(1) , ., .

(2) P={a,b,c,} Q={xx }

(1) A B

A B ,AB .,AB AB A B . AB xA , xB

(2) A B , A B ,A=B . A=B AB,BA

* * A A * U

A={a1,a2,a3,,an}

(1)A =2n ()(2)A =2n-1()(3)a1,a2,,am (mn)

=2n-m ()(4)a1,a2,,ak (kn) =2n-k ()(5)A s =nCs()(6) s A =n-1Cs-1(a1+a2++an)

(7)A

=1nC1+2nC2++nnCn=n2n-1 ()(8)A

=2n-1(a1+a2++an)

(1)(Powerset) A

2A orP(A) .,2A=P(A)={XXA}(2)(CartesianProduct)aA,bB (a,b) AB={(a,b)aA,bB} .

(1) AB={xxA orxB}(2) AB={xxA andxB}(3) A-B={xxA andxB}=ABc

(4) Ac={xxU andxA}=U-A

(1)n(AB)=n(A)+n(B)-n(AB)

(2)n(ABC)=n(A)+n(B)+n(C)-n(AB)-n(BC)-n(CA)+n(ABC)

(3)n(ABc)=n(AB)-n(B)=n(A)-n(AB)(4)n(AcBc)=n(U)-n(AB)

(1)AB,BC AC(2)AB AcBc

(3)AB AB=A,AB=B(4)(AB) A (AB)

(1) AB=BA,AB=BA(2) A(BC)=(AB)C,A(BC)=(AB)C(3) A(BC)=(AB)(AC)

A(BC)=(AB)(AC)(4) (AB)c=AcBc,(AB)c=AcBc

(5) A(BA)=A,A(BA)=A(6) (Ac)c=A,((Ac)c)c=Ac(7)

AAc=U,A=A,AAc=,A=

AU=U,AU=A,c=U,Uc=

Ak:k (1)AaAb=A a,b

(2)AaAb A a,b

(1) :, (2) : p(x) p(x)(3) p(x): (3) P(x): p(x) P(x)={xp(x),xU}

p(x),q(x) P,Q ,(1)p(x) q(x) PQ(2)p(x) q(x) PQ(3)p(x) Pc

(4)p(x) q(x) PcQ

p(x),q(x) P,Q ,(1)p(x) q(x) ,PQ.p(x) q(x) ,q(x) p(x) (2)p(x) q(x) q(x) p(x) ,PQ,QPp(x) q(x)

p q

q p

q p

p q

A a,b ab=cA A .

a+bR abR abR (a+b)+c

=a+(b+c) (ab)c=a(bc)(ab)c

=a(bc) a(b+c)=ab+ac . a+0=a,(0) a1=a(1) ae=ea=a -a 1a ax=xa=e

(1),:A+(B-C)=A+B-C,A-(B+C)=A-B+C(2),: (3): .

. (2x3+x2-4)(x-3)

2x3+x2-4=(x-3)(2x2+7x+21)+59

(1)(ab)2=a22ab+b2

(2)(a+b)(a-b)=a2-b2

(3)(xa)(xb)=x2(a+b)x+ab(4)(ab)3=a3b33a2b+3ab2

(5)(ab)(a2ab+b2)=a3b3

(6)(a+b+c)(a2+b2+c2-ab-bc-ca)=a3+b3+c3-3abc(7)(a+b+c)2=a2b2+c2+2(ab+bc+ca)

(1)ma+mb+mc=m(a+b+c)()

(2)a22ab+b2=(ab)2()(3)a2-b2=(a+b)(a-b)(4)a3b3=(ab)(a2ab+b2)(5)x2+(a+b)x+ab=(x+a)(x+b)(6)a4+a2+1=(a2+a+1)(a2-a+1)(7)a3+b3+c3-3abc=(a+b+c)(a2+b2+c2-ab-bc-ca)

