Tuzuvchilar: V. Jo`Rayev

To`plamlar ayirmasi quyidagi xossalarga ega:

1. 
   A B    A \ B  A
   

2. A  B  A \ B  

3. 
   A \ _B
   
4. 
   A \ _B
   

5. A \  A

B \ C

6.  \ A  

7. A \ A 

TO`PLAMLARNING DEKART KO`PAYTMASI

A va B to`plamlarning dekart ko`paytmasi deb, 1-elementi A to`plamdan, 2 – elementi B to`plamdan olingan _a; b_ ko`rinishdagi barcha tartiblangan juftliklar to`plamiga

aytiladi. Dekart ko`paytma A B

ko`rinishda belgilanadi:

A B _a; b_ a  A va b  B_.

Masalan:

A _2; 3; 4; 5_,

B _a; b; c_ bo`lsa,

A B  {_2; a_,_2; b_,_2; c_,_3; a_,_3; b_,

_3; c_,_4; a_,_4; b_,_4; c_,_5; a_,_5; b_,_5; c_} bo`ladi.

Sonli to`plamlar dekart ko`paytmasini koordinata tekisligida tasvirlash qulay.

Masalan: A  _2; 3; 4_, B  _4; 5_ bo`lsin, u holda A B _2; 4_,_2; 5_,_3; 4_,_3; 5_,_4; 4_,_4; 5_

bo`ladi.

Koordinata tekisligida shunday koordinatali nuqtalarni tasvirlaymizki, bunda A to`plam

Ox o`qida va B to`plam Oy o`qida olinadi.

A to`plamning B to`plamga tegishli bo`lmagan elementlaridan va B to`plamning A

to`plamga tegishli bo`lmagan elementlaridan tuzilgan to`plamn A va B to`plamlarning

simmetrik ayirmasi deb ataladi va A B kabi belgilanadi, ya’ni

A B  _ A \ B_{ }B \ A_.

Misol:

A  _1,2,3,4,5,6,7_,

B _6,7,8,9,10_ bo`lsa,

A B  _1,2,3,4,5_{ }8,9,10_ 

_1,2,3,4,5,8,9,10_ bo`ladi.

X chekli to`plam elementlar sonini to`plamni k elementli to`plam deb ataymiz.
n_ X _
orqali belgilaymiz. k ta elementli X

Misol: X to`plam 10 dan kichik tub sonlar to`plami bo`lsin:

X _2,3,5,7_. Demak,

X to`plamda 4 ta elementdan tuzilgan ekan va u quyidagicha belgilanadi

n_ X _  4 .

BERILGAN TO`PLAMNING QISM TO`PLAMLARINI TOPPISH UCHUN QUYIDAGI FORMULADAN FOYDALANAMIZ

Agar to`plam elementlari

ko`rinishda bo`ladi.

n_ X _  m

bo`lsa, u holda qism to`plamlar soni 2^m

Misol:

X  _1, 2,3_ to`plamning

n_ X _  3

ta elementi bor. Qism to`plamlari soni

2³  8

bo`ladi va ular quyidagilardir:

¹

_1, 2<sub>

</sub>1, 3_

²

_2, 3_

³

_1, 2, 3_

hamda 

1. A _2, 4, 6, 8, 10_

B _4, 8, 12, 16_

10.

A _0,1,2,3,4,5_

B _3,4,5,6,7_

2. A  _1, 3, 5, 7, 9, ..., 2n 1,..._

11.

A _x x  R, x  9_

B _3, 6, 9, ...3n,..._

3. A _x : _ x  2_ x  3_  0_

12.

B _x x  R, 0  x  8_

A _x x  R, 5  x 12_

B _x : _ x  2_ x  4_  0_

13.

B _x x  R,  5  x  7_

A _x x  N, x 12_

4. A _x : x²  4  0_

B _x : x  2  0_

5. A _x x  R, 0  x  9_

B _x x  R, 5  x 12_

14.

B _x x  N, 6  x 15_

A _x x  N,  2  x 12_

B _x x  N, 5  x 12_

6. A - raqamlar to`plami.

B - yigirmadan kichik juft sonlar to`plami.

