1-amaliy mashg’ulot. To’plamlar va ular ustida amallar. Eyler

Misоl 2. Strixlangan sohani ^A , ^B , C top`lamlar orqali tasvirlang. Bunda <sup>A</sup> ,
<sup>B</sup> , C to`plamlar bitta universumga tegishli.
Bu masalani yechishning ham bir nechta usullari mavjud.

1-usul: 2-usul:
( А ∩ В \ С) ∪ ( А ∩ С \ В) ∪ (В ∩ С \ А)


АВС

To‘plamlar ustida amallarning asosiy xossalari.

U universаl to‘plаmning ^A , ^B , C qism to‘plаmlаri uchun quyidаgi хоssаlаr o‘rinli (ba’zi xossalarning isbotini keltiramiz, qolganlari shunga o’xshash isbotlanadi. Isbotni Eyler-Venn diagrammasida bajarish ham mumkin):

Kоmmutаtivlik (o`rin almashtirish) xossasi: 1⁰)

A ∪ B  B ∪ A

2⁰ )
A ∩ B  B ∩ A

1⁰ –xossaning isboti:
x  A ∪ B
bo`lsa, u holda
x  A va
x  B
bo`ladi.

Shuningdek,
x  B ∪ x  A
bo`lsa,
x  B ∪ A
kelib chiqadi. Bundan

x  A ∪ B  x  B ∪ A
hosil bo`ladi. Bularni umumlashtirilsa,
A ∪ B  B ∪ A

kоmmutаtivlik xossasi isbotlanadi.

Аssоtsiyаtivlik (guruhlash) xossasi: 3⁰)

( A ∪ B) ∪ C  A ∪ (B ∪ C)

4⁰) ( A ∩ B) ∩ C  A ∩ (B ∩ C)

Distributivlik (taqsimot qonunlari) xossasi:

5⁰)
6⁰)
( A ∪ B) ∩ C  ( A ∩ C) ∪ (B ∩ C)
( A ∩ B) ∪ C  ( A ∪ C) ∩ (B ∪ C)

Yutilish qоnunlаri: 7⁰)
8⁰)
A ∩ ( A ∪ B)  A A ∪ ( A ∩ B)  A

De Mоrgаn qоnunlаri (Ogastes de-Morgan (1806-1871yy) Shotlandiyalik matematik va mantiqchi, mantiqiy munosabatlar asoschisi):

9⁰ – xossaning isboti:
9⁰)
10⁰)
А ∩ В  А ∪ В А ∪ В  А ∩ В

А ∩ В  x : x  ( A  B) x : x  ( A  B) x : (x  A)  (x  B);
A ∪ B  x : (x  A)  (x  B)  x : x  A  x  B x : (x  A)  (x  B).

0 vа 1 (bo`sh va universal to`plam) qоnunlаri:

11^0)	А ∩ А  А	12^0)	А ∪U  U
13^0)	А ∪ А  U	14^0)	А ∩ Ø=Ø
15^0)	А ∩ А  Ø	16^0)	U  Ø
17^0)	А ∪ Ø=A	18^0)	=U
19^0)	А ∩U  A	20^0)	A\A= Ø
Ayirishdan qutilish qonuni:	21^0)	A\ B  A ∩ B
Ikkilаngаn rаd etish qоnuni:	22^0)	A  A

To’plamlar ustida amallarning xossalariga e’tibor berib qaraydigan bo’lsak, ular juft – juft yozilgan va har ikkinchisi birinchi xossada amalni o’zgartirish bilan hosil qilingan deyish mumkin, masalan, ∪ amali ∩ ga, 

^to’plam U ^{ga almashtirib hosil qilingan. Xossalarning bunday mosligi}
ikkiyoqlamalik qonunlari deyiladi.

Murakkab ifоdаlаrni sоddаlаshtirish.

To‘plаmlаr ustidа аmаllаrning аsоsiy хоssаlаrigа asoslanib, to’plamlarning murakkab ifоdаlаrini isbotlash yoki sоddаlаshtirish mumkin.

Misоl 3.

AB  ( A ∪ B) ∩ A ∩ B
(1) ifodani isbotlang.

Yechilishi:

АВ  ( A \ B) ∪ (B \ A)

yoki Eyler-Venn diagrammasidan
АВ  ( A ∩ B) ∪ (B ∩ A)
tenglikni hosil qilish mumkin.
( A ∪ B) ∩ A ∩ B  (9⁰-xossadan foydalanamiz)
 ( A ∪ B) ∩ ( A ∪ B)  (2⁰-xossa)

 ( A ∪ B) ∩ ( A ∪ B)  (5⁰-xossa)  A ∩ ( A ∪ B)∪ B ∩ ( A ∪ B)( 5⁰-xossa)
 ( A ∩ A) ∪ ( A ∩ B)∪ (B ∩ A) ∪ (B ∩ B) (15⁰-xossa)   ∪ (B ∩ A)∪ ( A ∩ B) ∪ 
 ( A ∩ B) ∪ (B ∩ A).