= 12(a+b+c){(a-b)2+(b-c)2+(c-a)2}

(8) X

2

x2=X

()x

ax+b=0 x a=0,b=0

ax2+bx+c=0 x a=0,b=0,c=0

ax+by+c=0 x,y a=0,b=0,c=0

ax+b=a/x+b/ x a=a/,b=b/

ax+by=a/x+b/y x,y a=a/,b=b/

(1) ()=() (2) ()()=0 f(x),g(x) (3) .(4) f(x) (x-a) f/(a)x+f(a)-af/(a), f/(a)=f(a)=0

(1) f(x) (x-a) Q(x), R f(x)=(x-a)Q(x)+R(2)f(x) x-a f(a)

(3)f(x) ax+b f(- ba)(4)xf(x) x-a af(a)(5)f(x)g(x) x-a f(a)g(a)

(6)f(x)+g(x) ax+b f(- ba)+g(- ba) (1)f(x) x-a 0 f(x) x-a . f(a)=0 f(x)=(x-a)Q(x)

(2)f(x) (x-a)(x-b) f(x) x-a x-b . f(x)=(x-a)(x-b)Q(x) f(a)=0,f(b)=0

(3)f(x) (x-a)2 f(x)=(x-a)2Q(x) f(a)=0 f(x)=(x-a)Q/(x) Q/(a)=0

(1)1 (2)1 (3)2 A=BQ+R .

(4)n xn-an x-a . xn+an x+a .

(1)A=BQ A B ,B A (2)A=apbqcr() A N=(p+1)(q+1)(r+1) A S=(a0+a1++ap)(b0+b1++bq)(c0+c1++cr)

(1) n n! (2) 2! (3) 3! (4) 4! (5)100 4 (6)1000 8 (7)3 3 (8)9 9

(GCM) (LCM)

A=aG,B=bG .(,a,b G )

(1)L=abG= ABG(2)AB=abG2=LG(3) . Euclid

(4) .

Euclid A B R , A B B R . .

(1) AB =ACBC (C0) (2)

AB =

ACBC (C0)

(3) ABCB AC (B0) .(4)

AB+

CB=

A+CB

(5) AB-CD=

AD-BCBD (6)

AB-

CB=

A-CB

(7)AB+CD=

AD+BCBD (8)

AB

CD =

ACBD

(9) ABCD=

ADBC(C0) (10)

AB =

CD AD=BC

(1) (2) ()

CAB=

CB-A(1A- 1B)

(3) --- AB =Q+

RB (Q ,R )

(4) ABCD= AB

CD=

AB

DC=

ADBC

(1)a:b= c:d ab=cd ad=bc

(2)a:b:c= x:y:z ax=by=

cz

(3) ba=

dc=

fe=

b+d+fa+c+e=

pb+qd+rfpa+qc+re

(4) . x . =k .

(1)a0 ,a a(2) n an )n a )n a

(3) a2=a={a0: aa0:-a (4) 3 a3=a

(1)a0,b0 a0,b0 a0,b0 , ab= ab(2)a0,b0 , ab=- ab

(3)a0,b0 a0,b0 a0,b0 , ab=ab

(4)a0,b0 , ab=-ab

(1) ca+ b=

ca- b(a+ b)(a- b)=

c(a- b)a-b

(2) c3 a3 b=c(3 a2 3 a3 b+3 b2)

ab ()

(1) a+b+2 ab= a+ b(,a0,b0)(2) a+b-2 ab= a- b(,ab0)

(1)a+b m=0 a=0,b=0(2)a+b m=c+d m a=c,b=d(3)a+ m=b+d n a=b,m=n

(1) a+b,ab (2)x=a+b2 ,f(x) ()2=()2

(2)=0 f(x)=(2)Q(x)+==0

(3) a, b a b -a

(1) : -1 i .

i , i2=-1(i= -1)(2)in : 4

in={ i (n=4k+1)-1 (n=4k+2)-i (n=4k+3)1 (n=4k)(3) :z=a+bi z a0,b=0

z {a=0,b0 a0,b0

(1)a+bi=0 a=0,b=0(2)a+bi=c+di a=c,b=d

(1)z=a+bi z=a-bi(2) ,

= , = , ( )= (3) = , =-(4) . :(a+bi)+(a-bi)=2a :(a+bi)(a-bi)=a2+b2

(1) 1a+bi=a-bi

(a+bi)(a-bi)=a-bia2+b2

(2) a+bic+di=(a-bi)(c-di)(c+di)(c-di)=

(ac-bd)-(ad+bc)ic2+d2

x=a+bi

x=a+bi =0

-story

(1)x3=1 2++1=0,3=1,,+=-1,=1,2=

(2) 3=-1 2-+1=0,3=-1,,+=1,=1,2=-

(ax=b)

(1)a0 ,x= ba

(2)a=0 ,{b0 0x=b()-- .b=0 0x=0()-- . ( )

(1) =0 x .(2) x .(3) .