7. A _x x  R, x  5_

B _x x  R, x  3_

15. A - uchburchaklar to`plami.

B -teng tomonli uchburchaklar to`plami.

16. A _x x  Z,  6  x  6_

B _x x  Z, 0  x  8_

8. A _a,b,c,d,e_ B _d,e,n,m_

17. A -ikki xonali juft sonlar to`plami.

B - 6 ga karrali 100 dan kichik natural sonlar to`plami.

9. A _x x  R, 0  x  7_

B _x x  R, 6  x  9_

18.

A _x x  N,

B _x x  N,

x 10_

x  4_

19. A -to`g`ri to`rtburchaklar to`plami.

B -kvadratlar to`plami.

31.

A _x x  R, B _x x  R,

x  12_

 9  x  1_

20.

A _x x  R, 1  x  9_

B _x x  R, 5  x 12_
32. A - qish oylari to`plami

B ={Yanvar, Mart, Noyabr, Dekabr}

21. A - 3 ga karrali sonlar to`plami.

B - 5 ga karrali sonlar to`plami.

33.

A _x x  N,

B _x x  N,

x  1_

x  5_

22.

A _x x  R, B _x x  R,

 2  x  0_

1  x  5_
34. A - aylanalar to`plami

B - tekislikdagi figuralar to`plami

23. A - oxiri nol bilan tugallanadigan sonlar to`plami.

B - 5 ga karrali sonlar to`plami.

35.

A _x x  Z,

B _x x  Z,

x  0_

x  0_

24.

A _x x  N, 1  x  5_

36.

A _l,m,n,o, p_

25.

B _x x  R, x  3_

A _x x  R, x  7_

37.

B _o, x, y, z_

A _x x  N, x  4_

B _x x  R, x  9_

B _x x  N, 2  x  4_

26. A - daraxtlar to`plami

B - olma va o`rik daraxtlari to`plami.

38. A - 6 ga bo`linadigan sonlar to`plam

B - 2 ga va 3 ga bo`linmaydigan sonlar to`plami.

27.

A _x x  Z, x 1_

39.

A _x x  R, 18  x  25_

B _x x  Z, 0  x 11_

B _x x  R,  25  x  18_

28. A - O`zbekistondagi viloyatlar to`plami

B ={Buxoro, Toshkent, Xorazm, Samarqand}
40. A - 20 gacha bo`lgan natural sonlar to`plami.

B - 20 dan kichik bo`lmagan natural sonlar to`plami.

29.

A _x x  R,

B _x x  R,

x  0_

1 x  1_

41.

A _x x  N,

B _x x  N,

x  8_

x  3_

30. A - sinfdagi o`g`il bolalar to`plami

B - sinfdagi qiz bolalar to`plami.

42.

A _x x  R,

B _x x  R,

x  23_

x  13_

43. A - butun sonlar to`plami. 51. A - transport vositalari to`plami

B _0; 9; 13;

 5;

 7; 55_

B - velosiped va yengil mashinalar to`plami.

44.

A _x x  Z, 2,5  x  3,8_

52.

A _x x  R, 81  x  91_

B _x x  Z, 11 x  10_

B _x x  R, 5  x 15_

45. A - 11 ga bo`linuvchi sonlar to`plami

B  _111, 121, 131, 141,..., 291_

53.

A _x x  Z,

B _x x  Z,

x  1_

x  0_

46. A _x x  R, x  9_

B _x x  R, 10  x  68_

47. A - ikki xonali sonlar to`plami

B - 5 raqami bilan tugallanuvchi sonlar

54.

A  _18,21,24,...,48_

B - 3 ga karrali 50 dan kichik va 20 dan katta natural sonlar to`plami

to`plami.
48. A _x x  N, 3  x  4_

55. A _x x  N, x  22_

B _x x  N, x  22_

B _x x  N, 6  x  7_

49. A - qiz bolalar to`plami

B ={Dilnoza, Sherali, Nozima, Fayoz, Zarif}

56. A - romblar to`plami

B - kvadratlar to`plami.
57. A _x x  R,  8  x  48_

B _x x  N,  2  x  42_

50.

