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Misоl 2. Strixlangan sohani A , B , C top`lamlar orqali tasvirlang. Bunda A ,
B , 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)
АВС
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:
110)
|
А ∩ А А
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120)
|
А ∪U U
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130)
|
А ∪ А U
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140)
|
А ∩ Ø=Ø
|
150)
|
А ∩ А Ø
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160)
|
U Ø
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170)
|
А ∪ Ø=A
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180)
|
=U
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190)
|
А ∩U A
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200)
|
A\A= Ø
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Ayirishdan qutilish qonuni:
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210)
|
A\ B A ∩ B
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Ikkilаngаn rаd etish qоnuni:
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220)
|
A A
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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.
AB ( A ∪ B) ∩ A ∩ B
(1) ifodani isbotlang.
Yechilishi:
АВ ( A \ B) ∪ (B \ A)
yoki Eyler-Venn diagrammasidan
АВ ( 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.
AB ( 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
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X={a,b,c,d},
Y={b,c,d,e}
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1.1.10
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X={c,d,e,f},
Y={e,f,g,h}
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1.1.20
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X={e,f,g,h},
Y={h,a,b,c}
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1.1.1
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X={b,c,d,e},
Y={c,d,e,,f}
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1.1.11
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X={d,e,f,g},
Y={f,g,h,a}
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1.1.21
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X={f,g,h,a},
Y={a,b,c,d}
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1.1.2
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X={c,d,e,,f},
Y={d,e,f,g}
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1.1.12
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X={e,f,g,h},
Y={g,h,a,b}
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1.1.22
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X={g,h,a,b},
Y={b,c,d,e}
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1.1.3
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X={d,e,f,g},
Y={e,f,g,h}
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1.1.13
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X={f,g,h,a},
Y={h,a,b,c}
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1.1.23
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X={h,a,b,c},
Y={c,d,e,,f}
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1.1.4
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X={e,f,g,h},
Y={a,f,g,h}
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1.1.14
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X={g,h,a,b},
Y={a,b,c,d}
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1.1.24
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X={a,b,e,f},
Y={c,d,e,,f}
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1.1.5
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X={a,,f,g,h},
Y={a,b,g,h}
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1.1.15
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X={h,a,b,c},
Y={b,c,d,e}
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1.1.25
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X={b,c,f,g},
Y={d,e,,f,g}
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1.1.6
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X={a,b,g,h},
Y={a,b,c,h}
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1.1.16
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X={a,b,c,d},
Y={d,e,f,g}
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1.1.26
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X={c,d,g,h},
Y={e,g,h,a}
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1.1.7
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X={a,b,c,h},
Y={a,b,c,d}
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1.1.17
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X={b,c,d,e},
Y={e,f,g,h}
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1.1.27
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X={d,e,h,a},