([])

(1)x

(2) [x]=n nxn+1(3) [x]=x-h(,0h1)(4) [x+n]=[x]+n(,n )

(ax2+bx+c=0(a0)) (1) (x-)(x-)=0 x= ,x=

(2)

ax2+bx+c=0 x= -b b2-4ac

2a

ax2+2b/x+c=0 x= -b/ (b/)2-ac

a(3) 0 0 . i . .

xO

y

-1

-1

1

1

2

2 3

(1)ABC=0 A=0 B=0 C=0 2 :x2=t , A2-B2 (2)--x4+3x3-5x2+3x+1=0

. x+ 1x=t .

. t x+ 1x . x .

x . (x+1)()=0 ()=0 . x -1 .

(3) f(x)=0 p+qi p-qi f(x)=0 p+q m p-q m .

(1)2 ,3 ,, .

(2)

{ax+by+c=0a/x+b/y+c/=0 : .

aa/=

bb/=

cc/

: .

aa/=

bb/

cc/

(1){12 1 2 (2){22 { {12

1 2

(1) ( )( )= (2) A2+B2=0,A+Bi=0,A+B=0

A=0,B=0

ax2+bx+c=0(a,b,c ,a0) D=b2-4ac

(1) D0 {D0: D=0:(2) D0: ( D0) ( D0)

D ?(1) .D

(2) f(x,y)=0 , x,y . D0 (x-a)20 a .

(3),,x . ,D=0

(4) D D=0

(5) . D0

(6)3 .(x-a)(2)=0 2=0 D0 .

(1)ax2+bx+c=0(a0) ,

+=- ba,=ca,-=

D

(2)ax3+bx2+cx+d=0(a0) ,,

++=- ba,++=ca,=-

da

(3)a0xn+a1xn-1++an=0 n 1,2,,n

1+2++n=-a1a0,(k

)=(-1)kaka0

(1), 2

(x-)(x-)=0 x2-(+)x+=0(2),, 3 (x-)(x-)(x-)=0 x3-(++)x2+(++)x-=0

(3){+=x=y , 2 t2-xt+y=0

(1) .(2) .

.(3) f(x)+g(x)i=0 f(x)=0,g(x)=0 .

(1)

(2) D0 .

ax2+bx+c=0,a0 ,D=b2-4ac

,

(1) +>0,>0,D0(2) +0,D0(3) b,m>0 am>bm, am >bm

a>b,mbd, ad>bc

(6)nN,a>b>0 a2>b2,a>b,an>bn,a1n>b

1n

(1)a0,a=-a,-aaa(2)ab=ab(3)a-ba+ba+b

a

(x1,y1) ax+by+c=0

d= ax1+by1+ca2+b2

(1) :x2+y2=r2-- (0,0), r(2) :(x-a)2+(y-b)2=r2-- (a,b), r(3) :x2+y2+Ax+By+C=0

{x2+y2+ax+by+c=0y=mx+n (1) . D0(2). D0(3) . D0

r1,r2 d

(1) . dr1+r2(2) . d=r1+r2(3) . d=r1-r2(4) . r1-r2dr1+r2(5) . dr1-r2

(1)

x2+y2+Ax+By+C=0 (2)

(x-a)2+(y-b)2=r2 (3) x :(x-a)2+(y-b)2=b2

y :(x-a)2+(y-b)2=a2

x,y :(x-a)2+(y-a)2=a2

(4)A(x1,y1),B(x2,y2)

(x-x1)(x-x2)+(y-y1)(y-y2)=0(5) 3

x2+y2+Ax+By+C+k(x2+y2+A/x+B/y+C/)=0 k=-1 , .