A _x x  N, 1 x  12_

B _x x  N, x  8_
58. A - 36 ning bo`luvchilaridan tashkil topgan sonlar to`plami

B - 3 ga karrali sonlar to`plami

1. C^ ni toping?

U {ingliz tili harflari}, C {unli harflar}
2. U _x 0  x 12, x  _,

A _x  2  x  7, x  _ bo`lsa, A^ ni

B _x  3  x  9, x  _,

C _x  5  x  11, x  <sub>

</sub> A ni toping?
10. n_U _  15, n_Q^_  4 bo`lsa,

bo`lsa,

n_Q_  ?

toping?
3. C^ ni toping?

U  , C _x x  5, x 

11. U  , P  {25 dan kichik bo`lgan tub <sup>sonlar} bo`lsa, n</sup>^P ^{ni toping?}

12. U _x 0  x  8, x  _,

4. 
   U _x
   

A _x

0  x  8, x  _,

2  x  7, x  _,

A _x

2  x  7, x  _ bo`lsa,

A^  ?

B _x

5  x  8, x  _ bo`lsa, A

13. U _x

A _x

 5  x  5, x  _,

 2  x  7, x  _,

4. 
   C^ ni toping?
   

U {butun sonlar}

C {manfiy butun sonlar}

B _x

A^

 3  x  2, ni toping?

x  _

bo`lsa,

6. U x A x		0  x 12, x  ,  2  x  7, x  ,		14. U x 0  x  8, x  , B x 5  x  8, x   bo`lsa,	B^ ?
C x		 5  x  11, x  	bo`lsa,	15. U  , Q 2; 4; 5; 11; 12; 15	bo`lsa,
A^C 7. C^ni to U  , ^8.^n^U ^ 9. U x A x		ni toping? ping? C x x  2 yoki x  8, ^15,^n^P ^^6 bo`lsa,^n 0  x 12, x  ,  2  x  7, x  ,	x   P^  ?	Q^ni toping? 16. U x 0  x 12, x  , B x  3  x  9, x    ^A^B^ ^ni^to^p^ing^?	bo`lsa,

A _x

 2  x  7,

x  _,

17. U _x

 5  x  5,

x  _,

22. U _x

 5  x  5,

x  _,

A _x 1 x  5,

x  _

bo`lsa, A^ ni

A _x 1 x  5,

x  _,

toping?

B _x

 3  x  2,

x  _

bo`lsa,

18. 
    U _x
    

 5  x  5,

x  _,

A B^ ni toping?

A _x 1 x  5,

x  _,

23. U _x

0  x 12,

x  _,

B _x

A^

18. 
    U _x
    

A _x B _x

 3  x  2, ni toping?

0  x 12,

 2  x  7,

 3  x  9,

x  _
x  _,

x  _,

x  _

bo`lsa
bo`lsa,

C _x B _x B^

24. 
    U _x
    

B _x

 5  x  11,

 3  x  9, ni toping?
0  x 12,

 3  x  9,

x  _,

x  _
x  _,

x  _

bo`lsa,

bo`lsa,

 ^{A ni toping?}

n_B^_

ni toping?

20. U _x  5  x  5,

x  _,

24. 
    U _x
    

0  x  8, x  _,

A _x 1 x  5, x  _,

A _x

2  x  7, x  _,

B _x  3  x  2,

x  _

bo`lsa,

B _x

5  x  8, x  _ bo`lsa, A^

A^
21. U _x

C _x

n_C^_

ni toping?

0  x 12,

 5  x  11, ni toping?

x  _,

x  _

bo`lsa,

ni toping?

1. A  _1, 3_,

B _2, 4_

10.

A _x x 

,  3  x  4_

2. A _x : 0  x  2_  _0; 2_,

B _x x 

, 3  x  2_

B _x :1 x  3_  _1; 3_

11. To`plamlar dekart ko`paytmasining tasviriga qarab, to`plamlarni yozing.

3. A _x x  , 0  x  7_,

B _x x  ,  3  x  2_

4. 
   A _x
   

B _x

x   7  x  7_

x   3  x  3_

4. 
   To`plamlar dekart ko`paytmasining tasviriga qarab, to`plamlarni yozing.
   

12.

A _4; 5; 6; 7; 8; 9; 10; 11_,

B _5;  4;  3;  2; 1; 0; 1; 2; 3_

13.

