23 + . . .+ n3 = n2(n+1)2

4

g6]Nn ≥ 1

k/ X1

6|(n3 + 5n)k

/hgB1100n < 2n + 577

k/ 71

10n < (3 + (−1)n)n + 23r

¤Er313257YXO_ 〈an〉n∈N

JmYg6<Y257MXY2]edZW257Y^ h 25JIW4uN\XYXY257YZWI6rW3g6ILK|IEg64625\bWY7Mg6]NÀN\Yg67YZWIn ∈ N ôN\^> an+2 · an − a2

n+1 = (−1)n+1 rÚGr313257YXO_

BJmYg6<Y257g6ILK#IW54d>E^ 46257YD6S6:;FU>E^DJIEg6<Y625IWB=7Y^

Nb|4uNaF[IW^2]N\:;F

f4

yhS646A\X98=e < P(N)K P(N)

FON\AeGb`Y7g6]N0g6ILK|IW5467YZWIX ⊆ N

<fN\XO_6IEg6<Y2f(X) ⊆ X

rd13257YXO_ 〈An〉n∈N

JmYg6<Y257RB=7YACS6B=7Y46X?>L8;46257M<Yg67?êu4625ILK#N\4C>E^ÊXY2]eCZW257Y^<Y625IWB=VLK¯b

A0 = B

An+1 = An \ f(An).

/ NO1x3g6IK|IEg6462 8?b6Y7TXY2]eCZ 〈An〉n∈N

8;7Y:;F3<Y:;F[mYD6Sa8=e\X?>WbJXY<?>E52(∀n ∈ N)An+1 ⊆ An

r/hB1NCPiVWY^>`g6IEguNaF[A\ILK|I6baY7|g6]NMg6ILK|IW5467YZWIs46257YD6S6:;F[7YZWI

X ⊆ N <fN\XO_6IEg6<Y2f(X) 6= X

rc3g6ILK#ICg6462 8?b6Y7(∀n ∈ N)An 6= ∅

rñ r313257YXO_

UJmYg6<Y25746257YD6S6:;FU>E^<Y625IWB=7Y^,bJN

f : P(U) → P(U)FON\AaeyhS646A\X98=eGb

Y7ýg6]Nlg6ILK|IW54d>EX_X,Y ⊆ U

8;7Y=52X ⊆ Y

b F[If(X) ⊆ f(Y )

r#13257YX_〈An : n ∈ N〉 JmYg6<Y257TB=7YACS6B=7Y46X?>L8;46257<Yg67?êu4625ILK#N\4C>E^X?2]eCZW257Y^æ<Y625IWB=VK¯b

A0 = ∅

An+1 = f(An).

3g6ILK|IEg64625\buY7(∀n ∈ N)An ⊆ An+1

r

ñ%6n× û µ ¯ ? ¶C³[³v°³ : ®\°³ : õB³ µ ³pª¢¬Y®,¶ ? ¬ 8 [³Ý r313257YXO_

UJmYg6<Y257,g6ILK|IW54d>E^ <Y625IWB=7Y^,b46257YXO_

B ⊆ UIWB[N\<

f : U → Ur

13257YXO_ 〈An〉n∈N

cmYg6<Y257TB=7YACS6B=7Y46X?>8;4627<Yg67?êu4625IK#N\4d>E^XY2]eCZW257?^j<Y625IWB=VLK¯b

A0 = BAn+1 = An ∩ f [An]

.

/ NO1x¢IWAaN\\buY7(∀n ∈ N)An ⊆ B

r/hB13g6ILK#ICg6462 8?buY7#8;7Y=52

x ∈ B IWB[N\< f(x) = xb6F[I

x ∈ ⋂

n∈N Anr

/hX*1vkQ>EAaN\\b!Y7IEgGKMB=I\F[4uNx25^D652AN\X98=Nog6IxF[798@<D6S646AdF[S9/hB1s46257^S6:=2<fN.-XO_6IEg6<Y25\r

Ê++!

ê a? )ð.LK.ov+·IH 0/ |0/

B=<Y7Yg6:;FONLKM25^>F[7YB[N\<RAC255ANsF[B=S6g64625798;:=<?>EX_B=IW<YS6ÎLKRN\bWAdF[VWB=7RIWD6S6=XY25525=^>K|7K|XY<Y7Y=4625798;:=<?>GX_pB=IW<Yg6<Y2]NCPhN\XO_rN\XY<Y46257Y^>vIEgg6ILK|IEg6S¡ KM257YB=g6<Y7Y462]N#N\4dF[IWB[N.-9q|7YB=46:;FO7?254uNGb6IAdF[VWB;>E^%^VKM2¨-

525=^>K ¨MIW<Yg6<Y2]N\57`åGr6kQ>EA\IWB=<?>E:;FON\^>xK4625^j4uN\:;F[mYD6Sa8=e\X?>57YôNaFfrü »E½lÀ Çvâ Í ÏÀ Ò ÀuÐaÑ< À6¹ Å Ó!Á¾ ÉHË¢Ò ÓJÐdÁh» Å?Ç ÀÇ á ½wÐø£Ô+W&;U;e

X 6= ∅ +ÒRV+W&;U;eF : P(X)→ P(X)

ÊQ*ST+W&v;CRn)OU P a[Q *Ra!UQa.¿cUQST',^l+(∀A,B ∈ P(X))(A ⊆ B ⇒ F (A) ⊆ F (B)).

Y&Q%'+_=WRË+W& P &A0 ∈ P(X)

i#.),+W&;¿È¸.&F (A0) = A0.

ãx¾E¿ ¼ ¨MIW<?K#N\Y^>B=IEg6<Y2546m S = A ∈ P(X) : F (A) ⊆ A rR¢IW46257?K#N\F (X) ⊆ X

b6<fNaF[7Y^%B=IEg6<Y254uN S 8;7Y:;F346257YD6S6:;FONGrc13257YXO_

A0 =⋂

S.¢IWAaN\Y7Y^>WbuY7R8;7Y:;FMF[I<Y625VWBYb6AdF[VWB=7YZWI:=<YS6AaN\^>WbcXY<?>E2!Y7

F (A0) = A0

r g67?êu4625X98;2@D6B=<Y7YACB=I\8;S S6IWZWVW5462IW467YZWIlKR>E4625AaNGb#Y7

(∀A ∈ S)A0 ⊆ Ar#s57

yhS646A\X98=NF8;7Y:;FB=IW:=4ue\XfNGbKM25mYX /hA\IWB=<?>E:;FON8=e\XF[7Y<g67?êu462X98;2RB=IEg6<Y254d> S 1ôN\^>

(∀A ∈ S)(F (A0) ⊆ F (A) ⊆ A1rc|<?>E52

F (A0) ⊆⋂

S = A0,

<fNaF[7YÂ0 ∈ S

buXYIIW<Y4uN\XY<fNGbc?7F (A0) ⊆ A0

rã N\5798?bJg6]Ng6ILK#IW5467YZWI

A ∈ S ôN\^> F (A) ⊆ AbJ<fNaF[7Y^

F (F (A)) ⊆ F (A)r

s57sF[IIW<Y4uN\XY<fNGb6Y7F (A) ∈ S rEk÷:=<YXY<Y7YZWVW546IW=XY2ôN\^>

F (A0) ∈ SbGXYIDJICXY2]eCZdN

A0 =⋂S ⊆ F (A0)

buXYIC>uPiIg6IS6g6ILK|IEg646257Y462]NGr

ñ9!ø ð ª½ó_¶d³ : ¬ ? ¯ µ ª5® 8 õB³*´ý¯w±\²s±u¯ó\² ¿oÁh»C¹a¼ ºd» Ò Áh» â5Þ ÏxÀ ÒJÇ ¾c¹x»C¹ Ò¢Å?Ç »EÁ Ò Ð ^#E! ,Xc ,^lRV'OU;eÔS*ÊT+W ,[Qd,X

X+Y¿ P &;\^l+

|X| ≤ |Y | + |Y | ≤ |X| ¿ï |X| = |Y | Îãx¾E¿ ¼ <fNCPiIWY7Y462]NM2E¡ KR257YB=g6<Y7Y462]N@åGr ñ KM27Y^>WbLY7|25:;F[4625798=eRyhS646A\X98;7

f : X −→1−1 Y2g : Y −→1−1 X

ru1sN\:=<?>E^XY7Y57Y^8;7Y:;F3<Y4uN\57Y<Y257Y46257`62 8;7YA\X98;2h : X −→1−1

na Yr

¨MIW<?K#N\fN\^>vyhS646A\X98;mDJIWÎCXY4625XY<feF : P(X)→ P(X)

<fN\guN\4ueKM<YIWB=7Y^

F (A) = X \ g[Y \ f [A]].

ã ]Ng6ILK|IW54d>EX_A,A′ ⊆ X

D6B[NfKMg6<Y2K|7R8;7Y:;F34uN\:;F[mYD6Sa8=e\XY7B=IW<YS6ÎLK#N\46257%b

A ⊆ A′ ⇒ f [A] ⊆ f [A′]⇒ Y \ f [A] ⊇ Y \ f [A′]⇒⇒ g[Y \ f [A]] ⊇ g[Y \ f [A′]]⇒⇒ X \ g[Y \ f [A]] ⊆ X \ g[Y \ f [A′]]⇔ F (A) ⊆ F (A′)

b<fNaF[7Y^jyhS646A\X98=N

F8;7Y:;F3B=IW:=4ue\XfNGrJwZWICg646257`<T!7YôNaF[7Y^%ID6S646A\XY257T:;FONCP]>E^%25:;F[4625798;7

A0 ⊆ XbGFON\AC257TY7

F (A0) = A0b6XY<?>E52

A0 = X \ g[Y \ f [A0]].

