matem12_

gamomcemloba

inteleqti

12XII

guram gogiSvili, Teimuraz vefxvaZe,

ia mebonia, lamara qurCiSvili

maTematika

XII klasis saxelmZRvanelo

gamomcemloba inteleqtiTbilisi 2012

redaqtori Teimuraz vefxvaZe

grifi mieniWa 2012 wels ssip ganaTlebis xarisxis ganviTarebis

erovnuli centris (brZaneba ## 375, 18. 05. 2012) mier

2guram gogiSvili, Teimuraz vefxvaZe,

ia mebonia, lamara qurCiSvili

m a T e m a t i k a

I nawili

XII klasis saxelmZRvanelo

gamomcemloba inteleqti

Tbilisi 2012

Guram Gogishvili, Teimuraz Vepkhvadze,

Ia Mebonia, Lamara Kurchishvili

MaTHS XII

Part I

InTeLeKTI Publishers

Tbilisi 2012

gamomcemloba inteleqtiTbilisi, ilia WavWavaZDis gamz. 5. tel.: 225 05 22

www.intelekti.ge [email protected] [email protected]

intelekti Publishers5 ilia Chavchavadze Ave., Tbilisi, Georgia. Tel.: (995 32) 225 05 22

isbN 978-9941-439-62-9 (I nawili)isbN 978-9941-439-57-5 (saerTo)

gamomcemloba `inteleqti~, 2012.

3Cveno keTilo megobaro!

mogesalmebiT da gilocavT axali saswavlo wlis dadgomas!

imedia, Cveni saxelmZRvanelo karg megzurobas gagiwevT maTema-

tikis didebul samyaroSi mogzaurobisas. SevecadeT, praqtikuli

amocanebis ganxilviT dagarwmunoT maTematikis Seswavlis aucile-

blobaSi. gaxsovdeT, aq miRebuli codna da misi gamoyenebis unari

sxva sagnebis ukeT aTvisebaSic dagexmarebaT.

Tu pirveli gacnobisas romelime sakiTxi kargad ver gaiazreT,

ar daRondeT, gaagrZeleT muSaoba _ amave sakiTxs saxelmZRvaneloSi

kvlav araerTxel SexvdebiT da aucileblad daZlevT mas.

warmatebiT gevloT maTematikis Seswavlis gzaze.

guram gogiSvili, Teimuraz vefxvaZe,ia mebonia, lamara qurCiSvili

avtorTa madloba

axali saxelmZRvaneloebis Seqmna da maTi skolaSi danergva ganaTlebis mimdinare reformis mniSvnelovani nawilia.

eqspertTa, pedagogTa da saxelmZRvaneloebis avtorTa erToblivi Za-lisxmeviT Seiqmna VII-XII klasebis saxelmZRvaneloebi, romlebic damkvi-drda saSualo skolebSi. yoveli axali wignis werisas viTvaliswinebdiT maswavlebelTa mosazrebebs. madlobas movaxsenebT maT nayofieri Tanam-SromlobisTvis.

gansakuTrebuli madlierebiT gvsurs aRvniSnoT eqspertTa Rvawli: maT mier Sedgenili 2011-2016 wlebis erovnuli saswavlo gegmis mixedviT dawerili XII klasis axali saxelmZRvnelo. gaviTvaliswineT yvela cv-lileba erovnul saswavlo gegmaSi, eqspertebTan konsultaciebis dros gamoTqmuli mosazrebebi da wina wlebSi miRebuli gamocdileba.

madlobas movaxsenebT yvelas, vinc Tavis arCevans am wignze SeaCerebs. vfiqrobT es saxelmZRvanelo maT imedebs aucileblad gaamarTlebs.

rogor visargebloT wigniT

gameoreba. zogierTi saxis gantolebisa da utolobis amoxsna. simravlis asaxva

1

1. utolobis amoxsnis magaliTebi

2. utolobaTa sistema

amocanebi gameorebisTvis

3. iracionaluri gantolebis amoxsna

4. simravle. moqmedebebi simravleebze

5. ori simravlis dekartuli namravli


6. asaxva. Seqceuli asaxva

7. grafTa Teoriis elementebi


8. debulebaTa dasabuTebis xerxebi

9. kombinatorika

albaToba2

1. xdomilobaTa sivrce. xdomilobis albaToba

2. operaciebi xdomilobebze

3. damoukidebeli xdomilobebi


4. pirobiTi albaToba. xdomilobaTa namravlis albaToba


5. did ricxvTa kanonis Sesaxeb


algoriTmi. evklides algoriTmi3

1. algoriTmi


2. evklides algoriTmi

3. evklides algoriTmis gamoyeneba

4. ricxvTa Teoriis gamoyenebis Sesaxeb


sarCevi

funqcia4

1. ricxviTi funqcia. ricxviTi funqciis mocemis xerxebi

2. operaciebi funqciebze. funqciaTa kompozicia


1. wrfivi funqcia. wrfivi funqciis cvlilebis siCqare


4. polinomuri funqcia


5. zogierTi racionaluri da iracionaluri funqcia

6. maCvenebliani da logariTmuli funqciebis gamoyenebis

magaliTebi


geometriuli figurebi5

1. veqtorebi sivrceSi


2. koordinatebi sivrceSi. veqtoruli namravli


3. veqtorebis da koordinatebis gamoyeneba

4. vagrZelebT veqtorebisa da koordinatebis gamoyenebis

magaliTebis ganxilvas; geometriuli gardaqmnebis gamosaxva

dekartul koordinatebSi

5. moculoba


6. zedapiris farTobi


monacemTa analizi da statistika6

P1. populacia da SerCeva

Aamocanebi gameorebisTvis

2. SerCevis ricxviTi maxasiaTeblebi (mediana, saSualo, saSualo

kvadratuli gadaxra)

3. dawyvilebuli monacemebi. korelacia. gabnevis diagrama

vemzadebiT gamocdebisTvis7

1. samkuTxedis kuTxeebis jami. monakveTis SuamarTobisa da

kuTxis biseqtrisis Tvisebebi

2. racionaluri ricxvebi. ricxvis xarisxi

3. proporcia. proporciis ucnobi wevris povna. ricxvis dayofa mocemuli

SefardebiT. Pproporciuli da ukuproporciuli

sidideebi. Pprocenti

4. cvladiani gamosaxuleba.cvladiani gamosaxulebis

mniSvnelobis povna. igivurad toli gamosaxulebebi.

gamosaxulebaTa gardaqmnis magaliTebi

5. Semoklebuli gamravlebis formulebi. mamravlebad daSla Semoklebuli

gamravlebis formulebis gamoyenebiT

6. racionaluri gamosaxulebebis gardaqmnis magaliTebi

7. oTxkuTxedebi

8. msgavsi samkuTxedebi

9. kvadratuli fesvi, iracionaluri ricxvi. namdvili ricxvi.

namdvili ricxvis gamosaxva ricxviT wrfeze.

namdvili ricxvis moduli

10. kvadratuli fesvis Semcveli gamosaxulebis gardaqmna

11. n -uri xarisxis fesvi. ariTmetikuli fesvi12. piTagoras Teorema

13. racionalurmaCvenebliani xarisxi. Algebruli gamosaxulebis

gardaqmnis magaliTebi

14. marTkuTxa koordinatTa sistema sibrtyeze. or wertils Soris

manZilis gamosaTvleli formula

15. wrewiri. wre

16. kvadratuli gantoleba. vietis Teorema. vietis Teoremis

Sebrunebuli Teorema

17. kvadratuli funqcia. kvadratuli samwevri.

kvadratuli samwevris daSla mamravlebad

18. kvadratuli utoloba


testebi

cnobari

maTematikuri terminebis leqsikoni

sagnobrivi saZiebeli

maTematikuri niSnebi

logariTmuli cxrilebi

trigonometriuli cxrili

pasuxebi

6am niSnakiT drodadro SegaxsenebT, rom yvela amocana unda gadaita noT rveulSi da DSemdeg SeasruloT. wignSi Canawerebi ar unda gakeTdes!

vin iqneba pirveli. am niSnakiT warmodgenili amocanebis amox-snisas ecadeT, gamWriaxobasTan erTad sxa rti azrovnebac ga-moavlinoT.

rv

eul

Si!

gaufrTxildiT wigns!

masSi Tqvenma Canawerebma wigni SeiZleba sxvebisTvis gamousadegari gaxados!

rogor visargebloT wigniT

wigniT sargebloba rom gagiadvildeT, ramdenime rCevas mogcemT.

saxelmZRvanelos TiToeuli Tavi masSi mocemuli ZiriTadi sakiTxebis CamonaTvaliT iwyeba.

TiToeuli Tavi paragrafebisgan Sedgeba. paragrafebi, rogorc wesi, ornawiliania. pirveli nawili Teoriul masalas eTmoba. misi gadmocema praqtikuli magaliTebis ganxilviT mimdinareobs. am nawils Seswavlili masalis SejamebiT vasrulebT.

es niSnaki migvaniSnebs paragrafis meore nawilis dawyebas _ `sxva-da sxva~. aq istoriuli sakiTxebi da sxva damxmare masalaa mo-cemuli.

s

es niSnaki amocanebis CamonaTvals win uZRvis.a

SemogTavazebT jgufuri muSaobis proeqtebs _ es TanaklaselebT-an erToblivad muSaobis gamocdilebas Seg ma tebT. aq SeiZleba dag-WirdeT informaciis damo u ki de be li moZiebac da misi damuSavebac.

j

zogierTi paragrafis Teoriul nawilsa da amocanebs mosdevs sa-varjiSoebi, romliTac Seswavlil (maT Soris _ gasul wlebSic) sakiTxebs gaimeorebT. am nawilis dawyebas gvauwyebs es sagangebo niSnaki.

g

teqstis ukeT gagebis mizniT saWirod CavTvaleT zo gi er Ti si-tyvis warmomavlobis, misi msgavsi mniSv ne lo be bi sa an sinonimebis miTiTeba. aRniSnuli miTiTebebi Cvens leq si konSia mocemuli. am saTauriT gamoyofil Ca na werebs Tqven xSi rad SexvdebiT. igi Tqvens sityvaTa ma rag sac gaamdidrebs da ufro gasagebs gaxdis wignis Si na arss.

Cvenileqsikoni

i

7am TavSi gavimeorebT gasul wlebSi Seswavlil sakiTxebs sim-ravleebisa da maTze moqmedebebis, kombinatoruli formulebisa da grafebis gamoyenebis Sesaxeb. gameorebasTan erTad ufro Rrmad SeviswavliT am TemebTan dakavSirebul sakiTxebs.

gameoreba. zogierTi saxis gantolebisa da utolobis amoxsna. simravlis asaxva

1.1 utolobis amoxsnis magaliTebi

1

gavixsenoT, rom kvadratul utolobas kvadratuli funqciis Tvisebebis gamoyenebiT vxsnidiT. Tu samwevrs aqvs nuli (fesvi), maSin SeiZleba kvadratuli samwevris mamravlebad daSliTac visargebloT.

vTqvaT, x2+bx+c kvadratuli samwevris diskriminanti D>0 da samwevris nulebia x

1 da x

2. maSin

ax2+bx+c=a(x-x1)(x-x

2).

amitomax2+bx+c>0 da a(x-x

1)(x-x

2)>0 (1)

utolobebi tolfasia. tolfasia agreTve

ax2+bx+c

8amovxsnaT, magaliTad, x2-9x+200.

marcxena mxaris nulebia: 3, 7 da 8. am ricxvebis mixedviT gamoiyofa 4 Sualedi: (; 3), (3; 7), (7; 8) da (8; +). mocemuli namravli yovel SualedSi niSans inarCunebs (dadebiTia an uaryofiTi). ase, rom, mocemuli namravlis niSani TiToeul am SualedSi emTxveva am Sualedis raime wertilSi namrav-lis niSans. nebismierad SevarCioT TiTo wertili am SualedebSi. magaliTad, 0; 5; 7,5; 10 wertilebSi namravlis niSnebiT davadgenT saTanado SualedebSi am namravlis niSnebs:

pasuxi: (3; 7)(8;+ ).

SeniSneT, albaT, namravlis niSnebis monacvleoba Sualedebis cvlilebisas. amrigad, mocemuli saxis utolobis amosaxsnelad sakmarisia mxolod erT-erT SualedSi niSnis dadgena da Semdeg niSnebis monacvleoba. es aixsneba imiT, rom sazRvris wertilebze x cvladis `gavlisas~ mxolod erTi mamravli icv-lis niSans.

2) amovxsnaT utoloba: (x-3)(x-7)(x-8)0.gaviTvliswinoT, rom wina amocanis amonaxsnebs unda daematos x=3, x=7

da x=8 ricxvebi. saTanado wertilebi suraTze sagangebodaa gamuqebuli.

pasuxi: [3;7]][8; +).

9SevajamoT: ganvixileT intervalTa meTodiT kvadratuli utolobebis amoxsnis procesi. mocemulia ufro rTuli utolobebis amoxsnis mag-aliTebic.

3) amovxsnaT utoloba:

x x

x x

1 6

3 5 02

2

$+ -

- -^ ^

^ ^

h h

h h (3)

wiladis mricxvelisa da mniSvnelis fesvebis mixedviT xuTi Sualedi gamoi yofa:

winaswarve SevniSnoT, rom sazRvris wertilebidan amonaxsnTa simravles ar ekuTvnis 1 da 6 wertilebi, 3 da 5 wertilebi ki _ ekuTvnis (daakvir-diT am wertilebis moniSvnis wess). roca x0, maSin ra simravlea ax2+bx+c0 SemTxvevaSi?4. vTqvaT, a>0, maSin ra simravles qmnis ax2+bx+c>0 utolobis

amonaxsnebi D>0 SemTxvevaSi?5. (x-3)(x+2)(x-8)>0 utolobis amoxsnisas ramden ricxviT Sualeds

gamoyofT? daasaxeleT es Sualedebi.

10

1 vTqvaT, a>0 da x10 utolobis amonaxsnTa simravlea 1) (-; -1)(3; ) 2) (-; -1] [3; +) 3) (-1; 3) 4) [-1; 3].

4 x2+1>0 utolobas akmayofilebs 1) mxolod [1; 1] Sualedis ricxvebi 2) mxolod (1;1) Sualedis ricxvebi 3) x-is nebismieri mniSvneloba

4) mxolod (1; +) Sualedis ricxvebi.

5 vTqvaT, a>0 da x10 da x1

11

10 amoxseniT utoloba: x22. 1) x 2 2) [ 2 ; 2 ]

3) (; 2 ][ 2 ; +) 4) (; 2][2; +).