A _x x  B _x x 

4  x 10_,

9  x  10_

6. A _x : x_ x  3_  0_,

B _x : _ x  3_x 1_  0_

7. A _x : xR, x²  8x 15  0_,

B _x : x  N, x²  6x  0_

8. A _x : x  Z,  2  x  4_,

B _x : x  N, 1 x  7_

14. To`plamlar dekart ko`paytmasining tasviriga qarab, to`plamlarni yozing.

15.

A _x x 

2_,

9. A _x x  , x  2_,

B _x

x   3  x  12_

B _x x  , x  2_

16.

A _x x 

6_,

18.

A _x 1  x  5,

x  _,

B _x x 

 5_

B _x

 3  x  2,

x  _

17.

A  _1; 2; 3; 4; 5; 6_,

B _2; 1; 0; 1; 2; 3; 4_

19.

A _a,b,c,d,e_ , B _d,e,n,m_

20.

A _2, 4, 6, 8, 10_,

B _4, 8, 12, 16_

1. U  _1, 2, 3, 4, 5, 6, 7, 8_

bo`lsa, quyidagi to`plamni Venn diagrammasida tasvirlang va

A ni toping?

A  _1, 3, 6, 8_ va

B _2, 3, 4, 5, 8_

2. 
   A va B to`plamlarni Venn diagrammasida tasvirlang:
   

U _2, 3, 4, 5, 6, 7_,

A _2, 4, 6_,

B _5, 7_

2. 
   Quyida berilgan to`plamlarni Venn diagrammasida bo`yab tasvirlang:
   

1) A ; 2) B ; 3) C ; 4) B^ ;

5) A^ ; 6) C^ 7) A B ; 8) A B ;

9) B C ; 10) B C ; 11) A C ;

12) A C ; 13) A B C ;

^{14) A B C ; 15)}  ^{A B C} ^{ ;}

¹⁶⁾  ^{A B}

¹⁸⁾  ^{A B}^

_U ²⁰⁾  ^{A C} 

^{C ; 17)} ^{B C}  ^{A; C ; 19)} ^{B C} ^{ A;} B .

2. 
   U _x 1  x 10, x  _, A  {10 dan kichik bo`lgan toq sonlar} va B  {10 dan
   

kichik bo`lgan tub sonlar} bo`lsa, quyidagi to`plamlarni Venn diagrammasida tasvirlang:

1) A B ; 2) A^

B ; 3) A^

B ; 4) A B ; 5) A B^;

^{6) A B; 7)}  ^{A B}^{; 8)}  ^{A B}^{; 9) A}

2. 
   A va B to`plamlarni Venn diagrammasida tasvirlang:
   

B _3, 5_.

B^; 10) A^

U _3, 4, 5, 7_,

B^.

A _3, 4, 5, 7_,

2. 
   A va B to`plamlar Venn diagrammasida tasvirlangan. Quyidagi to`plamlarning elementlarini yozing:
   

1) A ; 2) B ; 3) A^ ; 4) B^ ; 5) A B ; 6) A B ; 7) A^ B ; 8) A^ B ;

9) _ A B_^; 10) _ A B_^; 11) A^

B^; 12) A^

B^; 13) A B^; 14) A B^;

7. U  _1, 2, 3, 4, 5, 6, 7, 8_

bo`lsa, quyidagi to`plamni Venn diagrammasida tasvirlang va

 ^{A ni toping?}

A  _1, 3, 6, 7, 8_ va

B _3, 6, 8_

8. 
   A va B to`plamlarni Venn diagrammasida tasvirlang:
   

U  _1, 2, 3, 4, 5, 6, 7_,

A _2, 4, 5, 6_ ,

B  _1, 4, 6, 7_ .

8. 
   Quyida berilgan to`plamlarni Venn diagrammasida bo`yab tasvirlang:
   

1. 
   A B ; 2) A B ;
   

3) A B^; 4) A^ B ;

5) A^ B ; 6) A B^;

⁷⁾  ^{A B}^{; 8)}  ^A

U

8. 
   A
   

va B to`plamlarni Venn diagrammasida tasvirlang:

U _2, 3, 4, 5, 6, 7_,

A _2, 4, 6_, B _3, 5, 7_.