B=<?>L8;^2 8;^>A1 = X \ A0 = g[Y \ f [A0]]

IWB[N\<B0 = f [A0]

2B1 = Y \B0

rÚ NaF9K|I:=DJIW:;F[B=<Y7YX\buY7g[B1] = A1

ruwg67?êu4625Sa8;^>F[7YB[N\<

h(x) =

f(x)

g6]Nx ∈ A0

g−1(x)g6]N

x ∈ A1.

z\7Y:;F@F[IoyhS646A\X98=Nvg6IW6B=<Y7IWACB=7Y=5IW4uNGbJJIA0 ∩ A1 = ∅ r¢IW4uN\gGF[I8;7Y:;F`62 8;7YA\X98=eGbJXYIK D6B=IW:;F9>:=DJIW:=VWpKR>E4625ANo<Ty N\AdF[SbcY762 8;7YA\X98=N\^2!:[eyhS646A\X98;7

f A0 : A0 −→1−1na B0IWB[N\<

g B1 : B1 −→1−1na A1

r

k ¨MIW<Yg6<Y2]N\57 Ý KM:=DcIW^46257Y525=^>I!7YôN\XY257v©`S6B[NaF[ILKM:=AC257YZWI,-;wIWB=4uN g 8=N\A\IIFUKM257YB=g6<Y7Y4625S<fN\Dc7?KM462]Na8=e\X?>E^25:;F[46257Y46257Ô/iKDJ7?KM4d>EXO_K#N\B=S646AaN\XO_B1H7Y57Y^7Y4dF[SôNaAp-:;>EôN\5467YZWIKXY<YmY=XY25ILKR>E^DJIWB=<fe\g6ACSrc1sN\g6:=<Y7YgJPXY<fN\:YbuC>xZWI:;yhIWB=^SJPiILKRN\\r

ü »E½lÀ Çvâ5Ì Ï çpË ¹À Ç ¾E¿ Å Ó!Á ß ¾c¹ Ò Ðø£Ô+W&;U;e 〈X,≤〉 Ê=Q*ST+W&ÃRV+W&O('," S*ÊT+W ,[Q&"UQS.;\*U+W ,Xc Ò*V ,[wS*aO.)O ,XZ#%RV',"Î«äÈ#.ºd¸T"Ð'EV ,R#O% *¿¢¸.&¯)p#*¸*%'Ðº#KÓUVU;e

A ⊆ X"$#

Q`%[w#%RV+WUQS.&RV+W&D`!d,[=R&¯XXÎEY&Q%'XS*ÊT+W ,[wS.&

X+_(RV+W& P &Ó&^_&"&RVï"$#.)*',"$#%^lRV'pÎ

s@%@=£T|fWz1!z|;Wz|¿@%©W£,.

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(∀A 6= ∅)((∀A,B ∈ A)(A 6= B ⇒ A ∩B = ∅)⇒ (∃S)(∀A ∈ A)|A ∩ S| = 1),

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AdF[VWB;>EXO_,K#N\B;F[IK#>E^257Y4625N\:[N\g6m ã IW6B=7YZWIo3DcIWB=<fe\g6AWILK#N\462]Nnb ^#î)p#*¸*!&i`! S*ÊT+W ,[=

X+_=WRË+W& P &$[Q&^#!U P #

RR#

X¿oi#.)p#Y¸.& 〈X,R〉 P &;S*ÊT+W ,[Q&"! pÊT[wS.&¯*V ,[wS*aO.)O ,XZ#%RV',"$Î

z\7Yg64C>E^<#<fN\:;F[IW:=ILK#N\!7YôNFOS©`S6B[NaF[ILKM:=AC257YZWI,-;wIWB=4uN8;7Y:;FFUKM257YB=g6<Y7Y46257#<|s¨-ZW7Y6B;>!254625IK|798?ba^VKM2]e\XY7\baY7MAN\YguNTD6B=<Y7Y:;F[B=<Y7Y5254625ILK#NsôN@uN\<Ym\rCn>DcIWAaN\Y7Y^>8;7Y:=<YXY<Y725464678;7YZWI<fN\:;F[IW:=ILK#N\46257\b|g6IFUKM257YB=g6<Y7Y462]NGb|AdF[VWB=7DJIa8=NfKM2hPiI:=25m,K ¨MIW<=-g6<Y2]N\57`åGr ¿oÁh»C¹a¼ ºd» Ò Áh» â é ^#! ,Xc ,^lRV'OU;e$S*ÊT+W ,[Qd,X

A+B"$#%"Ð'

|A| ≤ |B| ∨ |B| ≤ |A|.ãx¾E¿ ¼ z\7Y=52#AdF[VWB;>E<Y7<Y625IWB=VLK

A2B8;7Y:;FD6S6:;F9>WbF[IýF[7Y<fNv8;7Y:;F:=DJ7Pi4625IW4uN

KIEXY<?>CKM25:;FU>p:=DJIW:=VWr!NCPiVWY^>xKM25mYX\bJY7<Y625IWB;>A2B:[eo46257YD6S6:;F[7\r¨MIW<?K#N\Y^>

<Y625VWB

X = f : f8;7Y:;FMyhS646A\X98=epû ∧ g6IW^ (f) ⊆ A ∧ B=46Z (f) ⊆ B ∧ f 8;7Y:;F 1− 1,

Qf.hgÈ§.©Wxz§%z£*¥Ev**v.; ª ;*¥oxwlzz£.y ¥D£.y ¡m «£o ª* W©Wz|= £.y ÈW©W,v*£TTz° ª* ¡m|;e$xwv*=£.y ¦¨©i=w;¡ ¥E£,Ðþ~)¡ ª; ¤Æa¨.£,v*=|2z£Ti= £ ~¨ £.T¤gw±B ª W~©W|§.©WxzQ¡m|Z=¡mz|ïÓ%;l%z@¡m «§.©iQ«v*xzy xz yCýCz©Wy y.@K*y ©W~Q¢üz®Ez£,Í§p ª =xwIg=Ë¥ .)gZ¡m 2WÓ£.y z§%©W~;£,=£.y W©W,v*£.y (¡ xzVzú%zx (ª ©Wxz i=£*ylExoWzþ ª* ¡m|;Wo£.y ¢v. y u¢,v*Q*v*o§.©Wxzz§.©W;v*xzyg=ÿz®E*¤( y |Z\a¨.£ ª ¥ ¡ ¥ Ô z£ y ïv*©W.þy (¡v*jc,£.y ¥ ¡my!§%¡muw¥y Õa¨.£ ª ¥ ¡my_

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X:[e

DcIEg6<Y625IWB[N\^2u<Y625IWB=SA×B 1?rG¥@XY<?>CKM25=XY257\b X 8;7Y:;F#46257YD6S6:;F9>WrG¢IWAaN\Y7Y^>WbGY7s<Y625VWBS6DcIWB=<fe\g6AWILK#N\4d> 〈X,⊆〉 :=DJ7Pi462]N<fNCPiIWY7Y462]N!7YôNaF[S,©`S6B[NaF[ILKM:=AC257YZWI,-;wIWB=4uNGr13257YXO_ B=IEg6<Y254uNyhS646A\X98;2

L ⊆ XJmYg6<Y257PhN\6XYS6XO_67Y^ 2`46257YXO_

F =⋃Lr3

g67?êu4625X98;2¢:=S6^>,S6IWZWVW54625IW46798#KR>E4625ANvICgB[N\<YSbJY7F8;7Y:;FÌWZWB[N\462XY<Y7Y46257Y^ZWVWB=4d>E^

B=ICg6<Y254C>Lru¢IW<YIW:;FONa8;7T<fNaF[7Y^%:=D6B[NfKMg6<Y25\bcXY<?>

F ∈ X buXY<?>E52 b6XY<?>

• F 8;7Y:;F3<Y625IWB=7Y^jDuN\BRS6DJIWB=<fe\g6A\IK#N\4d>EXO_!b

• F 8;7Y:;FMyhS646A\X98=eGb

• yhS646A\X98=N F 8;7Y:;F3B=VWY46ILKRN\B;F[IW=XY25ILK#NGb

• g6I\^ (F ) ⊆ Ab

• B=46Z (F ) ⊆ Br

z\7Y=52x ∈ F bJF[I<Y4uN\XY<>Wb Y725:;F[4625798;7 f ∈ L bwFON\AC2HY7 x ∈ f rs57 f 8;7Y:;F`yhS646A\X98=eGbKM25mYX 8;798#7Y57Y^7Y4dFU>v:[eDuN\B[N\^2JS6DJIWB=<fe\g6A\IK#N\4d>E^2 b6XYIAWIW6XY<?>xg6ILK|VEgD6257YB;KM:=<Y7YZWI

DcIEg6D6S646AdF[Srq >xDJIWAN\<fN\Tg6B[S6Z\2!DJIEg6D6S646AdFMF[B=<Y7YuN:=D6B[NfKMg6<Y25\bcXY<?>

(∀〈x, y1〉, 〈x, y2〉 ∈ F ) y1 = y2.

3:;FON\5^><fNaF[7Y^Æg6IK|IW5467 〈x, y1〉, 〈x, y2〉 ∈ FrC¢IW46ILKM46257M<3g67?êu4625X98;2u:=S6^>S6IWZWVW¨-

4625IW46798`g6IW:;FON8;7Y^>f1, f2 ∈ L

b!FON\AC257Y7 〈x, y1〉 ∈ f1

2 〈x, y2〉 ∈ f2

r s57L8;7Y:;F

PhN\6XYS6XO_67Y^,bw<fNaF[7Y^f1 ⊆ f2

5S6f2 ⊆ f1

rwq|7Y<<Y^4625798;:=<Y7Y462]NoIWZWVW546IW=XY2¢ÎWY7Y^><fNCPiIW?>E\bwY7<fN\XO_6IEg6<Y2HD6257YB;KM:=<fNv7?K|7Y4CF[SuN\46IW=\rs57KRF[7YgG> 〈x, y1〉, 〈x, y2〉 ∈ f2

2DcIW46257?KRN\

f8;7Y:;F3yhS646A\X98=eGb6F[I

y1 = y2buXYIA\IW6XY<?>g6ILK#VCg,g6B=S6ZW257YZWIDcIEg6D6S646AdF[Sr

ã IWA6PhN\g646257K%F[7Y4:[N\^ :=DJIW:=VWDcIWAaN\<YSa8;7Y^>Wb Y7yhS646A\X98=NF8;7Y:;FoB=VWY46IK#N\B-

F[IW=XY25ILK#NGrkÏXY7Y5SQS6<fN\:[N\g646257Y462]NgGK|VEXO_IW:;FONaF[4625XO_QDJICg6D6S646AdF[VKÆKR>E:;FON\B=XY<?>l<fN.-SdK#N\?>E\bcY7

g6IW^(F ) =

⋃

f∈L

g6IW^(f)

IWB[N\< B=46Z(F ) =

⋃

f∈L

B=46Z(f).