11 amoxseniT utoloba: 0 utolobis amonaxsnTa simravlea (-; b)(a; +), maSin aucileblad

1) a0 da b4 an a4 an a0.

16 SeavseT cxrili gamosaxulebebis niSnebisa da sazRvris wertilebSi mniSvnelobaTa miniSnebiT.

(-; 0) 0 (0; 4) 4 (4; 5) 5 (5; +)x

x-4

x-5

x(x-4)(x-5)

cxrilis mixedviT amoxseniT utoloba: a) x(x-4)(x-5)0, b) -2x(x-4)(x-5)

12

17 SeavseT cxrili gamosaxulebaTa niSnebisa da sazRvris wertilebSi mniSvnelobaTa miniSnebiT.

(-; 6) 6 (6;+ )

x-6

x2+2

(x2+2)(x-6)

amoxseniT utolobebi: a) 3(x2+2)(x-6)>0, b) -3(x2+2)(x-6)0.

18 SeavseT cxrili gamosaxulebebis niSnebisa da sazRvris wertilebSi mniSvnelobaTa miTiTebiT.

(-; -5) -5 (-5; 0) 0 (0; 2) 2 (2; +)x-2

x

x+5

xx x

52

+-^ h

cxrilis mixedviT amoxseniT utoloba:

a) x

x x5

3 2 0$+-^ h , b)

xx x3 5

2 0$- +

-^

^hh

.

19 amoxseniT utoloba intervalTa meTodiT: a) 3(x-2)(x+3)0, b) -3(x-2)(x+1)0, g) -2,4(x+1)(x-1)(x+5)>0, d) 2(2x-3)(x+1,2)0, T) -3x(4x+3)(x-10)0.

20 amoxseniT utoloba intervalTa meTodiT: a) 5(x-4)(x+8)0 b) -3(x-2)(x+6)>0, g) (x2+1)(x-3,5)>0, d) (x2+2)(x+2)0, e) (x2+x+1)(x-5,2)(x+10)>0, v) 2(x-0,5)(x+12)(x2+3)0.

21 amoxseniT utoloba intervalTa meTodiT:

a) ,

,x

x x3 2

2 1 5 20$

-- +^ ^h h

, b) , ,

x xx x

6 82 10 11 1 02

2

- +- + - ,

anu >x xx x

2 41 3 0

- -- -

^ ^^ ^

h hh h

(5)

utolobis. amovxsnaT (5) intervalTa meTodiT:

pasuxi: (; 1)(2; 3)(4; +).

24 amoxseniT utoloba: a) (x-5)2(x+3)(x-4)0, b) (x-5)2(x+3)(x-4)>0, g) (x-5)2(x+3)(x-4)xx

xx

x1 2

02

++

--

, d) x x x4 4 1

11 4

12 2$- + -

.

26 amoxseniT utoloba:

a) x2-13x+12

x(x+2)>0, b)

x xx x

46 7 3 02

2

#+ ++ - ,

g) >x x x x2 5 3

12 3 9

1 02 2+ -+

+ -, d)

x x x7 121

93

2 2#- + -.


a) x+ x1 2, b) x+ x

1 >2, g) x+ x1 x x xx

4 12 94 9 03 2

2

- +- .

14

29 ipoveT funqciis gansazRvris are:

a) yxx12

2

=+

, b) yx

x 12

2

= + , g) yx xx

925

3

2

=-- , d) y x x

x925

3

2

=-+

.


a) yx x x

x6 9

43 2

2

=- +

- , b) yx x

x x x4

9 6 1 4 13

2 2

=+

- + -^ ^h h ,

g) yx x x1 10

12

=- +^ ^h h

, d) ,

yx x

x x x x2 5 3 2 3

3 42 2=

- - -

+ +^ ^

^ ^h h

h h.


a) x xx x

55 03

3

#+- , b) >

x xx x x3 3 023 2

-- + - ,

g) x

x 10 02

$--

r , d) x xx x

1210 02

3

$+ -- .

32 wylis avzs marTkuTxa paralelepipedis forma aqvs. fuZis erTi gverdi meoreze 5 dm-iT metia, paralelepipedis simaRleze ki _ 5 dm-iT naklebi. ra zomebi unda hqondes am avzs, rom misi moculoba iyos im kubis moculobaze meti, romlis gver-di parlelepipedis fuZis didi gverdis naxevaria?

33 konteiners marTkuTxa paralel-epipedis forma aqvs, romlis fuZis gverdebi metia konteineris simaR-leze, Sesabamisad, 2 m-iT da 6 m-iT. ra zomis gverdi eqneba kubis for-mis konteiners, romlis moculoba naklebia pirveli konteineris mocu-lobaze 8-jer mainc, simaRleebi ki orives toli aqvs.

j amoxseniT utoloba sxvadasxva xerxiT (geometriulad _ sakoordi-

nato wrfis gamoyenebiT; modulis Tvisebebis gamoyenebiT _ sxvadasx-va SemTxvevis ganxilviT; Tu SesaZlebelia, kvadratul utolobaze mi-yvaniT _ kvadratSi axarisxebiT).

a) |x|>3, b) |x|

15

1.2 utolobaTa sistema

ganvixiloT amocanebi, romelTa amoxsna utolobaTa iseTi sistemis amoxsnas ukavSirdeba, romelSic kvadratuli utolobac SeiZleba Sediodes.

amocana 1

marTkuTxedis formis fotosuraTis zomebia 12sm18sm. mas vawebebT marTkuTxe-dis formis muyaos furcelze ise, rom muy-aos arSia erTnairi siganis iyos. miaxloebiT (1 mm-mde sizustiT) ra zomebi unda hqondes muyaos furcels, rom arSiis farTobi ar iyos fotosuraTis farTobze naklebi da gaormagebul farTobze meti.

amoxsna. TuU arSiis siganes x-iT aRvniSnavT, maSin arSiis farTobi Sei-Zleba ase warmovadginoT:

4x2+212x+218x. pirobis Tanaxmad, gvaqvs: 4x2+24x+36x1218 (1) 4x2+24x+36x21218. (2)veZebT x-is im mniSvnelobebs, romlebic (1) da (2) utolobebs akmayo-

filebs _ anu, veZebT

4x2+24x+36x1218 4x2+24x+36x21218 (3)

utolobaTa sistemis amonaxsnTa simravles _ unda amoixsnas (3) sistema.(3)-is tolfasi sistemaa:

x2+15x-540 x2+15x-1080.gamovsaxoT am utolobebis amonaxsnTa simravleebi:

(; 18][3; +) da ;2

15 6572

15 657- - - +; E

sadac 2

15 6571

a = - - , 215 657

2a = - + .

(3) sistemis amonaxsnTa simravle _ (1) da (2)utolobebis amonaxs-nebis simravleTa TanakveTa _ ase warmovadginoT:

a1x-18 an 3xa

2.

x

x

12 sm

18 sm

16

Tumca, unda gaviTvaliswinoT isic, rom amocanis pirobis mixedviT x ar SeiZleba uaryofiTi iyos, maSasadame,

x[3; a2], a

25,3.

es niSnavs, muyaos furclis zomebi (miaxloebiT) unda iyos 18x24 sm-dan 22,6x28,6 sm-mde.

ganvixiloT utolobaTa sistemis amoxsnis magaliTebi.

magaliTi 1

magaliTi 2

amovxsnaT 3x>5

x2-5x+6>0

utolobaTa sistema. es sistema

x>35

(x-2)(x-3)>0

sistemis tolfasia. warmovadginoT misi TiToeuli utolobis amonax-snebis simravleebi;

maTi TanakveTaa (35 ; 2)(3; +) simravle.

pasuxi: (35 ; 2)(3; +).

amovxsnaT

x2-x+210 x2-10x+90 nebismieri x-isTvis; e i. am utolobis amonaxsnTa simravlea (;+). maSasadame, sistemis amonaxsnTa simravle emTxveva meore utolobis amonaxsnTa simravles.

imedia, ar gagiWirdebaT meore utolobis amoxsna.

35

x>35

x3

17

magaliTi 3

amovxsnaT utolobaTa sistema:

x 2-x+2

18

2 4x2 -4x+10 x2-4x>0

utolobaTa sistemis amonaxsnTa simravlea

1) (; 0)(4; +) 2) [21 ; +) 3) (;

21 ] 4) (; 0][4; +).

3 x2+x+10

x2-6x0 sistemis amonaxsnTa simravlea

1) 2) (; 0)(6; +) 3) (; 0][6; +) 4) [0; 6].

5 4x2-4x+10 x2-4x

19

10 |x2-5x|6 utoloba tolfasia sistemis

1) (x-6)(x+1)0 2) x2-5x6

(x-2)(x-3) 0 x2-5x-6 3) x2-5x6 4) (x-6)(x+1)0

x2-5x-6 (x-2)(x-3)0.

amoxseniT amocanebi

amoxseniT utolobaTa sistema:

11 a) 2x+50 b) 5x-40, x(x+1,2)0.

12 a) (x-1)(x+5)0 b) 3(x+4)(x-2)>0 g) x(3x+1)0 (x+7)(2x-1)0, (x+1)(3-x)0, x( 3-x)>0.

13 a) x2-2x-30 b) 5x2+19x-300 g) 4x2-8x-5>0 5x2-7x>0, x2+3x-40

2x2+x-10, 72x+5x2.

14 a) x2-x+2>0 b) 4x2-3x+1>0 g) 4x2+4x+10 4x2-4x+10, 9x2+6x+1>0, x2+2x4 e) x2-2x+10 v) 4x2-12x+9>0

x2-4x+40, 2x2>1, 5x2-9x-4. 15 a) x(x-70)>30(x+10)-3x b) x(x-1) x

1036 -

(x-10)2100(x+1)+20x, 2(3+x)>x(x+1), g) x(x+1)< x 4

183+ + d) 2(x+1)2+x>0

(x-1)2>1, 2x(x+3)(x-3)(x+3).

20


a) y x x2 3 42= + + - , b) yx x

x x3 10 3

1 42

2=- +

- - ,

g) yx

x

9 3

4 32

=- -

- +

^ h, d) y

x x

x x

1 2 15

12

2

=+ + -

+ -^ h ,

e) yx x

x x

6 9

11 9 22

2

=- +

- - , v) y xx

9 43

12= - -+

,

z)* yxx x

3

2

=-+ , T)* y

x

x

3 5

3=

+ -

+ .

17 amoxseniT ormagi utoloba: a) 4

21

23 motoriani navi mdinaris dinebis mimar-TulebiT 10km-is gavlis Semdeg miemarTeba sawinaaRmdego mimarTulebiT _ 6 km-is man-Zilze. mdinaris dinebis siCqare 1 km/sT-ia. ra sazRvrebSi unda icvlebodes navis sakuTari siCqare, rom am gzis gavlaze daxarjuli dro ar iyos meti 4sT-ze da ar iyos naklebi 3 sT-ze?

24 x-is ra mniSvnelobebisTvis aris WemSariti toloba:

a) x x9 922

2- = -^ h , b) x x x x7 6 7 622

2- + =- + -^ h ?

25 Tu turistebi yoveldRiurad dagegmilze 5km-iT met manZils gaivlian, ma-Sin 6 dReSi isini 90km-ze metis gavlas moas-wreben. winaaRmdeg SemTxvevaSi _ dReSi 5km-iT naklebis gavlisas _ 8 dReSi 90km-ze na-klebi manZilis gavlas. ramdeni km-is gavlas apirebdnen turistebi 1 dReSi?

26 ipoveT miaxloebiT sportuli moednisT-vis gamoyofili marTkuT xedis formis nakve-Tis zomebi, Tu moednis sigrZe da sigane 5 da 3-is proporciulia, farTobi ar aris naklebi 375m2-ze da ar aRemateba 500m2-s.

amoxseniT sxvadasxva xerxiT (sinjvis xerxi, kvadratuli utolo-bis Sedgena):

a) cnobilia, rom orniSna ricxvis erTeulebis cifri 4-iT metia aTeulebis cifrze. am orniSna ricxvisa da amave cifrebiT Cawerili sxva orniSna ricxvis namravli naklebia 1620-ze, xolo am ricxvebis jami metia 80-ze. ipove es orniSna ricxvi.

b) cnobilia, rom orniSna ricxvis erTeulebis cifri 3-iT nak-lebia aTeulebis cifrze. am orniSna ricxvis mis momdevno orniSna ricxvze namravli 2750-ze metia, xolo am orniSna ricxvebis jami 150-ze naklebia. ipoveT mocemuli orniSna ricxvi.

g) marTkuTxedis sigane 2 sm-iT naklebia sigrZeze, farTobi nak-lebia 224 sm2-ze, xolo diagonali metia 19 sm-ze. am marTkuTxedis gverdebis sigrZeebi (sm-ebSi) naturaluri ricxvebiT gamoisaxeba. ip-oveT es gverdebi.

j

22

amoxseniT amocanebi

1 ori momdevno mTeli ricxvis namravli naklebia 6-ze. ipoveT es ricxvebi.

2 ori momdevno luwi ricxvis namravli naklebia 48-ze, ariTmetikuli saSualo dadebiTi ricxvia. ipoveT es ricxvebi.

3 a-s ra mniSvnelobisTvis akmayofilebs x xx ax

14 9 02

2

#- ++ + utolobas

x-is mxolod erTi mniSvneloba? ipoveT es mniSvneloba.

4 ramdeni nuli aqvs x2-ax-1 kvadratul samwevrs?

5 ra gansxvavebaa y=x+2 da yxx

242=

-- funqciebis grafikebs Soris?

6 amoxseniT gantoleba: |x2+x+10|=x2+x+10.

7 dawereT raime utoloba, romlis amonaxsnia: a) (; 3)(5; +), b) [0; 6], g) (0; 6], d) [0; 6), e) (0; 6).

8 moWadrakeTa turnirSiO moWadrakeTa yoveli wyvili TiTo partias Ta ma Sobs. aRniSneT x-iT moWadrakeTa odenoba da gamosaxeT x-is mimarT kvadratuli samwevriT Catarebul SexvedraTa odenoba.

9 y=ax2+bx+c parabolas abscisaTa RerZTan erTi saerTo wertili aqvs _ (1; 0), ordinatTa RerZs kveTs (0; 2) wertilSi.

a) SeadareT a koeficienti 0-s, b) ipoveT c.

10 ipoveT manZili y=x2+4x+4 parabolis wverodan koordinatTa saT-avemde.