8. 
   U _x 1  x 10, x  _, A  {10 dan kichik bo`lgan toq sonlar} va B  {10 dan kichik
   

bo`lgan tub sonlar} bo`lsa, A va B to`plamlarning elementlarini yozing va Venn diagrammasida tasvirlang?

8. 
   A , B va C to`plamlar Venn diagrammasida tasvirlangan. Quyidagi to`plamlarning elementlarini yozing:
   

1) A ; 2) B ; 3) C ; 4) A B ; 5) A B ; 6) B C ; 7) B C ;

8) A

9) A C ; 10) A B C ; 11) A B C ; 12)

n_ A B C _ ;

13) n_ A_  n_B_  n_C _  n_ A B_  n_ A C _  n_B C _  n_ A B C _

U

1. _x x  N,  5  x  5_

to`plamni nechta usul bilan ikkita kesishmaydigan qism

to`plamlarga ajratish mumkin?

2. M _36, 29, 15, 68, 27_,

P _4, 15, 27, 47, 36, 90_,

Q _90, 4, 47_

to`plamlar berilgan.

M P , M Q , P Q , M larni toping.

3. A _2, 3, 4, 5, 7, 10_,

B _3, 5, 7, 9_ ,

C _4, 9, 11_

bo`lsin. Quyidagi to`plamlarda

nechtadan element mavjud:

1. 
   A b) A
   

c) A d) C

4. 
   _x x  N, x²  23_ to`plamning nechta qism to`plamlari mavjud?
   

4. 
   A _x
   

 5  x  10_,

B _x x  N, 3  x  15_

bo`lsa,

A \ B va

B \ A to`plam

elementlarini toping.

4. 
   _x x  N, 6  x²  40_ to`plamning nechta qism to`plamlari mavjud?
   
5. 
   A - 18 ning hamma natural bo`luvchilari to`plami, B - 24 ning hamma natural
   

bo`luvchilari to`plami bo`lsa, A to`plam elementlarini ko`rsating.

4. 
   _x x  N, 2  x²  44_ to`plamning nechta qism to`plamlari mavjud?
   
5. 
   P _a, b, c, d, e, f _, E _a, g, z, e, k_ to`plamlar birlashmasini toping.
   
6. 
   _x x  N,  2  x  5_ to`plamning nechta qism to`plamlari mavjud?
   
7. 
   _x x  N, x²  17_ to`plamning nechta qism to`plamlari mavjud?
   

TAKRORLASH UCHUN TEST

1. 
   A to`plam
   

x² 11x 18  0

tengsiz-

6. A _2; 3; 4; 5; 7; 10_,

B _3; 5; 7; 9_ ,

likning yechimlari to`plami, B to`plam ^esa x² 15x  50  0 ^{tengsizlikning} yechimlari to`plami bo`lsa, A to`plamni toping.

A) [2; 10] B) _5;10_

C _4; 9; 11_ bo`lsin. A to`plamning nechta elementi mavjud. A) 4 B) 8 C) 16 D) 9

7. 
   _x x  N,  2  x  5_ to`plamning nech-
   

C) _2;9_ D) _5;9_

ta qism to`plamlari mavjud?

A) 30 B) 8 C) 16 D) 32

^{2. A to`plam} _x²  9x  20_x²  5x  6_  0

tenglamaning ildizlari to`plami, B

7. 
   A to`plam y 
   

funksiya-

to`plam esa 1 x 12

tengsizlikni

ning aniqlanish sohasi, B to`plam esa

qanoatlantiruvchi tub sonlar to`plami ^{y }

funksiyaning aniqlanish

bo`lsin. A

A) _2,3,7,11_

to`plamni toping?

B) _2,3,5_

sohasi bo`lsin. A

1. 
    B)
   

to`plamni toping.

x  4, x  6

C) _2,3_

D) 

C) 2  x  2

D) 2  x  6

3. A to`plam 28 sonining barcha natural bo`luvchilari to`plami, B to`plam esa

42 sonining barcha natural bo`luvchilari

9. A - barcha tub sonlar to`plami, B - barcha juft sonlar to`plami bo`lsa, A to`plamni toping.

to`plami bo`lsa, A

to`plamni

^{A)  B)} ²

toping.