NaF[7Y^ <,!7YôNaF[S©@S6BONFOILKM:=AC257YZWI,-;wIWB=4uNpKR>E4625AaNGbY725:;F[4625798;7x7Y57Y^7Y4dFôN\AC:;>O-ôN\54d>

hK <Y625IWB=<Y7S6DJIWB=<fe\g6A\ILKRN\4d>E^ 〈X,⊆〉 r¢¢IW46257?KRN\ h ∈ X

b!KM25mYXôN\^>g6IW^

(h) ⊆ A ∧ B[46Z (h) ⊆ BrJkQ>E4625ANv:;FOe\gbwY7^S6:=2H<fN\XO_6IEg6<Y25AdF[VWB;>E@<F[B=<Y7YX_

DcIW4625Y:=<?>EX_pDcI\F[7Y46X98=N\54d>EX_K#N\B=S646AWVLK 2 r• g6I\^ (h) = A

b3 @p==z|ZTzÆ@=©W.£. ª §.y z©W xz2yOv*©W.þy%£.y Æ ª ,¥xw=¡ Z y u= ª =wv*cxÈ£.y ¥ D ª ,¥xw y ux¢=©W.£ ª y z|fW©Wxzw¥y |Ô

ñ9u5

• B=46Z (h) = Bb

• g6IW^ (h) A ∧ B=46Z (h) Br

¢IWAaN\Y7Y^>Wb@Y7IW:;FONaF[462D6B=<?>EDuN\g67YA-8;7Y:;F46257YÎWY52~KR>Wrs`gG>EC> JILKM257Y^÷25:;F[462]NCP]>a ∈ A \ g6IW^ (h)

2b ∈ B \ B[46Z (h) bEF[I<Y625VWB h′ = h∪ 〈a, b〉 C>uPid>yhS646AWX98=eB=VW=-46ILKRN\B;F[IW=XY25ILK#e4uN\57Yfe\Xfeg6I<Y625IWB=S

Xrus57

h h′buXYID6B=<Y7YXY<?>ôN\AC:;>EôN\546IW=XY2

yhS646A\X98;2hr

|<?>E52!^S6:=2<fN\XO_6IEg6<?25TD6B=<?>E4uN8;^4625798 8;7Yg67Y4,<`gGK|VEXO_,D6257YB;KM:=<?>EXO_,K#N\B=S646A\VKTrz\7Y=52Hg6IW^

(h) = AbJF[I

h : A −→1−1 B2H<¡ KM257YB=g6<Y7Y462]NvåGr ñ KR>E4625ANGbwY7 |A| ≤ |B| rz\7Y=52!<fN\MB=46Z

(h) = Bb6F[I

h−1 : B −→1−1 Ab6XY<?>E52 |B| ≤ |A| buXYI4uNa57YfNCPiIg6ILKM257Y=\r

1sN<fN\A\IW6XY<Y7Y46257xD6B=<Y7Yg6:;FONfKM25^>lFUKM257YB=g6<Y7Y46257vS6IWZWVW5462]N8;e\XY7o¡!KM257YB=g6<Y7Y462]N ì r å2 ì r Ö ×Gr3<YS6DJ7Pi46257Y46257@8;7YZWIg6ILK#ICg6SQ4uNDcIEg6:;FONfKM257D6B=<Y7Yg6:;FONfKM25IW467YZWI,:=<YAC25XYSlDcI,-<YIW:;FONfKM25^>8=N\A\IF[B=S6g6467\bJN\57@254CF[7YB=7Y:=Sa8=e\XY7T?KM25XY<Y7Y46257\r ¿oÁh»C¹a¼ ºd» Ò Áh» â ÕÂÏV&;\=^l+

AP &;S*ÊT+W ,[Q&"RV+W&;;)O ®ÓUQS. ,RV',"À¿¢

A× A ∼ AÎ

ºdÓ!Á Ð,¼ ¾C¿¾!¼ Ë1 ã 7?êu4625Sa8;7Y^>x<Y625VWBXK4uN\:;F[mYD6Sa8=e\X?>:=DJIW:=VWr

X = f : f8;7Y:;FMyhS646A\X98=e ∧ g6IW^ (f) ⊆ A ∧ B=46Z (f) =

g6IW^(f)× g6IW^ (f) ∧

∧ f 8;7Y:;F362 8;7YA\X98=e r¢IW46IKM46257sDJIWAaNa<YSa8;7Y^>WbGY7@<Y625VWB|XY<YmY=XY25ILK|IS6DJIWB=<feag6AWILK#N\4C> 〈X,⊆〉 :=DJ7Pi462]N<fN.-PiIWY7Y462]Np!7YôNaF[S©`S6B[NaF[IKM:=AC257?ZWI,-;wIWB=4uNGrHNaF[7Y^.K <Y625IWB=<Y7

X25:;F[4625798;77Y57Y^7Y4dF

ôN\AC:;>EôN\54d>gr

HIWAN\Y7Y^>WbuY7`g6IW^(g) ∼ A

b6XYI<fN\A\IW6XY<?>g6ILK|VEgrGk÷FU>E^jXY7Y5S:=A\IWB=<?>E:;FON\^><TDJIW462Y:=<Y7YZWIy N\AdF[S¼/hAdF[VWB;>vKR>EôN\ZdNIW:=IW6467YZWIg6ILK|IEg6SB1rÏV&;\^l+

A ⊆ B = ∅ ∧ A ∼ B ∧ A ∼ A× A, A ∪B ∼ (A ∪B)× (A ∪B)r

¡¢7YB[N\<\b|ZWgG>Ed> | g6IW^ (g)| < |A| b F[IlKR>E6257YB[N\^><Y625VWB B ⊆ A \ g6IW^ (g)b FON\AC2

Y7B ∼ g6IW^ (g)

2RA\IWB=<?>E:;FON8=e\X,<DJILKR>EY:=<Y7YZWIly N\AdF[S-g6IW=,D6B=IW:;F[I<Y4uNa8;g6Sa8;7Y^>yhS646A\X98;m

g′Ig6<Y257Yg6<Y2546257xg6IW^

(g) ∪ B b4uN\57Yfe\Xfeýg6IQ<Y625IWB=S X 2#D6B=<Y7YgJPiS6fNa8=e\XfeyhS646A\X98;m

gbuXYID6B=<Y7YXY<?>,ôN\AC:;>EôN\546IW=XY2

gr

ñ9O6 ð ª½ó_¶d³ : ¬ ? ¯ µ ª5® 8 õB³*´ý¯w±\²s±u¯ó\²

Ê++!GF9

t0/ +Ü / v +|0/ +Ê.NM

F Ý àx #'Y'¡ ØÝ r / NO1 FON\A /hB1 FON\A /hX*1|FON\A /hgB1#46257/h7*1|FON\A /iy;1 FON\A /hZ!1|46257 /h_B1#46257

åGr / NO1p

/hB1r

ì r / NO1 ⇒ /hB1 ∧, ⇔×Gr / NO1

p′ ∧ q ∧ r /hB1p′ ∧ q ∧ r′ /hX*1

p ∧ q ∧ r′/hgB1(p ∧ q ∧ r′) ∨ ((p′ ∨ q′) ∧ r)

ó rRFON\AÖfø r / NO1|46257YDJIWD6B[NfKM467 /hB1#46257YDcIWD6B[NLKM467 /hX*1#46257YDJIWD6B[NfKM467Ö\Ö r3©@F[VWB[e`g6B=IWZWmRKM:=AaN\<fNCPhN\d>^2uHN\462c:=25IW:;F[B[NGbWZWgG>Ed>E^ <fN\Dd>CFONCPC8=eTIg6B=IWZWmM4uN

uN\ZW4uNO3Ö ¤ErM¡¢7\b6KAdF[VWB;>EXO_8;7Y:;F3DuN\B=<?>E:;FON255IW=

p/hg6ILK#VCg8;7Y:;F32546g6S6A\X?>L8;4C>Ë1r

Ö ñ rRkQ>E4625ANF[I<T25g67Y^DJI\F[7Y4dF[46IW=XY2 N\F[7YB=4uNaFU>CKR>x2AWIW4625S646A\X98;2 rÖLÝ rRkQ>E4625ANF[I<T:;>E^7?F[B=252!N\F[7YB=4uNaFU>CKR>x2!A\IW4625S646A\X98;2 rÖ åGr ¬p⇔ p|p, p ∧ q ⇔ ¬(p|q)⇔ (p|q)|(p|q) b62F[grÖ ×Grs`gG>pg67?êu4625Sa8;7Y^>p:=DcV\8;4625AgGKMSuN\B=ZWS6^7Y4CF[ILKR>f/h46Dr 1bcF[Ixg6]Ng6IK|IW54d>EXO_S6:;FON\5IW4d>EXO_K#N\B;F[IW=XY265IWZW2XY<Y4C>EXO_<Y^257Y464d>EXO_<YguN\4625ILKR>EXO_

p2q<YguN\46257

pq^IWY7TDcI\F[7Y46X98=N\546257`D6B=<?>L8;^ILKRN\TgGKM257`K#N\B;F[IW=XY2Wb ø 2 Ö rc¨MVWY467TA\IW^6254uN\X98;7

ñ9!² 5 ¯ > ±\²ª5®a¯waªHª¢² 8 ¶W³dhó\²¯¶cªH¯w±³d¯w³!·K#N\B;F[IW=XY2J5IWZW25XY<Y4d>EX_o<Y^257Y464C>EXO_v<YguN\4625ILKR>EX_

p2q:[eXY<?F[7YB;>WrG gGK#N:=DcV\8-

4625AC2 2 :[e46257YB=VLKM46ILK#N\Y467\bRZWgG>g6]NlDJ7?KM4d>EX_K#N\B;F[IW=XY2@5IWZW25XY<Y4d>EXO_<Y^257?464C>EXO_p2q8;7Yg646Ip<Y7<YguN\

p q 2 p q 8;7Y:;FD6BONfKMg6<Y2K|7\b¢Ng6B=S6ZW257y NCPi:=<?>CK|7\rÖLó r ã ]NAaN\Yg67YZWIo<DJIW<YIW:;FONCP]>EXO_ Ö ñ :=DJVa8;4625A\VK4uN\57Y?>S6<fN\:[N\g64625\bcY7N\5JI467=-

ZdN\X98=NEbcN\5cIN\F[7YB=4uNaFU>CK#N46257|8;7Y:;F3D6B[<?7Y<46257YZWIg67?êu4625ILK#N\54uNGr