11 ra maxvil kuTxes qmnis wuTebisa da saaTebis maCvenebeli isrebi 21 saaTsa da 36 wuTze.

i

23

g

1 20 sm sigrZis da 15 sm siganis mqone marTkuTxedis sigrZec da si-ganec x sm-iT Seamcires.

a) gamosaxeT darCenili nawilis farTobi x-ze damokidebuli gamosax-ulebiT, b) daadgineT x-is dasaSvebi mniSvnelobebi.

2 1200 _ lariani saqoneli jer ianvarSi ga Zvirda, Semdeg _ maisSi, orivejer erTi da imave procentiT. aRniSneT x-iT procen-tebis ricxvi da gamosaxeT saqonlis axali fasi kvadratuli samwevriT. daasaxeleT am samwevris koeficientebi.

3 cnobilia, rom y=ax2 kvadratuli funq-cia zrdadia (; 0] Sualedze da |a-1|=3. ipoveT a.

4 cnobilia, rom (a2-4)x=a2+2a wrfiv gantolebas (x-is mimarT) uamra-vi fesvi aqvs. ramdeni nuli aqvs x2-ax+1 kvadratul samwevrs? ipoveT es nulebi.

5 cnobilia, rom (9a2-4)x=3a2-2a wrfiv gantolebas (x-is mimarT) fesvi ara aqvs. ipoveT y=ax2+2x kvadratuli funqciis nulebi.

6 ipoveT manZili y=-x2+4x-5 da y=x2-14x+39 parabolebis wveroebs Soris.

7 a-s ra mniSvnelobebisTvis ara aqvs utolobas amonaxsni (x-is mimarT)?

x

x ax a

3 1

3 2 602

2

#- +

+ + +^

^

h

h .

8 amoxseniT gantoleba:

a) |5x2+14x-3|=5x2+14x-3, b) |3x2+2x-5|=-3x2-2x+5,

g) |x2-3x+4|=-x2+3x-4. d) x 422

-^ h =x2-4.

9 amoxseniT utoloba: a) |x+5|(x-3)>0, b) |x-3|(x+5)>0, g) |x2+2x-3|(x+4)0, d) |x2-9|(x-10)

24


a) | |xx

39 0

2

#+- , b)

| |x xx x

22 7 6 0

2

$-- +

^ h,

g) | |x x

x x x4

5 5 022 2

#-

+ -^ h , d) | |

| |x

x6

5 1 0$+

+ + ,

e) ,x x

5 3

2 5 20#

-

- +^ ^h h , v) >x5 17 1

4 1 02

- -

- .

11 ipoveT k-s im mniSvnelobaTa simravle, romelTaTvisac

y x kx3 4 32= - + da y kx kx2 32= - +

funqciebi gansazRvrulia nebismieri x-sTvis.

12 ipoveT a-s im mniSvnelobaTa simravle, romelTaTvisac y=(a2+a)x2-6x+3 parabolis Stoebi zeviT aris mimarTuli, y=(2a2-7a-4)x2+x-4 parabolisa ki qveviT.

13 a-s ra mniSvnelobebisTvis aris y=(a2-1)x2+6x-1 kvadratuli funq-ciis grafikis Stoebi mimarTuli: a) zeviT, b) qveviT?

14 k-s ra mniSvnelobebisTvis ara aqvs y=kx2-6x+7 kvadratuli funq-ciis grafiks saerTo wertilebi abscisaTa RerZTan?

15 p-s ra mniSvnelobebisTvis iRebs y=x2-2px+1 funqcia: a) mxolod dadebiT mnSvnelobebs, b) mxolod uaryofiT mniSvnelobebs?

16 ramdeni nuli aqvs y=2x2+2kx-k-1 kvadratul funqcias?

17 marTkuTxedis perimetri 80sm-ia. aRniSneT marTkuTxedis sigane x-iT. a) gamosaxeT marTkuTxedis S farTobi x-is saSualebiT;

b) ipoveT x-is is mniSvneloba, romlisTvisac S funqcia Rebulobs udides mniSvnelobas.

18 120 m sigrZis meseriT SemoRobili marTkuTxedis formis nakveTs udidesi farTobi aRmoaCnda. ras udris am nakve-Tis gverdebis sigrZeebi?

19 a sm perimetris mqone marTkuTxedebs Soris romels aqvs udidesi farTobi?

25

20 warmovidginoT, rom ori sxeuli marTobul gzebze moZraobs. isini moZraobas erTdoulad iw-yeben. erT-erTi moZraobs A-dan da Sordeba gze-bis gadakveTis O wertils, meore uaxlovdeba O wertils. cnobilia, rom pirvelis siCqare 40km/sT-ia, meoris _ 60km/sT, OA=5km, OB=12 km. ram-deni sa aTis Semdeg iqneba sxeulebs Soris manZili umciresi? ipove es manZili.


a) x

x11

3 5 02 #+ +- , b)

xx

x1 33 0$

--

+,

g) >x x

x2

33

0+

++

, d) >x x

x2 12 1 1-

+ - .


a) yx

xx

x22

310

= +-

+-

,

b) yx

x x x3

3 2 29 1042=-- + - +

g) yx x

xx x1 3 2 4 7 2

12

=- +

-- -^ ^h h

d) | |

yx x

x x4

12

2

=-

+-+ .

26

1.3 iracionaluri gantolebis amoxsna

vTqvaT, Ox (abscisaTa) RerZze veZebT im A wertils, romlidanac B(1; 1) da C(5; -2) werti-lebamde manZilebis jami 5 erTeulis tolia.

Tu gavixsenebT or wertils Soris manZilis formulas, saZiebeli wertilis koordinatebs x da y-iT aRvniSnavT, maSin amocanis amoxsna miiyvaneba x-is yvela im mniSvnelobis povnaze, romlisTvisac

x x x x2 2 10 29 52 2- + + - + = . (1)

(1) gantoleba iracionaluri gantolebaa, erTucnobiani gantoleba, romelSic `fesvqveSa~ gamosaxuleba (gamosaxulebebi) x ucnobs Seicavs.

cxadia, igi tolfasia

x x x x2 29 25 10 2 22 2- + = - - +5-x x x x2 29 25 10 2 22 2- + = - - + (2)

gantolebis. (2)-is fesvebi iqneba im gantolebis fesvebic, romlebic misgan orive mxaris kvadratSi ayvaniT miiReba

x2-10x+29=25-10 x x2 22 - + +x2-2x+2

10 x x2 22 - + =8x-2;

5 x x2 22 - + =4x-1;

am gantolebis fesvebi

25(x2-2x+2)=16x2-8x+1 (3)

gantolebis fesvebicaa. (3) gantolebas ki erTi fesvi aqvs _ x= 73

, SemowmebiT davrwmundebiT, rom igi (1)-is fesvicaa.

axla am amocanis amoxsna _ A wertilis SerCeva _ sxva xerxiTac vcadoT. daukvirdiT sakoordinato sibrtyeze A da B wertilebis ganlagebas. ra

SemTxvevaSi iqneba AB da AC manZilebis jami umciresi? SeecadeT ipovoT gantoleba BC wrfis _ im wrfis, romelic SeiZleba

Caiweros y=kx+l saxiT da romelic B da C wertilebze gadis. rogor vipoviT am wrfisa da Ox RerZis gadakveTis wertilis abscisas?axla iracionaluri gantolebis amoxsnis sxva magaliTebi ganvixiloT.

magaliTi 1

amovxsnaT gantoleba:

8 x x2- = . (4)

es is SemTxvevaa, roca kvadratSi ayvanis Semdeg miRebuli gantolebis

yvela fesvi ar aris (4)-is fesvi. marTlac,

8-x2=x2, saidanac x=2 an x=2. x=2 ar aris (4)-is fesvi (asec vityviT: 2 gareSe fesvia)pasuxi: x=2.

y

x

B(1; )1

A x( ; )y

C(5; )2

1

2

5O

27

magaliTi 2

vTqvaT, marTkuTxedis perimetri 35 sm-ia, diagonali _ 12,5 sm. vipo-

voT marTkuTxedis gverdebi.

Tu erT-erTi gverdi x sm-ia, maSin meore iqneba , x156 25 2- sm da amocanis amoxsna miiyvaneba iracionaluri gantolebis amoxsnaze:

2 2 156,25 35x x 2+ - = , (5)

anu

, x x2 156 25 35 22- = - .

kvadratSi ayvanis Semdeg miRebuli kvadratuli gantolebis fesvebia x=7,5 da x=10. isini (5)-is fesvebicaa.

SevniSnoT, rom am gantolebis amoxsnisas SeiZleboda Tavidan agvecile-

bina iracionaluri gantolebis miReba. marTlac, erT-erTi gverdi _ x,

II _ x235 -b l. piTagoras TeoremiT x2+(17,5-x)2=(12,5)2. misi fesvebia

x=7,5 da x=10.

SevajamoT: ganvixileT iracionaluri gantolebebis amoxsnis magaliTebi.

1. ra xerxiT SeiZleba amoixsnas iracionaluri gantoleba?2. ras vuwodeT iracionaluri gantolebis gareSe fesvi?3. SeiZleba Tu ara iracionaluri gantolebis kvadratul gantole-

baze miyvanisas miviRoT gareSe fesvi?

4. SeiZleba Tu ara, rom iracionalur gantolebas ar hqondes fes-

vebi?

1 x 1- +2=0 gantolebas 1) erTi fesvi aqvs 2) ara aqvs fesvi

3) uamravi fesvi aqvs 4) ori fesvi aqvs.

a

SearCieT swori pasuxi

Semdegi amocanis amoxsnac miiyvaneba iracionaluri gantolebis amoxsnaze.

28

2 x5 2- =1x gantolebis kvadratSi ayvanisas miiReba mocemuli gan-tolebis gareSe fesvi _

1) x=-2 2) x=2 3) x=1 4) x=-1.

3 x x1 2+ = da x+1=2x 1) tolfasi gantolebebia

2) araa tolfasi gantolebebi

3) SeiZleba ar iyos tolfasi gantolebebi

4) zogjer ar aris tolfasi gantolebebi.

4 (x-2)2=2x-1 gantolebis yoveli fesvi aris 1) x-2= x2 1- gantolebis fesvi

2) 2-x= x2 1- gantolebis fesvi

3) |x-2|= x2 1- gantolebis fesvi

4) x-2=2x-1 gantolebis fesvi.

5 x-2= x2 1- gantolebis yoveli fesvi aris 1) (2-x)2=2x-1 gantolebis fesvi

2) (x-2)2=1-2x gantolebis fesvi

3) (x+2)2=2x-1 gantolebis fesvi

4) (x-1)2=2x+1 gantolebis fesvi.

6 x3 5 3- = gantolebis fesvia

1) 4 2) 314 3)

314 4)

37 .

amoxseniT amocanebi


a) x3 1 6+ = , b) x2 3 152- = ,

g) x x4 11 4- + =^ ^h h , d) x x2 3 6+ =^ h .


a) x x 2 22+ + = , b) xx

2

3 4-

= - ,

g) x x3= - , d) x x3 1 2 3+ = - ,

e) x

x x31 3 5

-+ = - .

29

9 tolfasia Tu ara gantolebebi:

a) x 3 4+ = da x+3=16, b) x22 x 2 1=0 da x2-2x1=0,

g) x16 25 392- = da x x4 5 4 5 39- + =^ ^h h ,

d) x16 25 392- = da x x4 5 4 5 39$- + = ?

10 amoxseniT gantoleba axali cvladis Semotanis xerxiT?

a) xx

11

6 5- +-

= , b) xx

4 2 1 112 13+ = ++

,

g) x xx x

5 2 62 66 172

2- - +

- -= , d) x x2 3 0- - = ,

e) x x x x3 3 4 3 22 2+ - + = +^ h, v) xx

xx

32 3

23 4

-+ +

+- = .

11 kvadratSi ayvanis gareSe aCveneT, rom gantolebas fesvi ara aqvs:

a) x x15 12 3- - - = , b) x x3 1 2 4 1 0+ + + + = ,

g) x3 1 1+ - = , d) x x3 4- = - ,

e) x x2 6 3 3- + - = , v) ,x x x4 5 4 9+ - + + = .


a) x x x8 16 162 2+ + = - , b) x x x1 1 322+ + - = ,

g) x x3 2 2 1 6- + = , d ) x x x2 10 2 52- + = - ,

e) x x x2 10 5 22- + = - .


a) xx

xx

22 1 2

2 12 3

+- +

-+ = , b) 2 3x

x

x

x

2

2 1

2 1

2$

+

-+

-

+= .


x x2 12 4- + - = .

amoxsna. gantolebis orive mxare aviyvanoT kvadratSi, miviRebT:

x x x x2 2 2 12 12 16- + - - + - = .

gamartivebis Semdeg kvlav aviyvanoT kvadratSi:

x x2 12 3- - = , (x-2)(12-x)=9, x2-14x+33=0,

x1=3, x

2=11.

SemowmebiT vrwmundebiT, rom orive ricxvi gantolebis fesvia.

pasuxi: x=3, x=11.

30


a) x x2 1 2 3- = - - b) 5x x2 1 1- + - = ,

g) x x1 2 2 1- + - = , d) x x2 2 2 2- + = ,

e) x x2 1 11- + = + , v) x x12 6- + - = .

16 (jgufuri muSaoba) amoxseniT sxvadasxva xerxiT (koordinatebis gamoyenebiT, kvadratSi ayvaniT). SeadareT amoxsnebi:

a) x x x4 10 34 5 22 2+ + + + = ,

b) x x x x1 6 4 2 52 2 2 2

- + - + - + - =^ ^ ^ ^h h h h ,

g) x x x x6 10 4 8 102 2 #- + + - + .


2x x x x2 5 16 89 72 2- + + - + = .

31

1.4 simravle. moqmedebebi simravleebze

1) Tu A aris ( )x x121 0+ - =d n gantolebis namdvili fesvebis simravle

(im fesvebis, romlebic namdvili ricxvebia); B ki _ amave gantolebis mTeli

fesvebis simravle, maSin A da B SeiZleba ase CavweroT:

( ) ,A x x x x121 0 R!= + - =d n* 4 , (1)

( ) ,B x x x x121 0 Z!= + - =d n* 4 . (2)

TiToeul SemTxvevaSi dasaxelebulia garkveuli (gansazRvruli) Tviseba. yvela obieqti, romelsac es Tviseba aqvs, qmnis Sesabamis simravles. TviT obieqtebi am simravlis elementebia. SesaZloa am Tvisebis mqone obieqti arc arsebobdes _ miiReba carieli simravle (). magaliTad, carieli simravlea

, .C x x x1 0 R2 != + =% /. (3)

Tu raime M simravle gansazRvrulia TvisebiT, romelic aqvs am simravlis yovel elements da ara aqvs yovel obieqts, romelic M-s ar ekuTvnis, maSin am Tvisebas M-simravlis maxasiaTebeli Tviseba ewodeba.