Y={g,h,a,b}
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1.1.8
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X={a,b,c,d},
Y={c,d,e,,f}
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1.1.18
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X={c,d,e,f},
Y={f,g,h,a}
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1.1.28
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X={e,f,a,b},
Y={h,a,b,c}
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1.1.9
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X={b,c,d,e},
Y={d,e,f,g}
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1.1.19
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X={d,e,f,g},
Y={g,h,a,b}
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1.1.29
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X={f,g,b,c},
Y={a,b,c,d}
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1.1.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.
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, db, c, d , e e, f , g, hb, c, d , e b, c, d , f , g, h
B AC
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
Murakkab to‘plamlarni soddalashtirishga doir topshiriqlar.
1.2.1
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𝑋̅̅̅𝖴̅̅̅𝑌̅ ∩ 𝑋̅̅̅̅𝖴̅̅̅𝑌̅ 𝖴 ̅𝑋̅̅∩̅̅̅𝑌̅
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1.2.15
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A/ B ∪ A/ C ∪ A/ B / C ∩ A ∩ B ∩ C
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1.2.2
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̅𝐴̅̅∩̅̅̅𝐵̅̅̅𝖴̅̅̅𝐴̅̅ ∩ ̅𝐵̅̅∩̅̅̅𝐴̅̅̅̅𝖴̅̅̅𝐵̅
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1.2.16
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(𝐴 𝖴 𝐵 𝖴 𝐶) ∩ (𝐴 𝖴 𝐵 𝖴 𝐶̅)
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1.2.3
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( A \ B ∪ A ∩ B) ∩ А
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1.2.17
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(𝐴 𝖴 𝐵 𝖴 𝐶) ∩ (𝐴̅ 𝖴 𝐵 𝖴 𝐶)
|
1.2.4
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(B\A) ∩ (𝐴̅ 𝖴B\A)
|
1.2.18
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(𝐴 𝖴 𝐵 𝖴 𝐶) ∩ (𝐴 𝖴 𝐵̅ 𝖴 𝐶)
|
1.2.5
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A B \ С A B \ С
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1.2.19
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𝐴̅ ∩ 𝐵̅ ∩ 𝐶̅ 𝖴 𝐴̅ ∩ 𝐵̅ ∩ 𝐶 𝖴 𝐴 ∩ 𝐵̅ ∩ 𝐶̅
𝖴 𝐴 ∩ 𝐵 ∩ 𝐶̅
|
1.2.6
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A B \ С A B \ С
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1.2.20
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𝐴̅ ∩ 𝐵 𝖴 𝐴 ∩ 𝐵̅ 𝖴 𝐴 ∩ 𝐵
|
1.2.7
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A B A B
|
1.2.21
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𝐴̅ ∩ 𝐵 𝖴 𝐴̅ ∩ 𝐵̅ 𝖴 𝐴 ∩ 𝐵
|
1.2.8
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𝐴̅̅̅̅𝖴̅̅̅𝐵̅̅̅𝖴̅̅̅𝐶̅̅ 𝖴 𝐴̅̅̅̅𝖴̅̅̅𝐵̅̅̅𝖴̅̅̅𝐶̅̅
|
1.2.22
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𝐴̅ ∩ 𝐵̅ 𝖴 𝐴 ∩ 𝐵̅ 𝖴 𝐴 ∩ 𝐵
|
1.2.9
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𝐴∆(𝐴∆𝐵)
|
1.2.23
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𝐴̅ ∩ 𝐵̅ 𝖴 𝐴 ∩ 𝐵̅ 𝖴 𝐴̅ ∩ 𝐵
|
1.2.10
|
A (B𝖴 𝐶) 𝖴 𝐴 ∩ 𝐵 ∩ 𝐶 𝖴 𝐴 ∩
𝐵 ∩ 𝐶̅
|
1.2.24
|
𝐴̅ ∩ 𝐵̅ ∩ 𝐶 𝖴 𝐴̅ ∩ 𝐵 ∩ 𝐶 𝖴 𝐴 ∩ 𝐵 ∩ 𝐶̅
𝖴 𝐴 ∩ 𝐵̅ ∩ 𝐶̅
|
1.2.11
|
А ∩ ( А ∪ В ∩ С) ∩ В ∩ С ∪ А ∩ В ∩ С
|
1.2.25
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𝐴̅ ∩ 𝐵̅ ∩ 𝐶̅ 𝖴 𝐴̅ ∩ 𝐵 ∩ 𝐶̅ 𝖴 𝐴 ∩ 𝐵 ∩ 𝐶
𝖴 𝐴 ∩ 𝐵̅ ∩ 𝐶
|
1.2.12
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С ∩ (С ∪ В ∩ А) ∩ В ∩ А ∪ С ∩ В ∩ А
|
1.2.26
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𝐴̅ ∩ 𝐵 ∩ 𝐶̅ 𝖴 𝐴̅ ∩ 𝐵 ∩ 𝐶 𝖴 𝐴 ∩ 𝐵 ∩ 𝐶
𝖴 𝐴 ∩ 𝐵 ∩ 𝐶̅
|
1.2.13
|
𝐴 ∩ 𝐵 𝖴 (𝐵̅ ∩ (𝐴 ∩ 𝐶̅ 𝖴 𝐴̅)) 𝖴 (𝐶̅
∩ 𝐵̅)
|
1.2.27
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𝐴̅ ∩ 𝐵̅ ∩ 𝐶̅ 𝖴 𝐴 ∩ 𝐵̅ ∩ 𝐶 𝖴 𝐴 ∩ 𝐵̅ ∩ 𝐶̅
𝖴 𝐴̅ ∩ 𝐵̅ ∩ 𝐶
|
1.2.14
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𝐴̅ ∩ 𝐵̅ ∩ 𝐶̅ 𝖴 𝐴̅ ∩ 𝐵 ∩ 𝐶̅ 𝖴 𝐴 ∩ 𝐵̅
∩ 𝐶̅ 𝖴 𝐴 ∩ 𝐵 ∩ 𝐶̅
|
1.2.28
|
𝐴 ∩ 𝐵 ∩ 𝐶̅ 𝖴 𝐴 ∩ 𝐵̅ ∩ 𝐶 𝖴 𝐴 ∩ 𝐵̅ ∩ 𝐶̅
𝖴 𝐴 ∩ 𝐵 ∩ 𝐶
|
Yuqorida keltirilgan soddalashtirishlarni amalga oshirish uchun quyida keltirilgan to‘plamlar ustida amallar xossalaridan foydalaning:
U-univyersаl to‘plаmning А, B, C to‘plаm оstilаri uchun quyidаgi хоssаlаr o‘rinli.
A ∪ B B ∪ A
A ∩ B B ∩ A
3. ( A ∪ B) ∪ C A ∪ (B ∪ C)
Kоmmutаtivlik 11.
12.
Аssоtsiаtivlik 13.
А ∩ А А А ∪ А U А ∩ А Ø
0 vа 1 qоnunlаri
4. ( A ∩ B) ∩ C A ∩ (B ∩ C)
14.
А ∪ Ø=A
5. ( A ∪ B) ∩ C ( A ∩ C) ∪ (B ∩ C)
6. ( A ∩ B) ∪ C ( A ∪ C) ∩ (B ∪ C)
distributivlik 15.
16.
А ∩U A
А ∪U U
7. A ∩ ( A ∪ B) A
8. A ∪ ( A ∩ B) A
Yutilish qоnunlаri
17.
18.
А ∩ Ø= Ø
U Ø
9.
10.
А ∩ В А ∪ В
А ∪ В А ∩ В
De Mоrgаn qоnunlаri
19. =U
20. A\ B A ∩ B
21.
A A
Ikkilаngаn rаd etish qоnuni
Do'stlaringiz bilan baham: |