F Ý àx #'Y'¡ Ö rA3 = A9, A4 = A5

ÚGrÀ/ NO1x = ∅

/hB1x = ∅, ∅ 5S6 x = ∅ 5S6 x = ∅, ∅

ñ r / NO1x = ∅ / B1

x = ∅Ý rÀ/ NO1

x = y = ∅/hB1x13257^oNDJIWD6B[NfKM46798ICg6DJILKM257Yg6<Y2/hJI

x = ∅, y = ∅ 46257ý:=DJ7Pi462]NK#N\B=S646A\VLK <fN\guN\462]NO1råGr313Dr ∅, ∅, ∅ rì rÀ/ NO1 1, 2 /hB1 1, 2×Gr /hgB1 0 ∪ [7,+∞)

/h7*1R

/hX*1(−∞, 0]/h_B1

R \ −2, 1, 2ÖWÖ rRk>E:;FNaB[X?<?>ýDcIWAaN\<fN\\bwY7g6]Nog6ILK|IW5467YZWIv<Y625IWB=S

Cbd8;7Y=52

A ⊆ C2B ⊆ C

bF[IA ∪B ⊆ C

rÖ ¤ErR£a¤Cb ÚGb ñ Ö ñ rÀ/hgB1

A = B/iy;1

A ⊆ B2B 6⊆ A

ÖÝ rÀ/ NO1 2 r −2,−1, 3 252 r 〈0, 3〉, 〈1, 3〉, 〈2, 3〉25252 r ∅, 〈−2, 3〉, 〈−1, 3〉, 〈−2, 3〉, 〈−1, 3〉/hB1|46257Ö åGrÀ/ NO1 2 r 〈1, ∅〉, 〈1, −2〉, 〈1, 4〉, 〈1, −2, 4〉 252 r ∅, 1/hB1|46257

ñ9 ÅÖ ×Grsn,IWY4uN<Y6S6g6ILK#N\

28 = 256B=VWY4d>EXO_,<Y625IWB=VLKTr613257@ôNKM=B=VEg4625XO_x<Y625IWB=S

8 r¤ ø r3©3IWB=<?>E:;FONa8=e\X`<`FON\J7Y5AC2!N\F[7YB=4uNaF9>CKR>vK#>uPhe\XY<fNa8=e\XY798?r¤aÚGr / NO1|<fN\XO_6IEg6<Y2 /hB1#46257`<fN\XO_6IEg6<Y2 /hX*1#46257`<fN\X_6ICg6<Y2

/hgB1|<fN\X_6ICg6<Y2 /h7*1#<fN\XO_6IEg6<Y2 /iy;1#46257`<fN\XO_6IEg6<Y2¤ ñ r

D = A \B, E = B

¤ Ý rD = A \ (B ∪ C)

¤aåGrsz\7Y=52A 6⊆ B

buF[IB=VLKM4uN\46257T46257TôNB=IW<?KM2]e\<fN\rJz\7Y=52A ⊆ B

buF[IB=IW<?KM2]e\<fN.-46257Y^8;7Y:;F3g6IK|IW54d>x<Y625VWB

Xb6FON\AC2!Y7

B \ A ⊆ X ⊆ Br

¤a×GrÀ/ NO1 ã ]Ng6ILK|IW5467YZWI7Y57Y^7Y4dF[SxôN\^>

x ∈ P(A) ∩ P(B)⇔ x ∈ P(A) ∧ x ∈ P(B)⇔ x ⊆ A ∧ x ⊆ B ⇔⇔ x ∈ A ∩B ⇔ x ∈ P(A ∩B)

/hB1 B=<?>EA6PhN\g67Y^,bGY7@462573ôNB[VLKM46IW=XY2w:[eg6ILK#IW54673gGKRN46257YD6S6:;F[7@<Y625IWB;>A2

BFON\AC257\b6Y7TfN\g67Y4p<T4625XO_46257`<fNLKM257YB[N:=25m@Kg6B=S6ZW25^,r

¤ ó r / NO1|46257 /hB1#46257Ú ø rÀ/ NO1#kl7T÷Y^>,g6ILK|IW54d>,7?57Y^7Y4CF

x ∈ P(A)rNaF[7Y^

x ⊆ ArJs57T<<fNCPiIWY7Y462]N

KM257Y^>WbY7A ⊆ B

bXY<?>E52D/h<D6B=<Y7YXO_6IEg64625IW=XY22546AC5S6<98;2m1MôN\^>x ⊆ B

rs57F[I<Y4uN\XY<?>WbuY7

x ∈ P(B)bEXYI6b6<YZWICg646257@<@g67?êu4625X98=e2546AC5S6<98;2 bCA\IW6XY<?>xg6ILK|VEgr

/hB1 kQ7T÷Y^>g6ILK#IW54d>7Y57Y^7Y4CFx ∈ A rak0F[7YgG> x ⊆ A

b\XY<?>E52 x ∈ P(A)r

0<fNCPiIWY7Y462]NKR>E4625AaNGbY7 x ∈ P(B)bXY<?>E52 x ⊆ B

r¡¢IQIW<Y4uN\XY<fNGbY7x ∈ B buXYI<YZWIEg646257`<Tg67?êu4625X98=e2546AC5S6<98;2 bGAWIW6X?<?>,g6ILK|VEgr

Ú Ö r31sN\57Y?>pD6B=<?>ED6S6=XY2546257KMD6B=IW:;FfbY725:;F[4625798;7<Y625VWBAbJFON\AC2HY7

A = P(A)2

B=IW<?K#N\?>E<Y625VWB x ∈ A : x ∈ x /iKyhIWB=^S657 x ∈ x D6257YB;KM:=<Y7 x 254CF[7YB=D6B=7=-F[Sa8;7Y^>s8=N\A\I7Y57Y^7Y4dFAbCNTg6B=S6ZW257\bW<YZWIEg646257M<M<fNCPiIWY7?46257Y^ 46257#KMD6B=IW:;Ffbf8=N\A\I

DJICg6<Y625VWBA1r

Ú\ÚGrÀ/hgB1(⇒)

NCPiVWY^>,46257KMD6B=IW:;FfbwY7A 6= ∅ 2 B 6= ∅ 2 A 6= B

rJk0F[7YgG>WbJ7Y<<Y^4625798;:=<Y7Y462]NIWZWVW546IW=XY2 b¢ÎWY7Y^>l<fNCPiIW?>E\bY725:;F[4625798;7v7Y57Y^7Y4dF

x ∈ A2

x 6∈ B r6s57 B 6= ∅ b6<fNaF[7Y^%25:;F[4625798;7 b ∈ B r FOe\gb6AWIWB=<?>E:;FON8=e\XT<T<fNCPiIWY7Y462]NGbôN\^> 〈x, b〉 ∈ A × B = B × A r!|<?>E52¢K:=<YXY<Y7YZWVW546IW=XY2 x ∈ B bJXYIxguNa8;7:=D6B=<Y7YXY<Y46IW=WbcAdF[VWB[NA\IW6XY<?>g6ILK|VEgrÚ ñ r313Dr

A = C 6= ∅ 2 B = ∅ r

ñ9!× 5 ¯ > ±\²ª5®a¯waªHª¢² 8 ¶W³dhó\²¯¶cªH¯w±³d¯w³!·Ú Ý r

(A×B)c = (Ac × Y ) ∪ (X ×Bc)

ÚWåGrM46257ÚW×Gr

A×B = x ∈ P(P(A ∪B)) : (∃a ∈ A)(∃b ∈ B)x = 〈a, b〉Ú ó r h IWB=^oN\54627o4625748;7Y=52|AWIWB=<?>E:;FON\^><xg67?êu4625X98;2|F[B=V\8;AC2#S6DcIWB=<fe\g6A\IK#N\46798Kg6S6XO_CS©`S6B[NaF[ILKM:=AC257YZWI6rcB[N\AdFU>EXY<Y46257FON\A$4oZWgG>^>E=525^>oIF[B=Va8;AaN\XO_,S6DJI,-

B=<fe\g6A\ILK#N\4C>EXO_l8=N\A\IlIW6257YAdFON\XO_B=IW<YB=VWY462]N8=e\X?>EXO_:;K#Ia8;7,KM:=DJVEPiB=<YmYg6467\r|¡¢7DcIEg6798;=XY2]N^IWY4uNDJIWZWIEg6<Y25`DcIEg6IW646257|8=N\AoK<fN\guN\4625S ó <¨MIW<Yg6<Y2]NCPiS ñ r

F Ý àx #'Y'¡ ;Ö rRkt46257YAdF[VWB;>EXO_KR>EDuN\g6AN\X_DJIW4625Y:=<Y7#ICg6DJILKM257Yg6<Y2G46257 :[e 8;7YgG>E4d>E^2GDcIWD6B[NLK2-4d>E^2 r/ NO1

x 6= y ⇒ x < y ∨ y < xk

/hB1 ¬∀n(¬2|n⇒ 3|n)k

/hX*1(∀x ∈ R)x2 ≥ 0

k/hgB1

x2 ≤ 0⇒ x = 0k

/h7*1n ≥ 2⇒ ¬(∃x, y, z ∈ N)xn + yn = zn

k/iy;1

(∀x ∈ Q)x2 6= 2k

/hZ!1(∀x ∈ R)(|x| = x ∨ |x| = −x) k/h_B1n ∈ N⇒ (∀k ∈ N)nk ∈ N k/h2m1(∃x ∈ Z)(∃k ∈ Z)x = 3k + 1

5S6(∃x ∈ Z) 3|(x− 1)

k/Ø8Q1

(∀A 6= ∅)(¬(∃x, y, z ∈ A)(x 6= y ∧ x 6= z ∧ y 6= z))k

/hAn1(∀x ∈ R)x · 1 = x

k/hm1

(∃y ∈ R)(∀x ∈ R)xy = xk

/h^1(∀n ∈ N)(∃k ∈ N) k > n

k/h4B1 ∀n(2 6 | n⇒ ∃k(2 6 | k ∧ k > n))

k/hI!1

(∃y ∈ R)x+ y = 0k

/hDB1(∀x ∈ R \ 0)(∃y ∈ R)xy = 1

k/n1

A 6= ∅ ⇒ (∃B)B Ak

/hB;1(∀A)(∃B)A B

k/h:;1

x ≥ 0⇒ (∀y < 0) y < xk

ñ9 ö/iF=1 ¬(∃x ∈ R)(∀y ∈ R) y ≤ x

k/hSB1

(∀A 6= ∅)(∃B)(B ⊆ A ∨ A ∩B = ∅) k/ Ë1(∀A ⊆ R)(A 6= ∅ ⇒ (∃B 6= ∅)B A)

k/iKo1

(∃x ∈ Z)(x ≤ 0 ∧ (∀y ∈ Z)(y ≤ 0⇒ y ≤ x))k

/V1 ∀A(A = ∅ ∨ ∅ ⊆ A)r

ÚGr / NO1 ã 5N3AaN\Yg6798H525X?<YC>TB=<Y7YXY<?>CKM25:;F[798J8;7?:=F525XY<YuNM4uNaF[S6B[N\54uN3ICg4625798!KM25mYAC:=<fN/ Na:ONaguNsB=X_625^7Yg67Y:[NO1r