Tu raime D simravle mocemulia P(x) maxasiaTebeli TvisebiT, maSin am sim-ravles ase CavwerT:

( ) .D x P x= % /.

cxadia, ( )x x121 0+ - =d n gantolebis fesvebia 1 da 2

1. axla (1) da (2)

formulebiT warmodgenil simravleebs maTi yvela elementis CamoTvliT Cavw-

erT:

; , { 1} .A B1 21= - = -) 3

A orelementiania, B _ erTelementiani, ( ) , nx x x x1 21

0 !+ - =e o* 4 sim-ravle carielia.

carieli simravlis magaliTia, agreTve, {x | 1 < x < 2, xn}, erTelementianisa _

,x x x21

23 Z# # !) 3.

eqvselementiania {x | x6, xn} simravle, romliTac SeiZleba warmovadgi-noT yvela SesaZlo Sedegi albaTobis TeoriaSi ganxiluli eqsperimentisa: vagorebT kamaTels da vakvirdebiT _ ra ricxvi mova.

Tqvenc daasaxeleT ramdenime magaliTi erT, or da samelementiani sim-ravleebisa;

32

gaixseneT carieli simravlis aRniSvna da daasaxeleT carieli simravlis ramdenime magaliTi.

daasaxeleT raime sasruli simravle da maxasiaTebeli Tviseba, romliTac is ganisazRvreba. SeiZleba Tu ara misi mocema yvela elementis CamoTvliT?

SeiZleba, Tu ara usasrulo simravlis mocema yvela elementis CamoTv-liT? Tu es SeuZlebelia, iqneb zogjer elementTa miniSneba gamoiyeneba?

vTqvaT, A simravle mocemulia maxasiaTebeli Tvisebis miTiTebiT: A={x | xn, x < 10}. SeecadeT A simravle misi elementebis CamoTvliT aRweroT. sasrulia Tu usasrulo es simravle?

vTqvaT, A simravle mocemulia maxasiaTebeli TvisebiT: A aris yvela or-niSna naturalur ricxvTa simravle, romelTa cifrebi erTnairia. kidev ra maxasiaTebeli TvisebiT SeiZleba aRvweroT A simravle? aRwereT igi yvela elementis CamoTvliTac.

kidev erTxel daasaxeleT naturalur, mTel, racionalur da namdvil ricxvTa simravleebis aRniSvnebi.

gaixseneT, ra SemTxvevaSi vambobT, rom mocemuli simravleebi tolia.

2) gavixsenoT, rom planimetriaSi ganxiluli yoveli SesaZlo figura si-brtyis wertilTa simravlis raime qvesimravles warmoadgens. analogiurad, raime sakiTxis Seswavlisas, xSirad, im simravleebs ganvixilavT, romlebic erTi garkveuli simravlis qvesimravlea; aseT simravles universaluri sim-

ravle vuwodeT. mas U-Ti aRvniSnavT xolme. vTqvaT, A misi qvesimravlea am simravlis elementebi U simravlis elementebicaa. magaliTad, Tu U natu-ralur ricxvTa simravlea, maSin A SeiZleba luw ricxvTa simravle iyos. Tu U sibrtyis wertilTa simravlea, maSin A SeiZleba iyos, magaliTad, raime wris wertilTa simravle.

axla gaixseneT Canaweris azri: AB, (`A aris B-s qvesimravle~, `A Sedis B-Si~, `B moicavs A-s~). amave azris mqonea `BA~ Canaweri. Tu, amave dros, ar-sebobs elementi, romelic ekuTvnis B-s da ar ekuTvnis A-s, maSin, rogorc gaxsovT, A-s ewodeba B-s sakuTrivi qvesimravle.

magaliTad, Tu A={0; 1; 2}; B={1; 2}; C={2; 3}. maSin B aris A-s sakuTrivi qvesimravle, B ar aris C-s qvesimravle, C ar aris B-s qvesimravle.

SeecadeT moiyvanoT kidev erTi magaliTi, roca ar sruldeba arc MN da arc NM.

cxadia, AB ar gamoricxavs BA; Tu orive damokidebuleba WeSmaritia, ma-Sin A da B simravleebi tolia A=B.

simravleebisa da maT Soris damokidebulebebis TvalsaCino gamosaxvis mizniT simravleebs brtyeli figurebiT (ZiriTadad, wreebiT) gamovsaxavdiT xolme _ e. w. venis diagramebiT (maT eiler-venis diagramebsac uwodeben). Tu AB, maSin A-sa da B-s Soris SesaZlo damokidebulebebs aseTi ori saxis dia-gramiT warmovadgenT:

33

a) b)

a) diagramis mixedviT A warmoadgens B simravlis sakuTriv qvesimravles, b) diagramis mixedviT ki A=B.simravleTa Soris AB damokidebuleba bevri ramiT waagavs ricxvebs Soris

ab damokidebulebas, kerZod, nebismieri A da B simravleebisTvis gvaqvs:1) AA; 2) Tu AB da BA, maSin A=B; 3) Tu AB da BC, maSin AC.

Tumca, aris gansxvavebac _ ricxvebSi (simravleebisgan gansxvavebiT) yovel a-sa da b-s Soris aucileblad sruldeba erT-erTi SemTxveva: ab, an b

34

AB=BAA(BC)=(AB)CAA=AA(BC)=(AB)(AC)A=AAU=U

AA'=U AA'= '=UU'= (A')'=A

(AB)'=A'B'(AB)'=A'B'

suraTze marTkuTxediT universaluri simravlea gamo-saxuli.

ra simravlea warmodgenili damuqebuli nawiliT?

axla simravleebze moqmedebebis zogierTi Tviseba warmovadginoT:

(1) (2) (3) (4) (5) (6)

imedia, ar gagiWirdebaT am Tvisebebis dasabuTeba.

warmogidgenT erT-erTi Tvisebis dasabuTebas: A(BC)=(AB)(AC).

vTqvaT, raime xA(BC), maSin xA da xBC; maSasadame, xA da x ekuTv-nis erT-erTs mainc B da C-dan, anu xB an xC; Tu xA da xB, maSin xAB da amitom x(AB)(AC). Tu xA da xC, maSin xAC, x(AC)(AB); am-rigad, A(BC)(AB)(AC);

analogiurad vaCvenebT: Tu raime x(AB)(AC), maSin xA(BC), anu (AB)(AC)A(BC).

amrigad, A(BC)=(AB)(AC).

xSirad mosaxerxebelia aRniSnuli Tvisebebis venis diagramebiT ilustri-rebac.

Tu am formulebSi movaxdenT da ; da U; da simboloebis urT-ierTCanacvlebebs, kvlav WeSmarit tolobebs miviRebT.

CawereT es axali formulebi da SeecadeT daasabuToT maTi WeSmaritoba.magaliTad, (4) da (5) formulebi miiRebs saxes:

A(BC)=(AB)(AC); AU=A.

axla `damatebis~ Tvisebebi warmovadginoT:

(7) (8) (9) (10) (11) (12) (13)

A

U

35

SeamCnieT, albaT, simravleTa gaerTianebis an TanakveTis damatebis Sed-genisas, yoveli simravle Sesabamisi damatebebiT icvleba, da simboloebi ki erTmaneTs Caenacvleba.

SeecadeT daasabuToT (7)-(13) Tvisebebi.

3) axla simravleebze moqmedebebis gamoyenebebi warmovadginoT. vTqvaT, S gvirilebis simravlea, P _ im yvavilebis simravle, romlebic min-

dorSi izrdeba.warmovadgenT oTxi tipis winadadebas am simravleebis Sesaxeb da iqve mivu-

TiTebT maTi Caweris sxvadasxva formas

roca, magaliTad, aq vwerT: `yvela S aris P~, an `zogierTi S aris P~, isini, Sesabamisad, warmogvidgenen winadadebebs:

`S simravlis yoveli elementi P-s elementicaa~, `S-is zogierTi elementi P-s elementicaa~.analogiurad aRviqvamT mesame svetis sxva winadadebebsac.Tu s warmogvidgens S simravlis nebismier elements, p ki _ P simravlis

nebismier elements, maSin es ori winadadeba, Sesabamisad; aseTi formiTac Cai-wereba:

`yoveli s aris p~; `zogierTi s aris p~.

analogiurad Camoyalibdeba danarCeni ori winadadebac.

msjeloba: `Tu yoveli A aris B da yoveli B aris C, maSin yoveli A aris C~, simravleebis gamoyenebiT martivad Caiwereba:

Tu A B da B C, maSin A C.

winadadeba: `zogierTi A aris B~ _ Semdegi oTxi diagramiT gamoisaxeba (1, 2, 3, 4):

a: yvela gvirila mindorSi iz-rdeba

S P`yvela S aris P~`Tu S, maSin P~`S-dan gamomdinareobs P~

b: `arc erTi gvirila mindorSi ar izrdeba~

S P = `arc erTi S ar aris P~

c: `zogierTi gvirila mindorSi izrdeba~

S P `zogierTi S aris P~

d: `zogierTi gvirila mindorSi ar izrdeba~

S P `zogierTi S aris P~

A

BA

B

A, B

1 2 3 4

A B

36

axla simravleebze moqmedebebis gamoyenebiT amovxsnaT praqtikuli amoca-na: saTxilamuro inventariT movaWre maRaziis menejerma SemTxveviTi SerCevis wesiT gamokiTxa 160 myidveli saTxilamuro sportis sayvareli saxeobebis Ses-axeb. gamokiTxulTagan oTxmocma daasaxela krosi, samocdaaTma _ swrafdaS-veba, ocma _ orive es saxeoba; sxva saxeoba aravis dausaxelebia.

gamovarkvioT:a) ramden myidvels ar dausaxelebia krosi an ar dausaxelebia swrafdaS-

veba? b) ramdenma daasaxela swrafdaSveba, magram ar daasaxela krosi?g) ramdenma myidvelma daasaxela krosi an swrafdaSveba?d) ramden myidvels ar dausaxelebia sayvareli saxeoba?gamokiTxulTa simravle U-Ti aRvniSnoT, krosis moyvarulebisa _ C-Ti,

swrafdaSvebis moyvarulebisa _ D-Ti. pirobis Tanaxmad, am simravleebis el-ementebis odenobebia: n(U)=160, n(C)=80, n(D)=70; amasTanave, n(CD)=20.

warmovadginoT es informacia venis diagramiT:

diagramis mixedviT, SearCieT zemoT CamoTvlili SekiTxvebis swori pasuxe-bi:

a 130

b 30+60+50

g 30

d 50

60

U AD

5020

30

SevajamoT: gavixseneT simravleebze moqmedebebi, warmodgenilia am moq-medebebis Tvisebebi da maTi gamoyenebis magaliTebi.

simravleebis geometriuli figurebiT (mag-aliTad, wreebiT) warmodgena da maTi gamo-yeneba debulebaTa dasabuTebis gamartivebisa da TvalsaCinoebis mizniT cnobili ingliseli

maTematikosis jon venis (1834-1923) saxels uka-vSirdeba. ufro adre, XVIII saukuneSi analogi-uri diagramebiT sargeblobda didi Sveicarieli

s

jon veni

37

maTematikosi leonard eileri (1707-1783). eileri

iyo 800-ze meti samecniero naSromisa da aRmoCe-nis avtori. magaliTad, mxo lod ricxvTa TeoriaSi

man 140 naSromi gamoaqveyna. maTematikis garda mas aqvs Sromebi fizikaSi, meqanikaSi, astronomiaSi; amitom igi ase moixsenieba: didi maTematikosi, fizikosi, meqanikosi da astronomi.

leonard eileri

1. ras ewodeba ori simravlis gaerTianeba?2. ras ewodeba ori simravlis TanakveTa?3. ras ewodeba ori simravlis sxvaoba?4. ra xerxebiT SeiZleba warmovadginoT yoveli sasruli simravle?5. sul mcire ramdeni qvesimravle aqvs yovel simravles?6. ra SemTxvevaSi vityviT, rom A simravle B simravlis qvesimravlea

(AB)?7. aris Tu ara carieli simravle yoveli simravlis qvesimravle?8. ra SemTxvevaSi vityviT, rom ori simravle tolia?

j

1 warmoadgineT venis diagramiT winadadeba: a) `yvela A aris B~; b) `zogierTi A aris B~; g) `zogierTi A ar aris B~; d) `arc erTi A ar aris B~.

2 SeadgineT maTematikuri debulebebi, rom-lebic zemoT mocemuli formiT SeiZleba Caiweros (upiratesoba miecema im jgufs, romelic met debulebas warmoadgens).

3 warmoadgineT zemoT mocemuli formis msj-elobaTa magaliTebi, romlebSic TiToeuli Seicavs or wanamZRvars da erT daskvnas. venis diagramebis gamoyenebiT SeamowmeT _ WeSmaritia msjeloba, Tu mcdari.

38

a


1 dedamiwis zedapiris im wertilTa simravle, romlebSic erTi da igive saSualo wliuri temperaturaa, aris

1) izobari 2) izoTerma 3) grZedi 4) ganedi.

2 dedamiwis zedapiris im wertilTa simravle, romlebsac erTi da igive grZedi aqvs, aris

1) ganedi 2) paraleli 3) izoTerma 4) meridiani.

3 mocemulia simravleebi: A={1; 21; 121; 15; 52; 512}; B={16; 2; 5}; C={1; 21; 15}; D={21; 15; 52}; Q={5; 12}.

maTTvis WeSmaritia 1) BA 2) mxolod CA 3) CA da DA 4) QC.

4 Semdegi gamonaTqvamebidan, romeli ar aris WeSmariti? 1) arsebobs simravle, romelsac mxolod erTi qvesimravle aqvs 2) arsebobs simravle, romelsac mxolod ori qvesimravle aqvs 3) arsebobs simravle, romelsac ara aqvs qvesimravle 4) arsebobs simravle, romelsac 2-ze meti qvesimravle aqvs.

5 vTqvaT, A aris safexburTo turnirSi monawile gundebis simravle, Y _ erT-erTi monawile gundi, y - Y gundis moTamaSe. Semdegi Canawerebidan, romeli ar aris swori

1) YA 2) YA 3) yY 4) {Y}A.