(1) (x1,y1)

{ :x1x+y1y=r2 :(x1-a)(x-a)+(y1-b)(y-b)=r2(2) m y=mx+b D=0

y=mxr m2+1 .(3) (x1,y1)

y-y1=m(x-x1) D=0

r=d= ax1+by1+ca2+b2

.

(x-a)2+(y-b)2=r2 (x1,y1)

l= (x1-a)2+(y1-b)2-r2

(1)yf(x),yf(x) yf(x)-- y=f(x) yf(x)-- y=f(x) xf(y)-- x=f(y) xf(y)-- x=f(y)

(2)f(x,y)0,f(x,y)0 f(x,y)=0 .

f(x,y)=0 A(x1,y1),B(x2,y2)

(1) f(x1,y1)f(x2,y2)0(2) f(x1,y1)f(x2,y2)0(3) A,B f(x1,y1)f(x2,y2)0

(1) .(20 .(3) .

y=f(x,a) a

a { :D0 :D0

(1) (2) f(x,y) f(x,y)=k .

(3)k .( :D=0, : )

x m,y n (1) (x,y) (x+m,y+n)

(2) y=f(x) y-n=f(x-m)

(3) y=f(x) y+n=f(x+m)

(x,y) y=f(x)

x (x,-y) -y=f(x)

y (-x,y) y=f(-x)

(-x,-y) -y=f(-x)

y=x (y,x) x=f(y)

y=-x (-y,-x) -x=f(-y)

x=a (2a-x,y) y=f(2a-x)

y=b (x,2b-y) 2b-y=f(x)

(a,b) (2a-x,2b-y) 2b-y=f(2a-x)

(1) X,Y X Y X Y f:XY . ,X ,Y ,{yy=f(x)} .

(2) f,g x f(x)=g(x) f,g . f=g .

X Y f:XY (1) X Y x1x2 f(x1)f(x2)

(2) {}={},Y={f(x)}

(3)

(4) --- (5) --- (6) --- x1,x2 x1x2

f(x1)f(x2) (7) --- x1x2 f(x1)f(x2)

X={x1,x2,,xr} Y={y1,y2,,yn} f:XY(1) =nr=nr()(2) =nPr()

(3) =n!()(4) r n

(5)ij f(i)f(j) =nCr(6)ij f(i)f(j) =nHr()

f:X R, g:X R (1) :(f+g)(x)=f(x)+g(x)(2) :(f-g)(x)=f(x)-g(x)(3) :(fg)(x)=f(x)g(x)

(4) :(fg)(x)= f(x)g(x)(5) :(kf)(x)=kf(x)

f:X Y, g:Y Z (gf)(x)=z=g(f(x))

(1)gf /= fg(2) (gf)h= g(fh)(3)fI= If=f

f:X Y f-1:Y X f .

.

(1)y=f(x) .( )(2)x y .(3)y x .(4) .( = )

(1)y=f(x) y=f-1(x) y=x (2)ff-1=Iy,f-1f=Ix( .)

(3)(f-1)-1=f(4)(gf)-1=f-1g-1( )(5)gf=Ix,fg=Iy g=f-1,f=g-1

( .)

(1)(gf)(x)=h(x) f(x)=(g-1h)(x)(2)(fg)(x)=h(x) f(x)=(hg-1)(x)

(1) (y=ax) a0 1,3 a0 2,4 . a y .

(2) (y=ax+b) x :y : a0:,a0:,a0:y=b(x )

(3) (ax+by+c0)

a=0,b0:y=- cb,a0,b=0:x=-ca

(1) 0 n n+1

(2) x : 0 ,y :x

(3) y=f(x) )x0 y=f(x))x0 y=f(-x)-- y

y=f(x) )y0 y=f(x))y0 -y=f(x)-- x

y=f(x) x .

y=f(x) )x0,y0(1) y=f(x)) x,y,

y=ax2 y=a(x-)2+ y=ax2+bx+c

(0,0) (,) (- b2a,- b2-4ac4a ) y x=a x=- b2a

(1) (m,n) y=a(x-m)2+n(2) y=ax2+bx+c(3)x (,0),(,0) y=a(x-)(x-)