/hB1x13257T2:=F[46257U8;7`4uNa8;^4625798;:=<fN525XY<YuNB=<Y7YXY<>CK325:;FONGr/ X1x9:;F[4625798;7`4uNa8;^4625798;:=<fN525XY<YuN4uNaF[S6B[N\54uNGr/hgB1 ã 5NAaN\Yg6798R525XY<YC>vXfNCPiA\ILKM2F[798#25:;FO462798;7`525XY<YuNg6I4625798#D6B=<Y7YXY2KM4uNGr/ 71, AaN\Yg6798x525XY<Yd>B=<Y7YXY<?>CKM25:;F[79846257YSa8;7Y^46798oÎWY4uNKR>EXY2]edZW4ue\D6257YB-KM2]N\:;F[7YAxAdK#N\g6B[NaF[ILKR>Wr

/iy;1xHIW^25mYg6<?>vgGKM257YôNB=VWY4d>E^2525XY<YuN\^2wB=<Y7YXY<>CK325:;FU>E^2!<fNLKM:=<Y7Î\Y4uN<Y4uN\57T÷YT525XY<YJm@KR>E^257YB=4ueGr

åGr313Drϕ(x) = (x = 0)

2ψ(x) = (x = 1)

buZWg6<Y257x ∈ R r

×GrJ/hgB1(∃x, y ∈ R)(x < y ∧ (∀q ∈ Q)(g ≤ x ∨ y ≤ q))/iy;1(∀x)(x ≥ 3 ∧ (∃y > x) y < 3)/hZ!1(∃x)(x 6= 2 ∧ (∃z) z < x) ∧ (∀y)(y = 2 ∧ (∀t) y ≤ t)

ó rÀ/ NO1 FON\A /hB1|46257 /hX*1#46257 /hgB1|FON\AÖ\Ö r31sN\57Y?>:=AWIWB=<?>E:;FON\,<g67?êu4625X98;2RAdK#N\4dF9>CêuANaF[IWB=VK%IWZWB[N\4625XY<YIW4C>EXO_2#¡ KM257YB-

g6<Y7Y462]NÚGr ñ rÖ ¤Er31sND6B=<?>EA6PhN\g

/ NO1ϕ(x) = (x > 0)

2ψ(x) = (x2 = −1)

b/hB1

ϕ(x) = (x = 0)2ψ(x) = (x 6= 0)

b/ X1

ϕ(x) = (x = 0)2ψ(x) = (x = 1)

rÖ ÚGr31sND6B=<?>EA6PhN\g

/ NO1ϕ(x, y) = (x 6= y)

/hB1ϕ(x, y) = (x = y)

Ö ñ r313257YXO_S =

⋃

i∈I Ai2P =

⋂

i∈I Air6k0F[7YgG>

/ NO1S = R P = ∅

ñp²Cø 5 ¯ > ±\²ª5®a¯waªHª¢² 8 ¶W³dhó\²¯¶cªH¯w±³d¯w³!·/hB1

S = (2, 4) P = 3/hX*1S = R P = ∅/hgB1S = [0, 3] P = [1, 2]/h7*1S = (−1, 2) P = 1/iy;1S = [0,+∞) P = [1, 2)/hZ!1S = (1

2, 9] P = (2, 4]/h_B1

S = [0, 2) P = ∅/h2m1S = 1 ∪ [2,+∞) P = ∅/Ø8Q1S = 〈x, y〉 ∈ R2 : x > 0 ∨ y < 0 ∪ 〈0, 0〉P = 〈x, y〉 ∈ R2 : x ≤ 0 ∧ y ≤ 0/hAn1S = 〈x, y〉 ∈ R2 : xy > 0 ∪ 〈0, 0〉P = 〈0, 0〉

Ö ×Gr3¢IEg6IW646257|8=N\AK<fN\guN\4625S ÖWÖ <T¨MIW<Yg6<Y2]NCPiSp¤ErÖLó rsnpN\^> ⋃ ∅ = ∅ buJI

x ∈⋃

∅ ⇔ (∃A ∈ ∅)x ∈ A,XYI6ba8=N\AoKM257Y^>Wb646257`<fN\X_6ICg6<Y2 g6]NfN\g6467YZWI

xr

1sNaF[IW^25N\:;F ⋂ ∅ d>uPiC>o<Y625IWB=7Y^ KM:=<?>E:;F[AC25X_<Y625IWB=VLKN/hIAdF[VWB;>E^ KM257Y^>WbEY746257`25:;F[4625798;7*1bGJIx ∈

⋂

∅ ⇔ (∀A ∈ ∅)x ∈ A,XYI<TA\IW57Y2G8;7Y:;F3D6B[NfKMgueg6]Ng6IK|IW5467YZWI

xr

¤ ø r313257YXO_Ap = x ∈ B : p|x g6]N p ∈ P r6k0F[7YgG>x^IWY4uNDJIWAaN\<fNa\bcY7

A = Ap : p ∈ P.

F Ý àx #'Y'¡ k 46257YAdF[VWB;>EXO_DJICg6D6S646AdFON\X_<fN\guN\ Ö 4 Ý DJIEguN\467 IEg6DcIKR257Yg6<Y2C46257:[e8;7YgG>E4C>E^2

DcIWD6B[NLKM4d>E^2 rÖ r / NO1

(∃t > 0)(∀x ∈ R) f(x+ t) = f(x)/hB1(∃M ∈ R)(∀x ∈ R) |f(x)| ≤M/hX*1(∀s > 0)(∃t > 0)(t < s ∧ (∀x ∈ R) f(x+ t) = f(x))

ñp²Vñ

/hgB1(∀x ∈ R)(∃y ≥ x) f(y) < g(y)/ 71(∃x ∈ R)(∀y ≥ x) f(x) ≥ 0/iy;1(∀M ∈ R)(∃x < M) f(x) = 0/hZ!1(∃x ∈ R)(∀y1, y2)(y1 ≥ x ∧ y2 ≥ x⇒ f(y1) = f(y2))5S6

(∃x ∈ R)(∃c ∈ R)(∀y ≥ x) f(y) = c

¤Er / NO1(∃x ∈ R) f(x) 6= f(x+ 1)/hB1(∃x ∈ R) f(x) > 1/ X1(∀M ∈ R)(∃x ∈ R) f(x) < M/hgB1(∃x1, x2 ∈ R)(x1 < x2 ∧ f(x1) > f(x2))

ÚGr / NO1(∃M ∈ R)(∀n ∈ N) |an| ≤M/hB1(∀m,n ∈ N)(m < n⇒ am < an) ∨∨ (∀m,n ∈ N)(m < n⇒ am > an)/ X1(∀M ∈ R)(∃n ∈ N) an ≥M/hgB1(∀n ∈ N)(∃m ≥ n) am > 0/ 71(∃n ∈ N)(∀m, k ∈ N)(n ≤ m ∧ m < k ⇒ am < ak)

ñ r / NO1 x ∈ R : f(x) < 0/hB1 x ∈ R : f(x) = 0/ X1 t ∈ R : t > 0 ∧ (∀x ∈ R) f(x+ t) = f(x)/hgB1 M ∈ R : (∀x ∈ R) f(x) ≤MÝ r / NO1

(∀n ∈ N)(∃x ∈ R) f(x) = an/hB1(∀n ∈ N) f(an) > 0/ X1 x ∈ R : f(x) = 0 ∧ (∃n ∈ N)x = an/hgB1 M ∈ R : (∀x ∈ R) f(x) ≥M ∧ (∃n ∈ N) an < M

ì r / NO1|46257 /hB1 FON\A /hX*1|FON\A /hgB1#46257 /h7*1|FON\A×Gr / NO1

(0,+∞)/hB1

(0,+∞)/hX*1 n2 : n ∈ N/hgB1

(0,+∞)/h7*1N+ /iy;1

Z/hZ!1 2 /h_B1N

/h2m1 P(N)

ó r31sN\57Y?>vB[I\<YDuNaF[B=<?>ETyhS646A\X98;m

F :∏

i∈1,2Ai → A1 × A2

<fN\guN\4ueKM<YIWB=7Y^F (f) = 〈f(1), f(2)〉 r

ñp²Gù 5 ¯ > ±\²ª5®a¯waªHª¢² 8 ¶W³dhó\²¯¶cªH¯w±³d¯w³!·ÖWÖ r / NO18;7Y:;F

1− 1bu46257|8;7Y:;FRU4uNOb6B=46Z

(f) = (7,+∞)/hB1x46257|8;7Y:;F1− 1

b646257|8;7Y:;FRU4uNOb6B=46Z(f) = [1,+∞)/hX*18;7Y:;F

1 − 1/hJI8;7Y:;F /h=XY25=57*1B=IW:=4ue\XfNO1b!8;7Y:;FU4uN¼/hJIlôNK@PhN\:=46IW=

ã N\B=JIWSV1/hgB1x46257|8;7Y:;F

1− 1ba8;7Y:;FRU4uN$/hJIôNK@PhN\:=46IW= ã N\B=JIWSV1

/h7*1x46257|8;7Y:;F1− 1

b646257|8;7Y:;FRU4uNOb6B=46Z(f) = [−1, 1]/iy;1v46257|8;7Y:;F

1− 1b646257|8;7Y:;FRU4uNOb6B=46Z

(f) = [−1,+∞)

ÖÝ rs0y N\AdF[Sb Y7yhS646AWX98=Nf : A → B × C

8;7Y:;Fo254a8;7YA\X98=eQ46257xKR>E4625ANGb Y78;798:=A6PhN\g6ILK|7TF[7YT:[e254a8;7YA\X98=N\^2 r

Ö ×Gr1−1

U4uNf [A] f−1[B]