6 Tu raime A, B da C simravleebisTvis AB da BC, maSin 1) CA 2) BA 3) CB 4) AC.

7 Tu raime A da B simravleebisTvis AB da BA, maSin 1) AB 2) A=B 3) ABA 4) ABA.

8 Tu raime A da B simravleebisTvis AB, maSin 1) AB=A 2) AB=A 3) AB=A' 4) AB=B'.

9 Tu raime A da B simravleebisTvis AB, maSin 1) BA 2) B=A 3) A'B' 4) BA.

10 suraTze warmodgenili A da B simravleebisTvis 1) A'B' 2) A'=B' 3) B'A' 4) BA.

AB

U

39

11 suraTze warmodgenili A, B da C simrav-le ebis Tvis

1) ABBC 2) ABC 3) BCA 4) AC.

12 suraTze warmodgenili A, B da C simravleebisTvis 1) ABC 2) AB=AC 3) ABC 4) ACC.

13 Tu U universaluri simravlea, maSin nebismieri A simravlisTvis U\A= 1) A 2) A' 3) U 4) .

14 A simravlis damatebis damateba _ (A')'= 1) A' 2) U 3) A 4) .

15 (AB)'= 1) A'B' 2) A'B' 3) 4) U.

16 ricxviT wrfeze moniSnuli AB monakveTis wertilTa simravlea

1) {x | xN; -1x5} 2) {x | xZ, x-1 an x5} 3) {x | xR, x-1 an x5} 4) {x | xR, -1x5}.

17 nebismieri A simravlisTvis A= 1) A 2) U 3) 4) A'.

18 nebismieri A simravlisTvis A= 1) A 2) U 3) 4) A'.

19 vTqvaT, A erTelementiani simravlea, A={a}; maSin 1) {a}A 2) {a}A 3) aA 4) a=A.

20 erTelementiani simravlis qvesimravleebis ricxvi aris 1) 2 2) 1 3) 0 4) 3.

21 vTqvaT, xR (R namdvil ricxvTa simravlea) da A aris x2+3=0 gantolebis fesvebis simravle, anu A={x | x2+3=0, xR}, maSin A=

1) {1} 2) {1} 3) {1; 1} 4) .

22 Tu simravles mxolod erTi qvesimravle aqvs, maSin is 1) carieli simravlea 2) erTelementiani simravlea 3) orelementiani simravlea 4) universaluri simravlea.

23 Tu P martiv ricxvTa simravlea, S _ Sedgenil ricxvTa simravle, maSin PS=

1) n 2) P 3) S 4) n\{1}.

A B C

A B C

-1 0 5

A B

40

24 Tu P martiv ricxvTa simravlea, T luw naturalur ricxvTa simravle, maSin PT=

1) 2) P 3) T 4) {2}.

25 nebismieri A da B simravleebisTvis, xAB niSnavs: 1) xA da x!B 2) x!A da xB 3) x!A da x!B 4) xA da xB.

26 nebismieri A da B simravleebisTvis, xA\B niSnavs: 1) xA da xB 2) xA da xB' 3) xA' da xB' 4) xA' da xB.

27 vTqvaT, A im moswavleebis simravlea, romlebic swavloben inglisurs; B im moswavleebis simravlea, romlebic swavloben germanuls; maSin im moswavleebis simravle, romlebic swavloben orive enas, aris

1) AB 2) AB' 3) A'B 4) AB.

28 vTqvaT, P tolgverda samkuTxedebis simravlea; Q _ tolferda sam-kuT xedebis; S _ tolferda marTkuTxa samkuTxedebis. a), b), g) da d) suraTebidan romel ze SeiZleba iyos isini gamosaxuli?

1) a) 2) b) 3) g) 4) d).

29 vTqvaT, A aris yvela mTeli dadebiTi ricxvis simravle, B _ yvela im mTeli dadebiTi ricxvis simravle, romlebic iyofa 6-ze. maSin

1) A=B 2) AB 3) BA 4) AB=.

amoxseniT amocanebi

30 vTqvaT, raime A, B da C simravleebisTvis A\B=C. daasabuTeT, rom maSin SeiZleba ar iyos WeSmariti toloba A=BC.

Q PS

QP

S

P

Q

S

S

P

Qa) b) g) d)

miTiTeba. gamoiyeneT kontrmagaliTi:

A={1; 2; 3; 4; 5; 6}, B={2; 4; 5; 6; 8}.

SearCieT kidev sxva kontrmagaliTi; warmoadgineT diagrama, romelic miuTiTebs dasaxelebuli

tolobis mcdarobaze.

41

31 mocemulia simravleTa wyvilebi: a) A da AB; b) A da AB; g) AB da AB. yovel wyvilSi erT-erTi simravle meoris qvesimravlea. CawereT es

mimar Teba simbolos gamoyenebiT.

32 mocemulia: A aris im rombebis simravle, romelTac kuTxeebi marTi aqvs, B _ kvadratebis simravle, C _ im marTkuTxedebis simravle, romlebsac toli gverdebi aqvs, D _ oTxkuTxedebis simravle, romelTac yvela kuTxe marTi aqvs, E _ marTkuTxedebis simravle. miuTiTeT toli simravleebi.

33 vTqvaT, A={x | xR, x-1,2}. ipoveT A da B simravleebis TanakveTa (gamoiyeneT ricxviTi wrfe).

34 vTqvaT, P={x | xZ; -4 x 6}; S={x | xn; 3 x 10}. ipoveT simravle P\S.

35 vTqvaT, K=Z\n. Semdegi gamonaTqvamebidan romelia WeSmariti? a) -3K; b) 0K; g) 0K; d) -3,5K; e) 2K; v) 12K.

36 vTqvaT, n naturalur ricxvTa simravlea, Z _ mTel ricxvTa simravle, C _ luwi naturaluri ricxvebis simravle, P _ martivi ricxvebis simravle. gamosaxeT venis diagramebiT es simravleebi.

37 vTqvaT, M aris amozneqili mravalkuTxedebis simravle, C _ amozneqili oTxkuTxedebis, K _ kvadratebis, P _ marTkuTxedebis, T _ samkuTxedebis. gamosaxeT es simravleebi venis diagramebiT.

38 erT-erTi skolis moswavleTa nawili mxolod inglisurs flobs, nawili _ mxolod germanuls, danarCenebi _ orive enas. moswavleTa 90% flobs inglisurs, 70% _ germanuls. moswavleTa ra procenti flobs orive enas?

39 vTqvaT, A aris 30-is martivi gamyofebis simravle, B _ 42-is martivi gamyo febis simravle. ipoveT AB.

40 vTqvaT, A da B universaluri simravlis is qvesimravleebia, romelTa TanakveTa ar aris carieli simravle. gamosaxeT venis diagramebiT (raime SesaZlo SemTxvevaSi):

a) (AB)'; b) AB'; g) A'B; d) AB'.

41 daasabuTeT: a) (A\B)\C=A\(BC); b) (A\C)\B=(A\B)(A\C); g) A(B\C)(AB)\C; d) (A\B)C(AC)\B.

42

42 vTqvaT, A, B da C raime simravleebia. daasabuTeT: a) A\B=AB'; b) (A\B)'=A'(AB).

43 vTqvaT, gvaqvs pirobiTi winadadeba: `Tu A, maSin B~ (1) gamosaxeT es gamonaTqvami diagramiT. aCveneT diagramis gamoyenebiT, rom, Tu aRniSnuli gamonaTqvami

WeSmaritia, maSin WeSmaritia gamonaTqvami: `Tu ara B, maSin ara A~, anu _ `Tu B', maSin A'~. (2) aCveneT, rom, roca gamonaTqvami `Tu A, maSin B~ WeSmaritia, maSin

gamonaTqva mebi: `Tu B, maSin A~, (3) `Tu A', maSin B'~ _ (4) SeiZleba mcdari iyos. moiyvaneT raime pirobiTi winadadeba ((1) tipis), romlis

WeSmaritobis simravle* emTxveva danarCeni samis ((2), (3), (4)) WeSmaritobis simravleebs, an _ ar emTxveva zogierTis WeSmaritobis simravles (romels SeiZleba ar daemTxves?).

44 miuTiTeT simravlis maxasiaTebeli Tviseba: a) yvela paralelogramis; b) yvela marTkuTxedis; g) yvela kvadratis; d) yvela tolferda samkuTxedis.

45 miuTiTeT raime saerTo Tviseba, romelic aqvs mocemuli simrav-leebidan TiToeulis yovel elements, garda erTi elementisa:

a) {samkuTxedi, kvadrati, trapecia, wre, wesieri eqvskuTxedi}; b) {4; 6; 18; 15; 158}; g) {2; 7; 13; 16; 29}; d) {1; 9; 67; 81; 121}; e) {1; 5; 10; 175; 180}; v) {Tbilisi, Telavi, foTi, londoni, zestafoni};

z) ; ; ;32

43

54

51) 3 .

46 mocemulia simravle: A={1; 2; 3; 4; 5; {4; 5}}. a) aris Tu ara {4; 5} am simravlis qvesimravle? b) aris Tu ara {4; 5} am simravlis elementi?

miTiTeba.

I magaliTi: `Tu naturaluri ricxvis kvadrati kentia, maSin es ricxvic kentia~;

II magaliTi: `Tu naturaluri ricxvi 5-iT bolovdeba, maSin is 5-ze iyofa~.

* WeSmaritobis simravle _ im obieqtTa simravle, romelTa mimarT gamoTqmuli wina-dadeba WeSmaritia.

43

47 ipoveT {x | xn, x20} simravlis kenti ricxvebiT Sedgenili qvesim-ravlis TanakveTa im ricxvebiT Sedgenil qvesimravlesTan, romlebic 3-ze gayofisas naSTSi iZleva 1-s.

48 suraTze gamosaxulia A, B da C simrav leebi. ra simravleebia gamosaxuli romauli cifre-

biT _ I, II, III, IV, V, VI, VII, VIII (gamosaxeT isini mocemuli simravleebis saSualebiT)?

49 gamoikiTxa 100 studenti. aRmoCnda, rom maT gan 28 flobs germanuls, 30 _ rusuls, 42 _ inglisurs, germanuls da ingli surs _ 8, germanuls da rusuls _ 10, inglisurs da rusuls _ 5. sa mi ve enas flobs 3 studenti.

a) ramdenma studentma ar icis arc erTi ena? b) ramdeni flobs mxolod rusuls? g) ramdeni flobs rusuls an inglisurs? d) ramdenia iseTi studenti, romelic arc inglisurs flobs da arc

germanuls?

50 vTqvaT, wina amocanaSi ganxiluli ga mo kiT xvis analogiuri

gamokiTxvis Sede gebi ase Ti iyo: gamoikiTxa 100 stu denti; rusuls flobs 18 studenti; flobs rusuls da ar flobs ger manuls _ 13; flobs rusulsa da ingli surs _ 9; germanuls _ 27; inglisurs _ 48; inglisursa da germanuls _ 4; arcerTs _ 24.

a) ramdeni flobs mxolod germanuls? b) ramdeni flobs rusulsac da germanulsac? g) ramdeni flobs inglisuri da germanuli enebidan erT-erTs mainc?

51 wina amocanebis analogiurad, 100 studentis gamokiTxvis Sedegebi aseTia: samive ena _ 5; germanuli da rusuli _ 10; inglisuri da germanuli _ 8; rusuli da inglisuri _ 20; germanuli _ 30; rusuli _ 28; inglisuri _ 50.

a) ramdeni studenti ar flobs am sami enidan arc erTs? b) ramdeni studenti flobs am sami enidan mxolod erTs? g) ramdeni studenti flobs am sami enidan mxolod ors?

A

B

C

I

II

III

VIII

VII

IV

VVI

miTiTeba. gamoiyeneT venis diagrama da gamosaxeT simravleebi stu-den tebisa, rom lebic floben germanuls, inglisurs, rusuls. amocanis piro be bis mixedviT miRebul 8 figuraSi CawereT studentebis Sesabamisi ode no bebi.

44

52 klasis 30 moswavlidan 19 cekvis wreze dadis, 17 _ simReris; 11 _ gitaraze dakvris; 12 _ cekvisa da simReris; 7 _ gitarisac da cekvisac; 5 _ simRerisac da gitarisac; 2 _ cekvis, simReris da gitaris. daadgineT:

a) aris Tu ara moswavle, romelic arc erT wreze ar dadis?

b) ramdeni dadis cekvis da ar dadis gitaris wreze?

g) ramdeni dadis mxolod erT wreze? d) ramdeni moswavle dadis mxolod or wreze?

53 vTqvaT, U universaluri simravlea; A, B, C, D _ misi qvesimravleebi.

daasabuTeT rom: a) U\(AB)=(U\A)(U\B), anu (AB)=AB; b) U\(ABC)=(U\A)(U\B)(U\C), anu (ABC)=ABC; dawereT formulebi im SemTxvevebisTvisac, roca qvesimravleebis

ricxvi aris: 4; 5. SeecadeT daasabuToT Tqveni varaudi.

54* (cnobili mezRaprisa da msjelobaTa dasa bu -Tebis Sesaxeb mravali amocanis avtoris lu is kerolis amocana).

brZolaSi 100 mekobridan 70-ma erTi Tva li dakarga, 75-ma _ erTi yuri, 80-ma _ erTi xeli, 85-ma _ erTi fexi. SeafaseT im mekobreebis odenoba, rom lebmac dakarges Tvalic, yuric, xelica da fexic.

55 SeadgineT wina amocanis analogiuri amocana da amoxseniT.

miTiTeba. b) Tu x ar ekuTvnis ABC simravles, maSin x ar ekuTvnis an A-s, an B-s, an C-s, anu erT-erTs mainc am simravleebidan. maSasadame, ekuTvnis an A-s, an B-s, an C-s damatebas.

miTiTeba. gamoiyeneT toloba:

U\(ABCD)=(U\A)(U\B)(U\C)(U\D);U _ universaluri simravlea, A, B, C, D _ misi qvesimravleebi.

CvenTan U-Si elementebis ricxvi aris 100. marcxena mxareSi elementebis ricxvia 100-n; n _ im mekobreebis ricxvia, romlebmac dakarges Tvalic, fexic, yurica da xelic. marjvena mxareSi elementebis ricxvi, cxadia, ar aRemateba sxvaobebSi elementebis odenobebis jams.

45

56 vTqvaT, n(P) aris P simravlis elementebis odenoba. daasabuTeT: n(AB)=n(A)+n(B)-n(AB).

57 daasabuTeT: n(ABC)=n(A)+n(B)+n(C)-n(AB)-n(AC)-n(BC)+n(ABC).