Bundan talab qilingan tenglikni hosil qilamiz.

AB  ( A ∪ B) ∩ A ∩ B.

Misоl 4.

A ∪ ( A \ B) ∪ ( A \ B)
ifodani soddalashtiring.

Yechilishi:

A ∪ ( A \ B) ∪ ( A \ B) (21⁰-xossa)= A ∪ (A ∩ B) ∪ (A ∩ B) 

(22⁰-xossa)  A ∪ ( A ∩ B) ∪ ( A ∩ B)  (10⁰-xossa)  A ∩ A ∩ B ∩ A ∩ B (9⁰-xossa)=
[ A ∩ ( A ∪ B)] ∩ ( A ∪ B) (22⁰-xossa)  ^( A ∩ A) ∪ ( A ∩ B)^∩ ( A ∪ B )  (15⁰-xossa).
 A ∩ B ∩ A∪ ( A ∩ B ∩ B )  A ∪ B ∩ B.
Mustaqil bajarish uchun masala va topshiriqlar


1.1. Eyler-Venn diagrammalariga doir topshiriqlar
Quyidagi misollarnig shartlarida Universal to‘plam U={ a, b, c,

d, e, f, g,h } da X va Y to‘plamlar berilgan bo‘lib,

X  Y , ^Y ,

^{X Y} ,
X ∩ Y _,

X \ Y
to‘plamlarni A, B, C lar orqali ifodalang va Eyler-

Venn diagrammalrida tasvirlang.

1.1.0	X={a,b,c,d}, Y={b,c,d,e}	1.1.10	X={c,d,e,f}, Y={e,f,g,h}	1.1.20	X={e,f,g,h}, Y={h,a,b,c}
1.1.1	X={b,c,d,e}, Y={c,d,e,,f}	1.1.11	X={d,e,f,g}, Y={f,g,h,a}	1.1.21	X={f,g,h,a}, Y={a,b,c,d}
1.1.2	X={c,d,e,,f}, Y={d,e,f,g}	1.1.12	X={e,f,g,h}, Y={g,h,a,b}	1.1.22	X={g,h,a,b}, Y={b,c,d,e}
1.1.3	X={d,e,f,g}, Y={e,f,g,h}	1.1.13	X={f,g,h,a}, Y={h,a,b,c}	1.1.23	X={h,a,b,c}, Y={c,d,e,,f}
1.1.4	X={e,f,g,h}, Y={a,f,g,h}	1.1.14	X={g,h,a,b}, Y={a,b,c,d}	1.1.24	X={a,b,e,f}, Y={c,d,e,,f}
1.1.5	X={a,,f,g,h}, Y={a,b,g,h}	1.1.15	X={h,a,b,c}, Y={b,c,d,e}	1.1.25	X={b,c,f,g}, Y={d,e,,f,g}
1.1.6	X={a,b,g,h}, Y={a,b,c,h}	1.1.16	X={a,b,c,d}, Y={d,e,f,g}	1.1.26	X={c,d,g,h}, Y={e,g,h,a}
1.1.7	X={a,b,c,h}, Y={a,b,c,d}	1.1.17	X={b,c,d,e}, Y={e,f,g,h}	1.1.27	X={d,e,h,a}, Y={g,h,a,b}
1.1.8	X={a,b,c,d}, Y={c,d,e,,f}	1.1.18	X={c,d,e,f}, Y={f,g,h,a}	1.1.28	X={e,f,a,b}, Y={h,a,b,c}
1.1.9	X={b,c,d,e}, Y={d,e,f,g}	1.1.19	X={d,e,f,g}, Y={g,h,a,b}	1.1.29	X={f,g,b,c}, Y={a,b,c,d}

1. 
   Eyler-Venn diagrammalri doir topshiriq(na’muna).
   

U={ a, b, c, d, e, f, g,h } da X={a,b,c,d} va Y={b,c,d,e} to‘plamlar berilgan

bo‘lib,

X  Y , ^Y ,
^{X Y} ,
X ∩ Y _,

X \ Y

to‘plamlarni A, B, C lar orqali

ifodalang va Eyler-Venn diagrammalrida tasvirlang.

1. Topshiriqning bajarilishi bo’yicha na’muna

X  Y  a, b, c, d b, c, d , e  a, b, c, d , e  f , g, h A  B  A  B  C
Y  b, c, d , e  a, f , g, h  A  B  C  A  B  A  B  C

X Y  a, b, c, db, c, d , e  e, f , g, hb, c, d , e  b, c, d , f , g, h 
 B  AC
X  Y  a, b, c, d b, c, d, e a, b, c, d a, f , g, h a

 A  B  C

X \ Y  a, b, c, d\ b, c, d , e  e, f , g, h\ a, f , g, h  e  A  B  C

2.