A) _2, 7, 14_

B) _4, 6, 7_

C) _2, 3, 5, 6, 7, ..._ D) _4, 12, 30, ..._

C) _1, 2, 7, 14_

4. A _2, 4, 6, ..._,

D) _7, 14_

B  _1, 3, 5, ..._

bo`lsa,

10.

^{A }^{a, b, c}<sup>to`plamning xosmas qism</sup> to`plamlari soni nechta?

A) 2 ta B) 6 ta C) 8 ta D) 4 ta

A to`plamni toping.

A) _2, 12, 30, ..._ B) _3, 7, 11, ..._

11.

A _x 0  x  5_, B _x 2  x  8_

C)  D) N  _1, 2, 3, ..._

bo`lsa, A ni toping.

A) _2; 13_ B) _5; 8_

5. 
   26 o`quvchining 14 tasi shaxmatga, 16 tasi shashkaga qiziqadi. Ham shashka- ga, ham shaxmatga qiziqadigan o`quv- chilar nechta?
   
   1. 
      5 ta B) 4 ta C) 6 ta D) 7 ta
      

C) _0; 8_ D) _2; 5_

12. _x x  N, x²  17_ to`plamning nechta qism to`plamlari mavjud?

A) 12 B) 16 C) 8 D) 32

13.

A _x 2  x  7_,

B _x 3  x  9_

17. 
    1 x  50 kesmadagi 3 ga; 4 ga; 7 ga
    

bo`lsa, A ni toping.

A) _2; 9_ B) _7; 9_

bo`linadigan barcha natural sonlar to`plami mos ravishda A , B , C bo`lsin.

C) _2; 3_

D) _3; 7_

A

toping.

to`lam elementlari sonini

14.

A _p  tub son 2  x  32_,

B _p  tub son 7  x  28_ bo`lsa, B to`plamning A to`plamgacha to`ldiruv- chisini – A ning B ga tegishli bo`lma- gan barcha elementlaridan tuzilgan to`plamni toping.

A) _2, 3, 5, 29, 31_ B) _29, 31_

C) _p  tup son 9  x  30_ D) 

A) 23 B) 19 C) 28 D) 35

17. 
    10-sinf o`quvchilarining hammasi 3 ta to`garakka qatnashadi. Matematika to`garagiga 18 ta, fizika to`garagiga 15 ta, adabiyot to`garagiga 14 ta o`quvchi qatnashadi. Ham matematika, ham fizika to`garaklariga 6 ta o`quvchi, ham matematika ham adabiyot to`garagiga 4 ta o`uvchi, ham fizika ham adabiyot to`garagiga esa 5 ta o`quvchi qatnashadi.
    

15. 1 x 100

kesmadagi 10 ga bo`linadi-

1 ta o`quvchi uchala to`garakka ham

gan barcha natural sonlar to`plami A bo`lsa, shu kemadagi 25 ga bo`linadi- gan barcha natural sonlar to`plami B

qatnashadi. 10-sinfda necha nafar o`quvchi bor?

A) 36 B) 40 C) 38 D) 33

soni nechta?					19.	^A^^a^,^b^,^c^to`plamning	barcha	qism
A) 14 B) 12		C) 4	D) 18			to`plamlari soni nechta?
16.	A x x  2k 1, k  Z to`plamning to`plamga to`ldiruvchisini toping? A) A x x  2k 1, k  Z A x x  k, k  Z A x x  2k, k  Z A x x  k 1, k  Z			Z	20.	A) 8 B) 27 C) 9 D) 6 ^A^^a^,^b^,^c^,^d ^to`plamning^barchaqism to`plamlari soni nechta? A) 64 B) 16 C) 81 D) 8

bo`lsa, A to`plamning elementlari

TAKRORLASH UCHUN TEST JAVOBLARI

1	2	3	4	5	6	7	8	9	10
A	B	C	D	B	A	D	A	B	A
11	12	13	14	15	16	17	18	19	20
C	B	D	A	B	C	C	D	A	B

QO`SHIMCHA MA’LUMOTLAR UCHUN

QO`SHIMCHA MA’LUMOTLAR UCHUN