58* wina ori amocanis Sedegebis mixedviT gamoTqviT varaudi ABCD simrav lis elementTa odenobis formulis Sesaxeb da SeecadeT

daasabuToT Tqveni varaudis WeSmaritoba.

59 gamosaxeT venis diagramis gamoyenebiT gamonaTqvami: a) Tu arcerTi a ar aris b da zogierTi c aris b, maSin zogierTi c ar

aris a;

b) Tu yvela a aris b da zogierTi c ar aris b, maSin zogierTi c ar

aris a;

g) Tu zogierTi a aris b da zogierTi b aris c, maSin zogierTi a

aris c.

romelia msjelobaTa am formebidan swori (ra SemTxvevebSi gamomdi-

nareobs wanamZRvrebidan daskvna)? aageT kontrmagaliTi, romelic gvi-Cvenebs, rom g) forma ar aris swori.

60 venis diagramebis gamoyenebiT aCveneT, rom mocemuli msjeloba ar aris swo ri (mocemuli wanamZRvrebidan ar gamomdinareobs dasaxelebu-li daskv na):

a) 47-e skolis yvela moswavle TbilisSi cxovrobs da kotetiSvi-lis quCaze mcxovrebi yvela moswavle TbilisSi cxovrobs; maSasadame, koteti Svilis quCaze mcxovrebi yvela moswav le 47-e skolis moswavlea;

miTiTeba. I xerxi: SeiZleba CavweroT:

n(ABC)=n((AB)C)da gamoviyenoT wina amocanis Sedegi:

n((AB)C)=n(AB)+n(C)-n((AB)C)= n(A)+n(B)-n(AB)+n(C)-n((AC)(BC))= n(A)+n(B)+n(C)-n(AB)-n(AC)-n(BC)+n(ABC).II xerxi: SeiZleba wina amocanis analogiuri msjeloba gamovi ye noT.

miTiTeba. elementebi, romlebic erTdroulad Sedis rogorc A-Si, ise _ B-Si, e. i. AB-Si, n(A)+n(B) jamis gamoTvlis dros orjer iangariSeba. maSasadame, am jamidan unda gamovakloT n(AB), rom miviRoT AB simravlis elementebis odenoba _ n(AB).

46

b) yvela saskolo saxelmZRvaneloSi gvxvdeba Sec-domebi; yvela saskolo saxelmZRvanelo ada mianebis mier aris dawerili; maSasadame, yve laferi, rac dawerilia adamianebis mier, Seicavs Secdomebs.

g) Cqari matareblebi K baqanTan ar Cerdeba; dRes K baqanTan arc erTi matarebeli ar gaCerda; maSasadame, yvela es matarebeli _ Cqari iyo.

d) Cveni klasis moswavleebidan arc erTi ar aris Tbilisis `dinamos~ fan-klubis wevri, radgan isini sawevro gadasaxadebsac ki ar ixdian. ma-Sasadame, yvela, vinc ixdis sawevro gadasaxads, aris Tbilisis `dinamos~ fan-klubis wevri.

e) yvela profesionali fexburTeli valde-bulia TamaSis dros daicvas fexburTis Ta maSis yvela wesi; Cveni klasis arc erTi mo swav le ar aris profesionali fexburTeli; maSasadame, Cveni kla-sis arc erTi moswavle ar aris valdebuli fexbur-Tis TamaSis dros daicvas TamaSis yvela wesi.

47

1.5 ori simravlis dekartuli namravli

suraTze sakoordinato sibrtye da masze ori wertil-ia (M

1 da M

2) gamosaxuli. M

1 wertilis abscisaa 1, or-

dinati _ 4; asec vambobdiT: M1 wertils (1; 4) ricxvTa

wyvili Seesabameba. misgan gansxvavebul M2 wertils _

(4; 1) ricxvTa wyvili. (1; 4) wyvili (4; 1) wyvilisgan gansxvavebulia, anu dalagebul wyvilebs ganvixilavT da ara 1 da 4 ricxvebis simravles _ gaxsovT, albaT, {1; 4} da {4; 1} simravleebi erTmaneTs emTxveva, toli simravleebia.

O

y

x

1

4

1 4

M1(1; 4)

M2(4; 1)

nebismieri ori A da B simravlisTvis SeiZleba ganvsazRvroT dalage-buli wyvili (a; b), sadac aA da bB. `dalagebuloba~ niSnavs: Tu a=c da b=d, maSin (a; b)=(c; b) da piriqiT, Tu (a; b)=(c; d), maSin a=c da b=d.

A da B aracarieli simravleebis dekartuli namravli AB ewodeba yvela dalagebuli (a; b) wyvilis simravles, sadac aA da bB.

magaliTi 1

magaliTi 2

vTqvaT, A={1; 2; 3}; B={2; 6}. maSin AB={(1; 2); (1; 6); (2; 2); (2; 6); (3; 2); (3; 6)}. ramdeni elementisgan Sedgeba AB simravle? SeadgineT BA dekartuli namravli. Seicavs Tu ara es simravle

(dalagebuli wyvilebis simravle) (1; 6) elements? tolia Tu ara AB da BA simravleebi?

vTqvaT, A={1; 2; 3}. SevadginoT dekartuli namravli A simravlisa A simravleze;

AA={(1; 1); (1; 2); (1; 3); (2; 1); (2; 2); (2; 3); (3; 1); (3; 2); (3; 3)}. am SemTxvevaSi SeiZleba ufro martivi aRniSvnac gamoviyenoT: A2

(AA=A2). ramdeni elementisgan Sedgeba AA? SeadgineT A simravlis orelementiani qvesimravleebis simravle.

ramdeni elementisgan Sedgeba es simravle? ra gansxvavebaa am simravlesa da A2-s Soris?

Tu A da B simravlidan erT-erTi mainc carielia, maSin dekartuli namrav-lic carielia _ AB=.

48

magaliTi 3

magaliTi 4

magaliTi 5

gavixsenoT, rom R-iT namdvil ricxvTa simravles aRvniSnavT; sakoor-dinato sibrtyeze, yoveli wertili erTaderTi dalagebuli (x; y) wyvilis (xR da yR) Sesabamisia da piriqiT _ yovel dalagebul (x; y) wyvils, anu R2-is elements am sibrtyis erTaderTi wertili Seesabameba; SeiZleba vTqvaT: gvaqvs urTierTcalsaxa Sesabamisoba sakoordinato sibrtyesa da R2-s Soris; zogjer asec amboben _ sakoordinato sibrtye aris R2 simravle.

vTqvaT, A={a1; a

2; a

3; ...; a

m}; B={b

1; b

2; ...; b

n); AB dekartuli namravli

SeiZleba cxrilis saxiT warmovadginoT (m striqoniTa da n svetiT): (a

1; b

1) (a

1; b

2) ... (a

1; b

n)

(a2; b

1) (a

2; b

2) ... (a

2; b

n)

- - - - -

(am; b

1) (a

m; b

2) ... (a

m; b

n)

ramdeni elementisgan (wyvilisgan) Sedgeba AB simravle?

erTi kamaTlis orjer gagorebisas _ roca gvainteresebs ra movida pirvel jerze da ra movida meoreze, gvaqvs elementaruli xdomilobe-bi: (x; y), 1x6; 1y6; SeiZleba vTqvaT, rom elementarul xdomilobaTa sivrce aris A={1; 2; 3; 4; 5; 6} simravlis Tavis Tavze dekartuli nam-ravli _ A2, igi 36 elementisgan Sedgeba.

SevajamoT: ganvsazRvreT ori simravlis dekartuli namravli: AB={(a; b)| aA; bB}; sadac (a; b) dalagebuli wyvilia; analo-

giurad ganisazRvreba sami an meti simravlis dekartuli namravli.

`dekartuli namravli~ _ es saxelwodeba dekar-tis pativsacemad Semoi Res.

rene dekarti (1596-1650) _ didi frangi filoso-fosi, maTematikosi, fizikosi da fiziologi. dekar-tis maTematikuri Sromebi ukavSirdeba filosofiasa da fizikaSi mis aRmoCenebs. man SemoiRo geometriaSi cvladi sididisa da funqciis cnebebi. bevri maTema-tikuri simboloc mis mieraa SemoRebuli. misi mrava-li mecnieruli miRwevidan gamorCeulad unda aRin-iSnos, rom man, fermasTan erTad, safuZveli Cauyara koordinatTa meTods, riTac geometriisa da algebris erTobliv Seswavlas mieca dasabami.

s

rene dekarti

49

1. ra gansxvavebaa {a; b} orelementian simravlesa da (a; b) dalage-bul wyvils Soris?2. ra SemTxvevaSia ori dalagebuli wyvili toli?3. ras ewodeba X da Y simravleTa dekartuli namravli XY?

a


1 Tu A simravle Sedgeba m elementisgan, maSin AA dekartul nam-ravlSi elementTa odenobaa

1) 2m 2) m+1 3) m-1 4) m2.

2 Tu A simravle Sedgeba m elementisgan, B _ n elementisgan, maSin AB namravlis elementTa odenobaa

1) mn 2) m2n2 3) m+n 4) mn-1.

3 vTqvaT, Z mTel ricxvTa simravlea. romeli formulaa swori?

1) (41 ; 1)ZZ 2) (-5; 1)ZZ

3) (21 ; 5)ZZ 4) ( 2; 1)ZZ.

4 sakoordinato sibrtyis wertilTa simravle SeiZleba ase CavweroT: 1) nn 2) RR 3) ZZ 4) QQ.

5 sakoordinato sibrtyis mTelkoordinatebian (mTel) wertilTa sim-ravle SeiZleba ase CavweroT:

1) nn 2) RR 3) ZZ 4) QQ.

6 suraTze erTeulovani kvadratia gaferadebuli; am kvad ratis wertilTa simravle SeiZleba ase CavweroT:

1) RR 2) [0; 1][0; 1]

3) [0; 1)[0; 1) 4) ; ;021

21 1#= =G G .

7 RR namravli SeiZleba ase CavweroT: 1) R2 2) Z2 3) n2 4) Q2.

8 [0, 1][0; 1] namravli SeiZleba ase CavweroT: 1) (0; 1)(0; 1) 2) (0; 1][0; 1) 3) [0; 1]2 4) (0; 1)2.

O

y

x

1

1

50

9 vTqvaT, A={1; 2; 3}. romeli formulaa araswori? 1) (1; 1)A2 2) (1; 2)A2 3) (3; 1)A2 4) (4; 1)A2.

10 vTqvaT, A={1; 2; 3}. B misi qvesimravlea. maSin B ar SeiZleba iyos 1) B={1; 2} 2) B={2} 3) B={1; 1} 4) B={2; 1}.

11 Tu (a; b) da (c; d) dalagebuli wyvilebia da a=c; bd; maSin 1) (a; b)=(c; d) 2) (a; b)(c; d) 3) (a; b) SeiZleba toli iyos (c; d) wyvilis 4) (b; a)=(d; c).

12 Tu (a; b) da (c; d) dalagebuli wyvilebi tolia, maSin 1) a=c; bd 2) ac; b=d 3) ac; bd 4) a=c; b=d.

amoxseniT amocanebi

13 vTqvaT, M={1; 2; 3; 4; 5}, N={6; 9; 8; 7; 0}. SeadgineT MN simravle.

14 SeadgineT A da B simravlee-bis dekartuli namravli:

A = {akaki wereTeli; vaJa-fSa ve la; nikoloz baraTaSvili; ilia Wav WavaZe};

B = {kacia-adamiani; gamz-rdeli; bax tri oni; Tornike eri-sTavi; qvaTa Ra Radi; merani; gan-Tiadi; SemoRameba mTa wmindaze}.

dekartuli namravlis elementebi ganalageT marTkuTxovani cxrilis sa xiT:

nawarmoebi

mwerali

daStrixeT cxrilSi is ujra, romelic Seesabameba wyvils _ pirveli `elementi~ aris meoris avtori.

15 vTqvaT, M qarTul anbanSi Tanxmovnebis simravlea, N _ xmovnebis. warmoadgineT MN da NM simravleTa ramdenime elementi.

51

16 vTqvaT, M={7; 20; 80; 100; 45; 38}, K={1, 19, 5; 40; 13}. SeadgineT MK simravlis is P qvesimravle, romlis wyvilebSi

pirveli elementi naklebia meoreze. M da K simravlis ra elementebi ar Sevida P simravlis arc erT

wyvilSi?

17 SeadgineT yvela orniSna ricxvis M simravle, romelSic aTeulebis Tanrigis cifri ekuTvnis A={9; 7; 1} simravles, erTeulebis _ B={5; 4; 0} simravles. ra kavSirs xedavT M simravlesa da AB dekartul namravls Soris?

18 gamosaxeT sakoordinato sibrtyeze A da B simravleebis dekartuli namravlis elementebi:

a) A=(2; 4; 6}, B={-2; 1}; b) A=B={-1; 0; 1}; g) A={x | xn, 3 x 7}, B={y | yn; 4 y 9}; d) A={x | xR, 3 x 7}, B={y | yR; 4 y 9}.

19 AB simravlis yvela elementi gamosaxulia sakoordinato sibrtyeze (gamuqebuli wertilebi; wertilTa daStrixuli simravleebi; d) sura-Tze daStrixulia I da IV meoTxedebi). ipoveT A da B.

20 gamosaxeT sakoordinato sibrtyeze XY, roca a) X={x | x=3}, Y={y | yR, 3 y 6}; b) X={x | xR; -1 x 4}, Y={y | y = 4}; g) X={x | xR; -1 x 2}, Y={y | yn, y 3}.

21 1, 2, 3, 4, 5 cifrebisgan (maTi gameorebac SeiZleba) SeadgineT yvela is orniSna ricxvi, romelic:

a) iwyeba 4-iT; b) bolovdeba 2-iT; g) erTnairi cifrebisgan Sedgeba; d) iwyeba 1-iT da mTavrdeba 5-iT; e) iwyeba 2-iT da mTavrdeba 3-iT;

rogor daukavSirebT am amocanis amoxsnas dekartuli namravlis cne-bas?

a) b) g) d

52

1 sul ramden aseuls Seicavs ricxvi oTxas ocdaori milion samas Tvrameti aTas xuTasi?

1) 422318 2) 4223185 3) 42231 4) 4223150.

2 podiumze gamosuli modelebidan yovel mexuTes TeTri qudi exura. modelebis ramden procents ar exura TeTri qudi?

1) 95%-s 2) 5%-s 3) 80%-s 4) 20%-s.

3 zooparkis binadarTa 41 -s

volierSi aqvs auzi, danarCenebis 32 nawili ki

sauzmeze xils miirTmevs. zooparkis binadarTa ra

nawils Seadgenen isini, romelTa volierSi auzi ar

aris da sauzmeze xils ar miirTmeven?