/ NO1 46257 46257[2, 6) [−1, 1] \ 0/hB1 FON\A 46257(1

2, 2) [0, 2]/hX*1 46257 FON\A

N+ 〈0, 3〉, 〈1, 2〉, 〈2, 1〉〈3, 0〉/hgB1x46257 FON\A 2n : n ∈ N 2n+ 1 : n ∈ N2/h7*1x46257 FON\A n2 : n ∈ N Z× 0 ∪ 0 × Z/iy;1 46257 FON\AZ 〈n, n2〉 : n ∈ Z/hZ!1 FON\A 46257 〈4n,−4n〉 : n ∈ Z 0/h_B1 FON\A FON\A 〈x, x〉 : x ∈ R 〈x, x〉 : x ∈ R

¤ Ö r31sN\57??>B=IW<?K#N\?>E46DrHyhS646A\X98;m<fN\guN\4uepKM<YIWB=7Y^f(x) = x2 5S6g6ILK|IW54ue25464ueGbGAdF[VWB[N46257|8;7Y:;F3254a8;7YA\X98=eð/h46Dr

f : 1, 2 → 1 1r¤ Ý r31sN\57??>B=IW<?K#N\?>E46DrHyhS646A\X98;m<fN\guN\4uepKM<YIWB=7Y^

f(x) = x2 5S6g6ILK|IW54ue25464ueGbGAdFOV\BON46257|8;7Y:;F3254a8;7YAWX98=eN\462!:=S6Bh8;7YA\X98=eA/h46Drf : 1, 2 → 1, 2 b6FON\AaeY7

f(1) = f(2) = 11r

¤\åGr31sN\57??>vKR>EAWI\4uN\B=IW<YS6ÎK#N\46257`46257@KMD6B=IW:;Ffr¤\×Gr / NO1|46257 /hB1 FON\A /hX*1 FON\A¤ ó r ã ]Ng6IK|IW5467YZWI7Y57Y^7Y4dF[S

xôN\^>

x ∈ A ∩ f−1[B]⇔ x ∈ A ∧ f(x) ∈ B ⇒ f(x) ∈ f [A] ∧ f(x) ∈ B ⇔⇔ f(x) ∈ f [A] ∩B ⇔ x ∈ f−1[f [A] ∩B]

rw4uN\57Y<?257Y46257#D6B=<?>EA6PhN\g6SDJIWAaN\<YSa8=e\XY7YZ\I6bdY7MÎWY7R46257#C>EMB=VLKM46IW=XY26KR>EôN\ZdNB=IW<YDuNaF[B=<Y7Y462]NyhS646A\X98;2!46257`JmYgue\XY798M254a8;7YA\X98=eGr

ñp²y5Ú Ö r313257YXO_

y1, y2 ∈ Y2y1 6= y2

r¢IW46257?K#N\vyhS646A\X98=Nf8;7Y:;FU4uNOb¢F[Iý25:;F[4625798=e

x1, x2 ∈ XbFON\AC257xY7

f(x1) = y1

2f(x2) = y2

r ¥@XY<?>CKM25=XY257\bx1 6= x2

r A\IWB=IyhS646A\X98=N

g f 8;7Y:;F 1− 1b6F[I

g f(x1) 6= g f(x2)rus57@F[IIW<Y4uN\XY<fN

g6IWA6PhN\g646257\b6Y7y1 6= y2

b6XYI4uN\57YfNCPiIg6IKM257Y=\rÚW¤Er31sN\57Y?>oKR>EAWIW4uN\DcIEg6IW6467TB=IW<YS6ÎK#N\46257\ba8=NaAoKDJIWD6B=<Y7Yg64625^%<fN\guN\4625SrÚ\ÚGr / NO1xHIW46257?K#N\oyhS646A\X98=N

f Nb¢KM2mYXoKR>E:;FON\B=XY<?>B=IW<YDuNaF[B=<?>EvFON\Aae,yhS646AWX98;m

g : Z→ Zb6AdF[VWB[N@8;7Y:;F

1− 12!B=46Z

(g) ⊆ N r/hB1v46257Ú ñ rÀ/ NO1#13Dr

N× 0. /hB1R13DrD = 0 r

Ú Ý r313257YXO_x1, x2 ∈ X

2x1 6= x2

rR`gG>Ed>C>uPiIf(x1) = f(x2)

b F[IlKRF[7YgG>g f(x1) = g f(x2)

r#s57ýF[I8;7Y:;Fp46257YÎWY52K|7\b#cI25g67Y4dF9>EXY<Y46IW=x8;7Y:;FB=VWY46ILK#N\B;F[IW=XY25IK#NGrcNaF[7Y^jyhS646A\X98=N

f8;7Y:;F

1− 1r

13257YXO_xF[7YB[N\<x ∈ X rEk0F[7YgG> x = g(f(x))

2f(x) ∈ Y b6XY<?>E52wyhS646A\X98=N g 8;7Y:;FU4uNOr

F Ý àx #'Y'¡ AÚGrsn,IWY4uNDcIW:PiS6?>E:=25m^7?F[ICguexIWD625:=SB=7Y]N\X98;24uNx<Y625IWB=<Y7:=A\IW6XY<YIW4d>E^µD6B=<?>DJIWÎCX?>vFON\c7Y5AC2 b646Dr/ NO1

X = a, b, c

R =

a b ca × ×b × × ×c × ×

XY<?>E52R = 〈a, a〉, 〈b, b〉, 〈c, c〉, 〈a, b〉, 〈b, a〉, 〈b, c〉, 〈c, b〉 r

ñ r313257\bT1 \ T2

8;7Y:;F3B=7Y]N\X98=eD6B=<Y7YXY2KM<?KMB=I\F[4ueGrÖfø r / NO1 FON\A /hB1 FON\A /hX*1#46257 /hgB1#46257 /h7*1#46257

/iy;1 FON\A /hZ!1|46257 /h_B1#46257 /h2m1 FON\A /Ø8Q1 FON\Aw625IWB;>o255IWB[N\<YILK#7\r/ NO1 A0, A1, A2, A3, A4

b6ZWg6<Y257Ai = x ∈ N : (∃k ∈ N)x = 5k + i = [i]R

g6]Ni ∈ 0, 1, 2, 3, 4 k/hB1 P,N buZWg6<Y257 P = x ∈ Z : 2|x b N = x ∈ Z : 2 6 | x k

ñp²!6 5 ¯ > ±\²ª5®a¯waªHª¢² 8 ¶W³dhó\²¯¶cªH¯w±³d¯w³!·/iy;1 Ac : c ∈ R buZWg6<Y257 Ac = 〈c, y〉 : y ∈ R k/h2m1 Az : z ∈ Z b6ZWg6<Y257 Az = 〈n,m〉 ∈ N2 : n−m = z k/Ø8Q1 A1, A2

b6ZWg6<Y257A1 = −4,−3,−2,−1 b A2 = 1, 2, 3, 4 r

ÖWÖ r/hB1xRy ⇔ [x] = [y]

buZWg6<Y257[x]IW<Y4uN\XY<fNXY<YmY=XfNCPiA\ILKM2FON<T525XY<Yd>

xk

/hX*1 〈x, y〉R〈z, t〉 ⇔ y − x = t− z k/hgB1 〈x, y〉R〈z, t〉 ⇔ x2 + y2 = z2 + t2r

Ö ¤Er / NO1R4FON\Acb

S, T4v46257

/hB1 2 r 1, 2, 3, 4, 5 252 r 3, 15/hX*1vFON\AÖ ÚGr

R ∩ S <fNfKM:=<Y7#8;7Y:;F3B=7Y]N\X98=eB=VLKM46IK#N\Y46IW=XY2 buN R ∪ S ^IWY7@46257@C>E`B=7Y]N\X98=eB=VLKM46ILK#N\Y46IW=XY2 rÖ ñ r/hB1 2, 3, 7, 8/hX*1 A0, A1, A2

b6ZWg6<Y257A0 = 0, 5, 10, A1 = 1, 4, 6, 9, A2 = 2, 3, 7, 8/hgB1vFON\A

ÖÝ r / NO1R = R−1

/hX*1 〈2, 2〉, 〈2, 3〉, 〈2, 4〉, 〈3, 2〉, 〈4, 2〉/hgB1 A1, A2, A3, A4buZWg6<Y257

Ai = 〈x, y〉 : min(x, y) = i g6]N i ∈ 1, 2, 3, 4Ö åGrM46257Öì r X×Y /T = [x]R × [y]S : x ∈ X ∧ y ∈ Y Ö ×Gr

NN

/T = Ak : k ∈ N buZWg6<Y257 Ak = 〈an〉n∈N : a0 = k g6]N k ∈ NÖLó r

T = R ∩ S¤ ø r

[0]=∗ = A ⊆ N : A4:=A\IW6XY<YIW4d>

¤W¤ErRkt46257YAdF[VWB;>EXO_KR>EDuN\g6AN\X_DJIW4625Y:=<Y7#ICg6DJILKM257Yg6<Y2G46257 :[e 8;7YgG>E4d>E^2GDcIWD6B[NLK2-4d>E^2 r/ NO1

(∀x ∈ X)(∃y ∈ X) y < x/hB1(∃a ∈ X)(∀y ∈ X) y ≤ a/hX*1(∃a ∈ X)(∀x ∈ A)(a ≤ x) ∧ (∃a ∈ X)(∀x ∈ A)(x ≤ a)

ñp².9/hgB1

(∀x ∈ A)(∃y ∈ B)x < y/ 71 ¬(∃x ∈ B)(∀y ∈ A)x ≤ y/iy;1(∀x ∈ A)(∀y ∈ B)x ≤ y/hZ!1(∀x ∈ B)(∀y ∈ X)(y ≤ x⇒ y = x)

¤ ñ r / NO1 /hB1 /hX*1

/hgB1 /h7*1 /iy;1

/hZ!1 /h_B1/h2m1

¡¢£〈2n : n ∈ N ∪ 3,〉 bx y ⇔ y|x

〈N,〉, x y ⇔ x|y〈N× N,〉,〈x, y〉R〈z, t〉 ⇔ x = z ∧ y ≤ t

¤ Ý r 〈2n : n ∈ N ∪ 6,〉, x y ⇔ x|y¤a×Gr / NO1x§57Y^7Y4CFM4uN8;^4625798;:=<?>oF[IyhS646A\X98=N

f ∈ NN<fN\guNa4uNKM<YIWB=7Y^

f(n) = 0/ X?<?>E523yhS646A\X98=N:;FON\57pB=VLKM4uN<Y7YB=I!1r#§57Y^7?4CFU>^oN\AC:;>E^oN\5467p46257p25:;Fz-4625798=eGr