Y

f-1y=f(x)x=f-1(y)

(,) (,)

ya0

a0O

O

a0a0

x

yy

xx O

f:X

(1) f(-x)=f(x) y ()y=x2,y=x4,y=cosx,(2) f(-x)=-f(x) ()y=x,y=x3,y=sinx,y=tanx,

f(x)=ax2+bx+c ,

a0,D=b2-4ac,ax2+bx+c=0 ,

a0 D0 D=0 D0

f(x)0 x,x x f(x)0 x f(x)=0 x= , x=

ax2+bx+c=0 ,

(1) m D0,f(m)0, +2 m

(2) m D0,f(m)0, +2 m

(3) m,n

D0,f(m)0,f(n)0,m +2 n

(4) m f(m)0

(1)y= ax (a0)

:x,y a

(2)y= ax-m+n (a0)

:x=m,y=n (m,n)

(3)y= ax+bcx+d

:x=- dc,y=ac

(1)=0 y :x=m(2)

{ x {y=0y= y=() y=ax+b+ cx+1 x=-1,y=ax+b

(1)

y=a(x-)2+ a0 x= a0 x= .

mxn y=a(x-)2+ , .

()20, ( )0,D0(),-1sinx1,log >0 .

(2) 2=0 x D0

y= 22 D0

y .(3)

A0,B0 A+B2 AB

(, A=B )

A0, 1A 0 A+1A 2

(4)Cauchy-Schwartz

(a2+b2)(x2+y2)(ax+by)2(, xa=yb )

(5) .max(x,y),min(x,y), .

60

rad=180

1rad= 1801= 180rad

rad= =lr

l=r ,S= 12r2= 12rl r

S

l

=2n+ (n=0,1,2,) , -=360n -=360n+180 +=360n +=360n+180

sin= yr, cos=xr

tan= yx, cosec=ry

sec= rx, cot=xy

x

y

O r

P(x,y)

x .

(+)

tan+

sin+

cos+x

y

Ox

y

all+

x x x

x

y= -xy

y=- -x

y= x

y=- x

0 30 45 60 90

sin 0 1212

32

1

cos 1 3212

12 0

tan 0 131 3

(1)

cosec= 1sin ,sec=1cos ,cot=

1tan

(2)

tan= sincos , cot=cossin

(3)

sin2+cos2=1,1+tan2=sec2,1+cot2=cosec2

90n

(90n) () ( )

() () .

:

:{cossintancot

f(x+p)=f(x): p

y=sinx y=cosx y=tanx

2 2 1 1 -1 -1 (0,0) (0,1) (0,0) () () ()

y=rsin(ax+b)+c y=rtan(ax+b)+c 2a

a r+c -r+c

y=rsinax x

- ba y

c

y=rtanax

x - ba

y c

(1) sin y cos x .

(2) sin,cos .

asinA =

bsinB=

csinC =2R (, )

sinA= a2R, sinB=b2R, sinC=

c2R

sinA :sinB:sinC=a:b:c

1

{a=bcosC+ccosBb=ccosA+acosCc=acosB+bcosA 2

{a2=b2+c2-2bccosAb2=c2+a2-2cacosBc2=a2+b2-2abcosC

cosA= b2+c2-a22bc

cosB= c2+a2-b22ca

cosC= a2+b2-c22ab

(1) a h

S= 12ah

(2) a

S= 34a2

(3) a,b

S= 12absin

(4) a,b,c ( )

S= s(s-a)(s-b)(s-c)(,s= 12(a+b+c))

(5) r

S=sr(,s= 12(a+b+c))

(6) R

S= abc4R =2R2sinAsinBsinC

(1) a,b S=absin

(2)

S= 12lmsin

(3) .

(1)

m, n ,

ehdt l

(2) m, n ,

O

-1

2

y1

x O

-1

2

y1

x O

y

x

(1)

(2)

nPr= n!(n-r)!

nPn=n!, nP0=1, 01

(1)

(2)

nCr = nPrr! =

n!(n-r)!r!

nCr = nCn-r nC0 = 1, nCn = 1