1) 41 2)

32 3)

61 4)

1211 .

4 marTkuTxedi miRebulia sami toli kvadratis erTmaneTze midgmiT. TiToeuli kvadratis farTobia 25 mm2. ipoveT marTkuTxedis perimetri.

1) 3 sm 2) 6 sm 3) 4 sm 4) 40 sm.

5 SeiZleba Tu ara, rom AB da BA namravlebs mxolod oTxi saerTo elementi hqondes? dadebiTi pasuxis SemTxvevaSi moiyvaneT A da B simravleTa raime magaliTi.

i

53

1.6 asaxva. Seqceuli asaxva

vTqvaT, X im luwi naturaluri ricxvebis simravlea, romlebic ar aRemate-ba 20-s, Y ki _ 20-ze naklebi kenti martivi ricxvebis simravle:

X={2; 4; 6; ... ; 20}, Y={3; 5; 7; 11; 13; 17; 19}.

x iyos X-is nebismieri elementi, y ki _ Y-is. SevadginoT yvela iseTi (x, y) dalagebuli wyvilebis simravle, romlebSic x iyofa y-ze. miviRebT _

M={(6; 3); (10; 5); (12; 3); (14; 7); (18; 3); (20; 5)}.

amrigad, winadadebiT (TvisebiT, wesiT): `x iyofa y-ze~ aRwerilia erT-erTi Sesabamisoba X da Y simravleebs Soris. am Sesabamisobis dros X-Si aRmoCnda elementi, romelsac ar Seesabameba arc erTi elementi Y-dan.

axla X da Y-s Soris aseTi Sesabamisoba ganixileT: `x naklebia y-ze~ da

warmoadgineT yvela aseTi (x, y) wyvilis simravle _ K. am Sesabamisobisas yovel x-s X-dan Seesabameba Tu ara raime y elementi

Y-dan?

xom ar Seesabameba romelime x-s erTze meti elementi Y-dan?albaT SeniSneT, rom M da K simravleebi, romlebiTac dasaxelebuli ori

Sesabamisobaa mocemuli, qvesimravleebia XY simravlis _

XY={(x, y) | xX, yY}.

axla zogierTi saxis Sesabamisobebze SevCerdebiT: vTqvaT, X samkuTxedebis simravlea, Y aris namdvil ricxvTa R simravle.

Tu xX da yY, maSin winadadebiT `y aris x-is farTobi~ ganisazRvreba iseTi Sesabamisoba, roca X simravlis yovel elements Y-is erTaderTi elementi Seesabameba.

Tu am Sesabamisobas f-iT aRvniSnavT, maSin CavwerT: f : X Y. kerZod, roca Y=X, maSin f aris X-is asaxva X-Si.

f : X Y asaxvisas Y-is is elementi, romelic x-s Seesabameba ase Caiwereba: f (x). mas f asaxvisas x-is saxe ewodeba, Tu f (x)=y, maSin x-s ewodeba f asaxvisas y-is winasaxe.

maSasadame, X-is Y-Si f asaxvisas yoveli xX elementisTvis arsebobs iseTi erTaderTi y elementi, romelic x-s Seesabameba: y=f (x); Tumca SesaZlebelia, rom raime y ar iyos arc erTi x-is Sesabamisi an erTze meti wina saxe hqondes (magaliTad, erTi da imave ricxviT uamravi samkuTxedis farTobi SeiZleba gamoisaxos).

Sesabamisobas, romlis dros X simravlis yovel elements See-sabameba Y simravlis erTaderTi elementi, ewodeba X simravlis Y simravleSi asaxva.

54

am suraTze warmodgenilia X da Y simravleebs Soris Sesabamisobis oTxi ni-muSi. simravleTa elementebi wertilebiTaa gamosaxuli. X-is yoveli elementi isriTaa dakavSirebuli Tavis Sesabamis elementTan Y-dan.

imedia am oTxi Sesabamisobidan amoicnobT:

Sesabamisobas, romelic asaxvas ar warmoadgens; X-is Y-Si asaxvas.aRvniSnoT f (X)-iT f : X Y asaxvisas X-is yvela elementis saxeTa simravle _ f (X)={y | y=f (x), xX}.

f (X)-s uwodeben X-is saxes f asaxvisas ( f-is mniSvnelobaTa simravles), TviT X-s ki f-is gansazRvris ares. magaliTad, Tu X=R da f : R R asaxva f (x)=x2 formuliTaa gansazRvruli, maSin f (X)=[0; +), f (X)R.

Tu f (X)=Y, maSin f-s vuwodebT X-is Y-ze asaxvas (`ze~-asaxvas). magaliTad, `ze~-asaxvas warmogvidgens asaxva f : A B, Tu A={-2; -1; 0; 2; 4}, B={0; 1; 4; 16} da asaxva gansazRvrulia f (x)=x2 formuliT, xA. marTlac, f (A)={0; 1; 4; 16} da f (A)=B. am SemTxvevaSi zogierTi yB aris A-s erTze meti elementis Sesabamisi.

Tu X={1; 2; 3}, Y={1; 4; 9; 10}, maSin f (x)=x2 formuliT ganisazRvreba X simrav-lis Y-Si iseTi asaxva, roca X-is gansxvave-bul elementebs gansxvavebuli saxeebi aqvs. aseTi saxis asaxvebs X-is Y-Si Seqcevad asax-vebsac uwodeben.

ganvixiloT kidev erTi magaliTi: Tu f asaxva [a; b] Sualedis yovel x ricxvs

Seusabamebs AB wiris (x; y) wertils, maSin es asaxva Seqcevadia _ gansxvavebul x

1 da x

2 wertilebs gansxvavebuli M

1 da M

2

wertilebi Seesabameba. amasTanave, wiris yoveli wertili aris [a; b] Sualedis erTaderTi ricxvis Sesabamisi.

vTqvaT, f aris X-is Y-Si Seqcevadi asaxva. maSin SeiZleba ganvsazRvroT f (X)-is X-ze iseTi asaxva, romelic f (X)-is yovel y elements `daabrunebs~ y-is wi-

nasaxeSi, anu X-is im erTaderT x elementSi, romlisT-visac f (x)=y.

f (X)-is X-ze aseT asaxvas f-is Seqceuli (Sebrunebu-li) asaxva ewodeba da f -1-iT aRiniSneba.

amrigad, Tu f Seqcevadi asaxvaa, maSin f -1( f (x))=x, xX, f -1( f (X))=Xda yoveli gansxvavebuli y

1 da y

2-sTvis (y

1=f (x

1) da

X Y X Y X Y X Y

X f Y

55

y2=f (x

2)), f -1(y

1)f -1(y

2), anu x

1x

2 (amas Tqven iolad daasabuTebT sawinaaRmde-

gos daSvebis me TodiT). f-is gamomsaxvel diagramaSi `isrebis SebrunebiT~ f -1-is diagrama miiReba.

X simravlis Y simravleze Seqcevad asaxvas X da Y simravleTa urTierT-calsaxa Sesabamisobasac uwodeben.

Tu f nebismieri asaxvaa X simravlisa Y-Si da yY, maSin f -1(y)-iT aRiniSneba y-is yvela wina saxeTa simravlec (sruli wina saxe) _

f -1(y)={x | xX, f (x)=y}.cxadia f -1(y) SeiZleba iyos carieli simravlec, an usasrulo simravlec.

moiyvaneT saTanado magaliTebi.daubrundiT paragrafis dasawyisSi warmodgenil diagramebs da gamoarkvi-

eT:

warmoadgens Tu ara romelime maTgani urTierTcalsaxa Sesabamisobas? Tu aseTi Sesabamisoba ver aRmoaCineT Tavad moiyvaneT aseTi Sesabamisobis ori maga liTi.

SevajamoT: visaubreT or simravles Soris Sesabamisobis Sesaxeb; ganvsaz-RvreT X-is Y-Si asaxva _ Sesabamisoba, roca X-is yovel elements Y-is erTaderTi elementi Seesabameba; ganvsazRvreT e. w. `ze~-asaxva _ X-is Y-ze asaxva, roca Y-is yoveli elementi X-is erTi romelime elementis saxe mainc aris.

vTqvaT, f aris X-is Y-Si asaxva da X-is gansxvavebul elementebs Y-is gansxvavebuli elementebi Seesabameba, maSin f-s ewodeba Seqcevadi _ Sei-Zleba ganvsazRvroT f -1 asaxva f (X)-isa X-ze. X simravlis Y simraleze Seqcevadi asaxva urTierTcalsaxa Sesabamisobaa.

asaxvis kerZo saxeebia: ricxvTa D simravleze gansazRvrulia f fun-qcia, romelic D-s yovel elements Seusabamebs erTaderT f (x) ricxvs (nebismieri simravleebis SemTxvevaSic zogjer terminebs: `asaxvas~ da `funqcias~ erTi da imave azriT gamoviyenebT); geometriuli gardaqmne-bi _ sibrtyis Tavis Tavze Seqcevadi asaxvebi (magaliTad, paraleluri gadatana).

X da Y simravleebs Soris Sesabamisobis ganxilvisas, ar aris gamor-icxuli SemTxveva: X=Y, anu _ Sesabamisoba X simravlesa da imave X sim-ravles Soris, kerZod, SeiZleba gvqondes asaxva X-isa amave X-Si.

asaxvebis im kerZo SemTxvevebisTvis, romlebic paragrafSia warmodgeni li sxva saxelwodebebsac iyeneben: f : X Y asaxva, romelic X-is gansxva ve bul elementebs Y-is gansxvavebul elementebs Seusabamebs inieqciaa; `ze~-asax va _ siureqcia; X da Y simravleTa urTierTcal-saxa Sesabamisoba _ bieqcia.

s

56

1. ra saxis Sesabamisobas ewodeba X simravlis Y simravleSi asax va?2. ra saxis Sesabamisobas ewodeba X simravlis Y simravleze asaxva?3. ra SemTxvevaSi vityviT, rom X-is Y-Si f asaxva Seqcevadi asaxvaa?4. ra SemTxvevaSi vityviT, rom f aris urTierTcalsaxa Sesabamisoba

X da Y simravleebs Soris?

a


1 Tu f aris X simravlis asaxva Y simravleSi, maSin X-is yovel ele-ments Seesabameba Y-is

1) erTaderTi elementi 2) ori elementi mainc 3) sami elementi 4) aranakleb oTxi elementisa.

2 X aris wrewirebis simravle, Y aris namdvil ricxvTa simravle; f : X Y asaxvaa, romelic yovel wrewirs Seusabamebs mis sigrZes. f asaxva aris

1) X-is Y-Si asaxva 2) X-is Y-ze asaxva 3) X-is Y-Si Seqcevadi asaxva 4) urTierTcalsaxa Sesabamisoba X-sa da Y-s Soris.

3 vTqvaT, A={0; 5; -7; 13}; B={1; 2; 3}. Sesabamisoba gansazRvrulia AB-s C, D, E da F qvesimravleebiT; romeli maTgani warmogvidgens A simravlis B-Si asaxvas?

1) C={(0; 1), (5; 2), (5; 3), (-7; 1), (13; 3)} 2) D={(0; 3), (5; 3), (-7; 3), (13; 2)} 3) E={(0; 1), (5; 2), (-7; 3)} 4) F={(0; 1), (0; 2), (-7; 3), (13; 3), (5; 3)}.

4 suraTebze sakoordinato sibrtyis wertilebiT mocemulia Sesabam-isobebi X={1; 2; 3; 4; 5} da Y={0; 1; 2; 3} simravleebs Soris; romeli maTgani ar warmogvidgens X-is Y-Si asaxvas?

1) 2) 3) 4)

57

5 vTqvaT, XZ, X , Y aris dadebiTi mTeli ricxvebis kvadratebis simravle, f : X Y aris Sesabamisoba _ X-is yovel elements Seesabam-eba misi kvadrati.

es asaxva Seqcevadia, roca, magaliTad, 1) X=Z 2) X={x | xZ, x-2} 3) X={x | xZ, x-10} 4) X=n.

6 oTxkuTxedebis ra X simravlisTvis warmoadgens asaxvas Semdegi Se-sabamisoba: `X-is yovel elements masSi Caxazuli wrewiri Seesabameba~?

1) roca oTxkuTxedebis mopirdapire kuTxeebis jami 1800-ia 2) roca oTxkuTxedebis simravle emTxveva paralelogramebis simravles 3) roca oTxkuTxedebis mopirdapire gverdebis jami tolia 4) roca oTxkuTxedebis simravle emTxveva marTkuTxedebis simravles.

amoxseniT amocanebi

7 aris Tu ara asaxva: a) Sesabamisoba, romelic yovel wrewirs masze Semoxazul samkuTxeds

Seusabamebs; b) Sesabamisoba, romelic yovel mTel ricxvs Seusabamebs misi 5-ze

gayofisas miRebul naSTs; g) Sesabamisoba, romelic yovel marTkuTxeds mis farTobs Seusabam-

ebs?

8 vTqvaT, m mocemuli naturaluri ricxvia. yovel mTel ricxvs Sevu-sabamoT m-ze gayofisas miRebuli naSTi.

aris Tu ara es Sesabamisoba asaxva? ra aris misi gansazRvris are, mniSvnelobaTa simravle?

aris Tu ara es asaxva Seqcevadi asaxva? ra ricxvebisgan Sedgeba am asaxvisas 0-is wina saxe?

9 aRvniSnoT r-iT (r=0, 1, 2, 3, 4) im mTeli ricxvebis simravle, romelTa 5-ze gayofisas miRebuli naSTia r.

SeadgineT simravle X={ ; ; ; ; ;0 1 2 3 4 5 }.

vTqvaT, Y={ 0; 1; 2; 3; 4}. ganvixiloT f Sesabamisoba X da Y simravleebs Soris: f (r)=r (r=0; 1; 2;

3; 4);

aris Tu ara f Sesabamisoba asaxva? aris Tu ara f Sesabamisoba `ze~-asaxva? aris Tu ara f Seqcevadi asaxva; urTierTcalsaxa Sesabamisoba?

58

10 X simravlis elementebia gamonaTqvamebi: p: `yvela naturaluri ricxvi luwia~; q: `yvela martivi ricxvi kentia~; r: `arsebobs marTkuTxedi, romelic kvadratia~; s: `zogierTi namdvili ricxvi iracionaluri ricxvia~; t: `yvela namdvili ricxvi iracionaluri ricxvia~.