/hB1 fk : k ∈ N baZWg6<Y257 yhS646A\X98=N fk 8;7Y:;F<fN\guN\4uNsKM<YIWB=7Y^ fk(n) = k/hXY<?>E52

8;7Y:;F3F[IB=ICg6<Y254uNyhS646A\X98;2 :;FONCP]>EXO_B1r

F Ý àx #'Y'¡ Ö r / NO1

f : B → A, f(n) = 3n+ 1

ñp²C² 5 ¯ > ±\²ª5®a¯waªHª¢² 8 ¶W³dhó\²¯¶cªH¯w±³d¯w³!·/hB1

f : A→ B, f(n) = 5n

/hX*1f : A→ B, f(n) =

n2

+ 1ZWgG>

4|n−n

2

ZWgG>4|(n− 2)/hgB1

f : B → A, f(x) = 〈x, 1− x〉/h7*1

f : A→ B, f(x) =

x2

ZWgG>x = (1

2)ng6]NDJ7?KM467YZWI

n ∈ NxKD6B=<Y7YXY2KM4d>E^%KR>EDuN\g6ACS

¤Er / NO1z\7Y=52f : A −→1−1

na B2g : C −→1−1

na DbwF[IvyhS646A\X98=N

h : A× C → B ×D<fN\guN\4uNKM<YIWB=7Y^h(a, c) = 〈f(a), g(c)〉 8;7Y:;F362 8;7YA\X98=eGr/hX*1,z\7Y=52

f : A −→1−1na B

bJF[IvyhS646A\X98=NF : P(A) → P(B)

<fN\guN\4uNoKM<YIWB=7Y^F (A) = f [A]

8;7Y:;F362 8;7YA\X98=eGrñ rRk>E:;FNaB[X?<?>:=A\IWB=<?>E:;FON\|< F[7YZWI6bLY7

A∩B ⊆ A ⊆ A∪B 2 A∩B ⊆ B ⊆ A∪BIWB[N\<`<T¡ KM257YB=g6<Y7Y462]N#N\4CF[IWB[N.-9q|7?B[46:;F[7Y254uNErÝ r / NO1|46257 /hB1 FON\A /hX*1|46257/hgB1 FON\A /h7*1|46257 /iy;1|46257

ì rs¥@<Y4uN\X?<Y^>A =

⋃

n∈NAnr`¥@XY<?>CKM25=XY257\bsg6]NAaN\Yg67?ZWI

m ∈ N <fN\XO_6IEg6<Y2|Am| ≤ |A|

rwNCPiVWY^>4627TKMD6B=IW:;FfbJY7M8;7Y:;Fm0 ∈ N

buFON\AC257Y7 |Am0| = |A| rs57@KM257Y^>Wb6Y7 |Am

0| < |Am

0+1|buX?<?>E52 |A| < |Am

0+1|r D6B=<Y7YX?<Y46IW[\r

F Ý àx #'Y'¡ Ö rsz\7Y=52

f(x, y) = f(x′, y′)bHF[Iý<Y4uN\XY<?>WbY7

2x(2y + 1) = 2x′

(2y′ + 1)rs57

KRF[7YgG>2x|2x′(2y′ + 1)

22x

′ |2x(2y + 1)bJXY<?>E52

2x = 2x′ r g FOe\g8;S6:=<?>E6A\Ig6IW:;FON8;7Y^>WbuY7

x = x′2y = y′

rJNaF[7Y^æyhS646AWX98=Nf8;7Y:;F3254a8;7YA\X98=eGr

3:;FON\5^>vF[7YB[N\<g6ILK|IW5467n ∈ N rJ13257YX_ x JmYg6<Y2574uN89KM25mYAC:=<feo525XY<Yue4uNaF[SC-B[N\54ue@FON\AaeGbCY7

2x|(n+1)rG1M257YX_

y = 12(n+1

2x − 1)rG¢IW46257?K#N\ n+1

2x

^S6:=2cC>E525XY<?ue46257YDuN\B=<?>E:;FOeGbHF[I

y ∈ N rÚ NaFUK|I:=D6B[NfKMg6<Y25\bY7 f(x, y) = nbX?<?>E52

yhS646A\X98=Nf8;7Y:;F3:=S6Bh8;7YA\X98=eGr

×Gr313DrX = 4n : n ∈ N ∪ 2n+ 1 : n ∈ N r

ó r / NO1x1sN\57Y?>xB=IW<?K#N\?>ETyhS646A\X98;mf : S → Q3 <fN\guN\4ueKM<YIWB=7Y^

f(K(〈a, b〉, r)) = 〈a, b, r〉.si¬À£.y ª WÔxcmy z©iv*xzz£.y . ª W~©Wc|2~Qy_Oz¡mw¤( y p

¡m ¢ y ¥x §.y z©W x x y yxzþ uwv*£.y §.y z©W xz|2yO y ¥x,=|2yO£,;W.©i= £|2yOy p|xy

*Wp|x h

p|y

ñp² Å

/ X113257YX_ S cmYg6<Y257,B=IW<?K#N\fN\4d>E^t<Y625IWB=7Y^,r ã ]NýAaN\Yg67YZWIK ∈ S S6:;FON.-]N\^>D6S646AdF 〈aK , bK〉 ∈ K bJFON\AC2HY7 aK , bK ∈ Q rw¡¢7YB[N\<B=IW<?K#N\fN\^>yhS646AWX98;m

f : S → Q2, f(K) = 〈aK , bK〉./hgB1x13257YX_ S cmYg6<Y257TB=IW<?KRN\fN\4d>E^<Y625IWB=7Y^,rEk0F[7YgG>S =

⋃

n∈N

Sn,

ZWg6<Y257 Sn 8;7Y:;F`<Y625IWB=7Y^XY2]eCZWVLK525XY<Y4uNaF[S6B[N\54d>EXO_ýIvgJPiS6ZWIW=XY2 n rJs57Sn = Nn

buXYIKR>E:;FON\B=XY<fNg6I:=A\IW6XY<Y7Y462]Ng6IK|IEg6Sr/iy;1xq|7Y<-<Y^4625798;:=<Y7Y462]NIWZWVW546IW=XY2ÎWY7Y^><fNCPiIW?>E\b`Y7yhS646A\X98=N

f8;7Y:;F

/ =XY2[7*1B=IW:=4ue\XfNGr ¥@<Y4uNaXY<Y^>D6B=<Y7Y<Df

<Y625VWBs8;798D6S646AdF[VLK46257YXY2]eCZ,-PiIW=XY2 r6k0F[7YgG>xg6]NAN\Yg67YZWI

a ∈ Df25:;F[4625798=egGKR257TZWB[N\4625XY7

da = limx→a−

f(x)2ga = lim

x→a+f(x)

2!ôN\^>da < ga

ru13257YXO_Ia = (da, ga)

r6k0F[7YgG>vyhS646AWX98=NF : Df → Ia : a ∈ Df, F (a) = Ia

8;7Y:;F62 8;7YA\X98=eGbaN Ia : a ∈ Df8;7Y:;FB=IEg6<Y254ueMDuN\B[N\^2dB=IW<Phe\XY<?4C>EXO_D6B[<?7=-

g6<Y2]NCPiVK I\FUK#N\B;F9>EXO_r/hZ!1x13257YX_ |A| = ℵ0

2S = X ⊆ A : |X| < ℵ0

r6k0F[7YgG>

S =⋃

n∈N

Sn,ZWg6<Y257

Sn = X ⊆ A : |X| = n g6]Nn ∈ N.

s57 |Sn| ≤ |An| r/h_B1x13257YX_ S JmYg6<Y257B=IW<?K#N\fN\4d>E^µ<Y62IWB=7Y^,r -AN\Ygue52F[7YB[e T ∈ S KM2]e.-Y7Y^>F[B=<?>D6S646AdFU>pT1 , p

T2

2pT3bWAN\YgG>IÌW6SKM:=DcVEPiB=<YmYg64C>EXO_KR>E^257YB-

4C>EXO_buK:=DJI\:[V\pIWD625:[N\4d>x4uNDJIW4625Y:=<?>E^%B;>E:=S646ACS

pT1 pT2

pT3¤¥§¦

k0F[7YgG>xyhS646A\X98=Nf : S → Q6, f(T ) = 〈xT1 , yT1 , xT2 , yT2 , xT3 , yT3 〉,ZWg6<Y257

pTi = 〈xTi , yTi 〉g6]N

i ∈ 1, 2, 3 b\8;7Y:;F3254a8;7YAWX98=eGr

ñp²C× 5 ¯ > ±\²ª5®a¯waªHª¢² 8 ¶W³dhó\²¯¶cªH¯w±³d¯w³!·ÖWÖ r3¢IW46257?K#N\

N =g6IW^

(f) =⋃

n∈©W£.þ

(f)

f−1[n],

F[Ix<<fNCPiIW?7Y46257\b!Y7 | B=46Z (f)| < ℵ0

2(∀n ∈ N) |f−1[n]| < ℵ0

D6B=IK#N\g6<Y2g6I:=D6B=<Y7YXY<Y46IW=XY2 r

Ö ¤Er |A| ≤ ℵ0

/h<fN\SdKRN\?^>WbY7 A 8;7Y:;FvPhN\6XYS6X_67?^ Kj<?625IWB=<Y7xXY<YmY=XY25ILK#IS6DJI,-B=<fe\g6A\ILK#N\4C>E^ 〈P(N),⊆〉 1r

Ö ÚGrs|I4uNa89KR>EY798RD6B=<Y7Y525XY<fN\54uNGrÖ ñ rM46257Öì r / NO1z\7Y=52

f : A −→1−1na B

2g : C −→1−1

na Db6F[IyhS646A\X98=N

F : AC → BD, F (ϕ) = f ϕ g−1

8;7Y:;F362 8;7YAWXU8=eGr/hB1 h S646A\X98=N F : (A×B)C → AC×BC <fN\guNa4uNRKM<YIWB=7Y^ F (ϕ) = 〈ϕ1, ϕ2〉

bZWg6<Y257

ϕi8;7Y:;F

i- FOe:=A6PhN\g6ILKReyhS646A\X98;2

ϕb\8;7Y:;F362 8;7YA\X98=eGr

/hX*1 h S646A\X98;m F : (AB)C → AB×C g67?êu4625Sa8;7Y^>x4uN\:;F[mYD6Sa8=e\XYI6r

F (ϕ)(b, c) = ϕ(c)(b)g6]N

b ∈ B 2c ∈ C.