Y simravlis elementebia 0 da 1; Y={0; 1}. yovel gamonaTqvams SeusabameT 0 an 1: Tu is WeSmaritia _ 1, winaaR-

mdeg SemTxvevaSi _ 0; aris Tu ara es Sesabamisoba asaxva? warmoadgineT is wyvilebis

saxiT. aris Tu ara es asaxva X-is Y-ze asaxva, Seqcevadi asaxva?

11 X aris sibrtyis wertilTa simravle; l wrfea sibrtyeze. yovel M wertils Sevusa-bamoT misi simetriuli M

1 wertili l wrfis

mimarT. aCveneT, rom es Sesabamisoba asaxvaa; ipoveT AB monakveTis saxe am asaxvisas; arsebobs Tu ara AB monakveTis iseTi

wertili, romelic Tavis saxes emTxveva (asax-visas AB-s uZravi wertilia)?

12 vTqvaT, A={3; 6; 9}, B={0; 1}. SeadgineT simravleebi: AB, BA. daamyareT raime urTierTcalsaxa Sesabamisoba AB da BA simrav-

leebs Soris.

13 vTqvaT, X=Y={0; 1; 2; 3; 4; 5}; Sesabamisoba mocemulia formu-

liT: y x32= = G , (xX, yY) _ y aris y x

32= = G-is mTeli nawili. aris Tu ara

es Sesa bamisoba X-is X-Si asaxva?

14 R namdvil ricxvTa simravleze gansazRvrulia Sesabamisoba for-muliT: a) y=3x; b) y=x2.

aris Tu ara es Sesabamisoba R R asaxva; Seqcevadi asaxva; urT-ierTcalsaxa Sesabamisoba?

15 mocemulia sxvadasxva Sesabamisoba nn simravlisa n-Si: a) (x; y) nn elements Seesabameba xy; b) (x; y) nn elements Seesabameba x da y-is udidesi saerTo

gamyofi; g) (x; y) nn elements Seesabameba x+y. TiToeuli SesabamisobisTvis daadgineT, aris Tu ara igi asaxva

nn n?

59

16 ricxvTa ra simravle SeiZleba iyos X, rom Sesabamisoba (x; y) y : x iyos asaxva XX simravlisa X-Si?

17 mocemuli Sesabamisobebidan romelia asaxva X simravlisa Y-Si; `ze~-asaxva?

a) X wreebis simravlea, Y=R, yovel wres misi farTobi Seesabameba; b) X wreebis simravlea, Y _ dadebiTi namdvili ricxvebis simravle.

yovel wres Seesabameba misi farTobi; g) X={x | -5 x 7, xR}; Y=R; f (x)=x2; d) X={x | -5 x 7, xR}; Y={x | 0 x 49, xR}, f (x)=x2;

18 vTqvaT, X saxelm-wifoTa sim rav lea, Y ki _ qala qebis simravle da gvaqvs f Sesabamisoba _ f : x y, `x saxelmwifos deda-

qala qi aris y qalaqi~ (Sesa-bamisobis Caweris aseT sax-es momavalSic gamoviye nebT xolme). aris Tu ara es Sesa-ba misoba asaxva?

19 gaixseneT asaxvis raime magaliTebi, romlebic TqvenTvis aris cno-bili geometriuli masalidan.

20 vTqvaT, f : x x2-5x+4 aris asaxva R-isa R-Si. ipoveT a) [3; 7] monakveTis saxe; b) 4-is sruli wina saxe; g) [-2; 4]-is sruli wina saxe. 21 mocemulia asaxvebi:

a) yovel wrewirs sibrtyeze SevusabamebT mis centrs; b) yovel erTeulovan wrewirs sibrtyeze SevusabamebT mis centrs; g) yovel kvadrats sibrtyeze SevusabamebT mis perimetrs (kvad-

ratebis simravlis asaxva dadebiT namdvil ricxvTa simravleSi). romeli asaxvaa Seqcevadi?

60

1.7 grafTa Teoriis elementebi

1) grafTa Teoriis sawyisebis Seswavla gasul wlebSi daviwyeT. grafebs logikuri amocanebis amoxsnisas, simravleebs Soris Sesabamisobebis TvalsaCinod warmodgenisas viyenebdiT. gaviRrmavoT Cveni codna am mimarTulebiT.

gavixsenoT, rom grafs geometriuli figuriT warmovadgendiT _ wertilebiTa da am wertilTa SemaerTebeli wirebiT (wiboebiT) Sedgenili figuriT, romelSic SeerTebuli SeiZleba iyos: mocemul wertilTa (wveroebis) yoveli wyvili, an mxolod maTi nawili, raime wertili _ Tavis TavTan; amasTanave; raime ori wertili SeiZleba ramdenime wiriTac iyos SeerTebuli.

magaliTi 1

magaliTi 3

magaliTi 2

suraTze gamosaxulia qimiu ri nae-r Tis C

2H

6 (meTanis) struqtu ru li

formula da Sesabamisi grafi. atomebi wertilebiT aris gamosa-

xu li, kavSirebi _ wirebiT.

mravalwaxnagas wvero eb sa da wiboebs Soris kav Sirebic SeiZleba sib r -t ye ze grafiT ga mo v sa xoT:

suraTebze pira mi disa da ku bis SemTxve vebia gamo saxuli. SeecadeT aRweroT gamosaxvis procesi.

suraTze benzolis struqtu ruli formula da Sesabamisi grafia gamo-saxuli.

C C HH

H H

H H

CH

CH

C

C

C

C

H

H

H

H

61

magaliTi 5

magaliTi 6

magaliTi 4

suraTze warmodgenili grafi saavto-mobilo gzebis qsels warmogvid gens (6 qalaqs Soris).

grafiT simravleebs Soris Sesabamisobebic SeiZleba gamovsaxoT. amjerad wertilTa simravle ori nawilisgan Sedgeba. erTi nawili pirveli simravlis elementebia, meore nawili _ meoris.

pirveli simravlis elementebi X1, X

2, X

3, X

4

wertilebiTaa warmodgenili, meore simravlis elementebi _ Y

1, Y

2, Y

3 wertilebiT.

es Sesabamisoba sibrtyeze orientirebuli grafiTaa warmodgenili.

Tu gvaqvs Sesabamisoba raime X simravlesa da amave X-s Soris, maSin grafi SeiZleba maryuJsac Seicavdes, anu wibos, romelic raime wveros Tavis TavTan aerTebs.

romelime ori A da B wvero SeiZleba SeerTebuli iyos orive isriT _ A B da B A.

vTqvaT, frenburTelTa turnirSi 6 gundi mona wileobs; aRvniSnoT es gundebi ase: A, B, C, D, E, F. turniri erT wred Catarda.

am suraTiT e. w. orientirebuli grafia warmodgenili, romelic cxrilis Sesabamisadaa Sedgenili _ isrebi mimarTulia mogebuli gundidan wagebulisken.

a B C D e F

a 1 1 1 0 0

B 0 1 1 1 1

C 0 0 1 1 1

D 0 0 0 0 0

e 1 0 0 1 1

F 1 0 0 1 0

1 aRniSnavs mogebas,

0 _ wagebas;

SexvedraTa Sedegebi suraT ze cxriliTaa warmodgenili.

B

C

DE

F

A

A

B

CD

E

F

X1

X2

X3

X4

Y1

Y1

Y1

62

magaliTi 7

miaqcieT yuradReba

vTqvaT, X={2; 3; 4; 5; 6; 7; 8; 9} da X-is yovel x elements See sa bameba amave X-is y elementi (erTi an ram-de ni me), Semdegi Tvisebis mixedviT: `x iyo fa y-ze~. saTanado grafi suraTzea gamosaxuli~.

E orelementian simravleTa simravlea _ martiv grafSi wveroTa yovel wyvils erTaderTi wibo SeiZleba akavSirebdes erTmaneTTan; amasTanave, ar arsebobs maryuJi _ martiv grafSi yoveli wvero Tavis TavTan ar aris da-kavSirebuli. Tu am or SezRudvas movxsniT, miviRebT ufro zogad cnebas _ grafis cnebas. am SemTxvevaSi SeiZleba gvqondes maryuJi da e. w. jera-di wibo _ wibo meordebodes _ aq E aris wyvilebis raRac mimdevroba (wyvilebi SeiZleba gameordes). orientirebul grafSi TiToeuli wyvili dalagebuli wyvilia.

2 3 4 5

6 7 8 9

2) axla grafis cneba aRvweroT simravleTa Teoriis gamoyenebiT. Tavda-pir velad martivi grafi ganvsazRvroT.

magaliTad, suraTze gamosaxulia martivi grafi, romel Sic

V={A, B, C, D, K, F, G, H, I} _ wveroebis sim-ravlea;

E={{A, B}, {A, C}, {B, D}, {C, K}, {K, F}, {K,

G}, {H, I}} _ wiboebis simravlea.

{A, B} wiboze vityviT, rom is akavSirebs (aerTebs) A da B wveroebs. amrigad Cven warmovadgineT TqvenTvis ukve cno-bili cnebis _ grafis _ axleburi gansazRvreba erT kerZo SemTxvevaSi.

martivi grafi aris simravleTa wyvili: (V; E), sadac1) V aris aracarieli simravle, mis elementebs wertilebs an

wveroebs vuwodebT; 2) E aris V-s raime orelementiani qvesimravleebis simravle. am

elemen tebis TiToeul wyvils (orelementian qvesimravles) grafis wibo ewodeba.

grafi aRvniSnoT _ G asoTi, vwerT: G=(V; E).

A

B

D

F

C

K

G I

H

A

B

C

63

A B

C D

A

B

C

D

A BC

D

BA C D

ganixileT wina nawilSi mocemuli magaliTebi da gansazRvreT _ romel maTganSia warmodgenili martivi grafi, orientirebuli grafi, grafi jeradi wiboebiT?

grafis, kerZod, martivi grafis zemoTmoyvanili gansazRvrebis mixedviT SeamCnevT, rom erTi da imave grafis sxvadasxvanairi warmodgena SeiZleba; su-raTze mocemulia grafis sxvadasxva gamosaxuleba:

grafis gansazRvrebidan vRebulobT, rom SeiZleba romelime wvero ar iyos mocemuli wibos bolo _ ar iyos am wibos incidenturi. Tu wvero arc erTi wibos incidenturi ar aris, mas izolirebuli wvero ewodeba. Tu G grafis romelime wvero n cali wibos incidenturia (n cali wibo iyris Tavs am wvero-Si), maSin n-s am wveros xarisxi ewodeba.

grafis warmodgena SeiZleba saTanado marTkuTxovani cxriliT. Tu wveroe-bia X

1; X

2; ...; X

n, maSin grafs warmovadgenT cxriliT, romelSic raime i-uri

striqonisa da k-uri svetis gadakveTaze warmodgenili ricxvi gviCvenebs xi da

xk wveroebis SemaerTebeli wiboebis odenobas.

magaliTad, suraTze warmodgenil grafs Seesabameba Semdegi cxrili:

ipoveT am cxrilis yovel striqonSi (an svetSi) ricxvebis jami. SeadareT miRebuli ricxvebi grafis Sesabamisi wveroebis xarisxebs. ra

varaudis gamoTqma SeiZleba? SeecadeT daasabuToT gamoTqmuli varaudi.

ra grafis SemTxvevaSi gvaqvs mTavar diagonalze (suraTze gamoyofilia mTavari diagonali) mxolod nulebi?

warmoadgineT raime martivi grafisTvis analogiuri cxrili. ra saerTo Tvisebebi eqneba nebismieri martivi grafisTvis Sedgenil aseT cxrils?

X1

X2

X3

X4

1 2 3 4

1 1 1 1 0

2 1 0 2 1

3 1 2 0 1

4 0 1 1 0

64

3) axla grafTa Teoriis mniSvnelovani debuleba _ eileris Teorema Camo-vayaliboT. winaswar kidev ramdenime cneba gavixsenoT.

ganvixilavT sasrul grafebs _ wveroebis simravle sasrulia.jaWvi (sawyisi M

1 da bolo M

n wveroebiT) ewodeba gansxvavebuli wiboebis

raime M1M

2, M

2M

3, ..., M

n-1M

n mimdevrobas. Tu M

1 da M

n erTmaneTs emTxveva

jaWvs cikli ewodeba.

am suraTebze A da B, C da D, K da L wertilebi jaW viT ukav Sirdeba erT-ma neTs, K da L-is dama-kavSirebeli jaWvi cikls war moadgens.

Tu grafis wveroTa yoveli wyvilisTvis arsebobs maTi damakavSirebeli jaWvi, maSin grafs bmuli ewodeba.

nebismier grafs vuwodoT brtyeli grafi, Tu SeiZleba misi sibrtyeze gamosaxva ise, rom wiboebs wveroebis garda saerTo wertilebi ar hqondes, grafis wveroebs Seesabamebodes sibrtyis sxvadasxva wertilebi da es wertilebi grafis wiboTa Sesabamisi wirebisTvis SeiZleba mxolod boloebs warmoadgendes.

suraTze gamosaxulia brtyeli bmu-li grafi, romelic sibrtyes yofs 5 nawilad. V nawili usasruloa _ is gaferadebulia. TiToeuli nawili grafis waxnagia (waxnagis nebismieri ori wertili SeiZleba SevaerToT wiriT, romelic mTlianad am waxnagSia). am grafs 8 wvero da 11 wibo aqvs:

8+5-11=2;

SeamowmeT es Tviseba magaliT 3-Si ganxiluli grafebisTvis.

winaswar vigulisxmoT, rom mocemuli grafi sibrtyeze gamovsaxeT. yoveli brtyeli grafisTvis es gamosaxva SeiZleba ise ganxorcieldes, rom wveroebis, waxnagebisa da wiboebis odenobebi ar Seicvalos.

AB

DC K L

I

II

III

IV

V

bmuli grafi ar aris bmuli, Tumca igi Sedgeba ori bmuli nawilisgan.

eileris Terema

vTqvaT, G brtyeli bmuli grafia, v, e da f, Sesabamisad, aris am grafis wveroebis, wiboebisa da waxnagebis odenobebi; maSin

v + f - e=2.

65

eileris Teoremis dasabuTebis procesis warmosadgenad grafSi iseTi cvlilebebi SevitanoT, romlebic v+f-e gamosaxulebis mniSvnelobas ar cvlis. magaliTad,

pirvel SemTxvevaSi `amovagdeT~ erTi wibo, amiT waxnagebis ricxvi erTiT Semcirda, wveroebis ricxvi ucvlelia. meore SemTxvevaSi amovagdeT `dakiduli~ wvero saTanado wibosTan erTad, waxn

matem12_

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