N\SdKRN\Y^>WbHY7xg67?êu4625X98=N8;7Y:;FDJIWD6B[NfKM4uNGbcIF (ϕ) : B × C → A

bNϕ(c) : B → A

r h S646AWX98=N F 8;7Y:;F362 8;7YA\X98=eGrÖ ×Gr3¢IEg6IW646257|8=N\Ao¡ KM257YB=g6<Y7Y46257 ì r ÖLó rÖLó rÀ/h2m1

RN ∼ (NN)N ∼ NN×N ∼ NN ∼ R¤ ø rsz\7Y=52

f : A −→1−1na C

b6F[I

f (A \B) : A \B −→1−1na f [A \B] = f [A] \ f [B] = C \ f [B].

s57f [B] ∼ B ∼ D

b6XY<?>E52C \ f [B] ∼ C \D /hcI

f [B], D ⊆ C1r

¤ Ö r3©3IWB=<?>E:;FON\^><y N\AdF[SbY7s8;7Y=52¢yhS646A\X98;7f, g : R → R

:[evXY2]eCZEPi72¢FON\AC257\bwY7f Q = g Q

bF[If = g

r¢¥@<Y4uN\XY<fNF[I6bY7yhS646A\X98=NF : A → RQ

<fNaguN\4uNKM<YIWB=7Y^

F (f) = f Q8;7?:=F3254a8;7?AWX98=eGr6s57

RQ ∼ RN ∼ R r

ñp² ö

¤\¤ErMU:;F[4625798;7 KM257Y57 AWIW46:;F[B=S6A\X98;2GB=IEg6<Y254d>I3FU>EXO_K@PhN\:=46IW=XY2]N\XO_rW¥sF[I 8;7Yg64uNs<|4625XO_rk257Y^>8;S6,<,<fN\guN\462]N ó /hgB1b Y7,<Y625VWB S :=A\IW6XY<YIW4d>EX_0XY2]eCZWVLK%525XY<Y4uN.-F[S6B[N\54d>EXO_8;7Y:;FD6B=<Y7Y525XY<fN\54d>Wr#3:;FON\5^>g6IK|IW54uel62 8;7YAWX98;m

ϕ : N −→1−1na S

rN\SdK#N\Y^>WbwY7:=AWIWB=I

f ∈ NN8;7Y:;F/h46257Y:=A\IW6XY<YIW4d>E^1@XY2]edZW257Y^525XY<Y4uNaF[SC-

B[N\54d>EXO_bJF[Ivg6]NAaN\Yg67YZWIn ∈ N 8;7YZ\IxIWcXY25mYXY257 f 0, 1, . . . , n 8;7Y:;F`:=A\IWC-XY<YIW4d>E^µXY2]eCZW257Y^525XY<YQ4uNaF[S6B[N\54d>EXO_ /Ø8;7Y:;FF[I

n + 1D6257YB;KM:=<?>EX_lKR>EB[N\<YVK

XY2]eCZWSf1r

ã ]NAN\Yg67YZWIf ∈ NN

B=IW<KRN\Y^>x<Y625VWB

Af = f 0, 1, . . . , n : n ∈ N.¥@XY<?>CKM25=XY257\b

Af8;7Y:;FD6B=<Y7Y525XY<fN\54d>TIWB[N\<!8;7Y=52

f 6= gbLF[I

Af 6= AgrW|I3KM25mYXY798?b

8;7Y=52k ∈ N 8;7Y:;F34uNa8;^4625798;:=<fe525XY<YueFON\AeGbuY7 f(k) 6= g(k)

b6F[I

Af ∩ Ag = f 0, 1, . . . , n : n < k/Ø8;7Y=52

k = 0bEF[I

Af ∩Ag = ∅ 1bG<fNaF[7Y^ Af ∩Ag8;7Y:;FR<Y625IWB=7Y^æ:=A\IW6XY<YIW4d>G^,r

wg67?êu4625Sa8;^>oF[7YB[N\<A = ϕ−1[Af ] : f ∈ NN.

¢IW46257?K#N\

f 6= g ⇒ Af 6= Ag ⇒ ϕ−1[Af ] 6= ϕ−1[Ag]/hcI

ϕ8;7Y:;F362 8;7YA\X98=eO1

,

F[I |A| = |NN| = c

rc¢IW4uN\gGF[I |ϕ−1[Af ]| = |Af | = ℵ0IWB[N\<

|ϕ−1[Af ] ∩ ϕ−1[Ag]| = |ϕ−1[Af ∩ Ag]| = |Af ∩ Ag| < ℵ0,

XYIAWIW6XY<?>A\IW46:;F[B=S6AWX98;m\r¤aÚGr ã ILK|VEgF[7YZWIy N\AdF[SxKR>EAWIWB=<?>E:;F[Sa8;7A\IW46:;F[B=S6A\X98;mB=7YACS6B=7Y46X?>L8;4ue%jr

3:;FON\5^>g6ILK|IW54ue62 8;7YA\X98;mϕ : N→ N× N r6N\SdK#N\Y^>4uN8;D6257YB;KTbCY7 8;7Y=52

ϕ1 : N→ N8;7Y:;FoD6257YB;KM:=<feQ:=A6PhN\g6ILK#eyhS646AWXU8;2

ϕbF[IQg6]NAN\Yg67YZWI

n ∈ N^oN\^> |ϕ−11 [n]| = ℵ0

/hcIϕ8;7Y:;F62 8;7YA\X98=eO1r wg67?êu4625Sa8;7Y^>pF[7YB[N\<B=7YACS6B=7Y4C-

X?>L8;46257TXY2]edZ525XY<Yx4uNaF[S6B[N\54C>EXO_ 〈an〉n∈N

r13257YXO_

a0 ∈ A0

cmYg6<Y257g6ILK|IW5467\r¢NCPiVWY^>pF[7YB[N\<\b!Y7^oN\^>8;S6KR>E6B[N\467525XY<Yd>

aig6]N

i ≤ nrGk0F[7YgG>xKR>E6257YB[N\^>

an+1 ∈ Aϕ1(n+1) \ ai : 0 ≤ i ≤ n.

¬Yy uw¥(¡2i ª y ¥ ª £ W©W ª ¥ ¡ ¥ Z¡m « xwvxzy = I.

ñ Å ø 5 ¯ > ±\²ª5®a¯waªHª¢² 8 ¶W³dhó\²¯¶cªH¯w±³d¯w³!·z\7Y:;F F[I3<fNfKM:=<Y7|ÎWY52~K|7\bfJI3<Y625VWB!< AdF[VWB=7YZWIMKR>E6257YB[N\^>

an+1b=8=N\AWIMB=VWY4625XfN

<Y625IWB=S46257Y:=A\IW6XY<YIW467YZWI2`:=A\IW6XY<YIW467YZWI6b8;7Y:;F,<fNfKM:=<Y7l46257YD6S6:;FU>Â/ N4uNLK|7?F46257Y:=A\IW6XY<YIW4d>Ë1rwg67?êu4625Sa8;^>oF[7YB[N\<Tg65NAaN\Yg67YZWI

n ∈ N

Bn = ai : i ∈ ϕ−11 [n].

N\SdK#N\Y^>WbCY73<MAWI\46:=F[B=S6A\X98;2cKR>E4625ANGbCY7MKR>EB[N\<?>XY2]edZWS 〈an〉n∈N

:[eDuN\B[N\^2B=VWY467\rNaF[7Y^,buDJIW46257?K#N\ |ϕ−1

1 [n]| = ℵ0bcF[I |Bn| = ℵ0

r ã N\5798?bcg6]Ng6ILK#IW¨-4d>EXO_

n,m ∈ N bd8;7Y=52 n 6= mbcF[I

Bn ∩ Bm = ∅ bcJI ϕ−11 [n] ∩ ϕ−1

1 [m] = ∅ rkQB=7Y:=<YXY257\b8;7Y=52ai ∈ Bn

bF[Iϕ1(i) = n

bXY<?>E52Í/h<xAWIW46:;F[B=S6A\X98;2m1ai ∈ An

rNaF[7Y^

Bn ⊆ Anr

F Ý àx #'Y'¡ Ö rÀ/h7*1 AWIWB=<?>E:;FON\<Y7T:=XO_67YôNaF[Sϕ(0) ∧ ϕ(1) ∧ ϕ(2) ∧ ϕ(3) ∧∧ (∀n ≥ 1)(ϕ(2n)⇒ ϕ(2n+ 2) ∧ ϕ(2n+ 1)⇒ ϕ(2n+ 3))⇒⇒ (∀n ∈ N)ϕ(n)

r¤Er ã ]N

n = 02n = 1

8;7Y:;F3g6IW6B=<Y7\rwNCPiVWY^>vY7an+2 · an − a2

n+1 = (−1)n+1 r|_6XY7Y^>0DcIWAaN\<fN\\bRY7an+3 · an+1 − a2

n+2 = (−1)n+2 r#s57<ý<fNCPiIWY7Y462]N2546g6S6A\X?>L8;467YZWIKR>E4625AaNGbuY7an+3 · an+1 − a2

n+2 = −(an+2 · an − a2n+1)⇔

⇔ an+3 · an+1 − a2n+1 = a2

n+2 − an+2 · an ⇔⇔ an+1(an+3 − an+1) = an+2(an+2 − an)⇔⇔ an+1 · an+2 = an+2 · an+1,XYIQAWIW6XY<?>g6ILK|VEgACB=IWACS2546g6S6A\X?>L8;467YZWI /hIW:;FONaF[462]NB=VLKM46ILKRN\Y46IW=,KR>EA\I,-B=<?>E:;F[Sa8;7g67?êu4625X98;mTXY2]edZWS h 25JIW4uNaXYXY257YZWI!1r

Ý rÀ/hX*1TkQ>E:;FON\B=XY<?>Wbd>B = U

IWB[N\<oyhS646A\X98=Nfd>uPhN62 8;7YAWX98=eGbN\57o46257v25g67Y4C-

FU>EXY<Y46IW=XY2]eGr¢k0F[7YgG>ôN\^>(∀n ∈ N)An = U

bHXY<?>E52 ⋂

n∈NAn = UbHN\57

25:;F[4625798;7x ∈ U b6FON\AC2Y7 f(x) 6= x

r

wstęp do matematyki b jan kraszewski

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