(A ∩ B)∪ (B ∩C)
A ∩ (B ∪C)∩ (-A ∪ -B ∪-C)
Set identities
A∪ ∅ =A ; A ∩U = A
A∪A=A ; A∩A=A
A∪-A=U ; A ∩ -A = ∅
--A = A
A ∪U = U ; A ∩ ∅ = ∅
A ∪ B = B ∪ A ; A ∩ B = B ∩ A
A ∪(B ∪C)= (A ∪ B)∪C ; A ∩ (B ∩C) = (A ∩ B)∩
A∪(B ∩C)= (A∪ B)∩ (A∪C); A∩ (B ∪C) = (A∩ B)∪(A∩C)
-(A∪B)=-A∩-B ;-(A∩B)=-A∪-B
例一.
Let A, B and C be sets. Use laws of algebra of sets to simplifythe following set expressions.
(a) (A ∩ B)∪ (B ∩C)
(A ∩ B)∪ (B ∩C)
= (A ∩ B)∪ (C ∩ B)
= (A ∪C)∩ B
(b) (A ∩ B)∪ (A ∩ B ∩-C ∩ D)∪ (-A ∩ B)
(A ∩ B)∪ (A ∩ B ∩-C ∩ D)∪ (-A ∩ B)
= (A∩ B)∪ (-A∩ B)∪ (A∩ B ∩-C ∩ D)
= ((A∪ -A)∩ B)∪ (A∩ B ∩-C ∩ D)
= (U ∩ B)∪ (A ∩ B ∩-C ∩ D)
= B ∪ (A∩ B ∩-C ∩ D)
= B
Application of Venn Diagrams in Counting
In a survey of 160 passengers, an airline found that 48 preferredwine with their meals, 78 preferred mixed drinks, and 66preferred ice tea. In addition, 12 enjoyed wine and mixeddrinks, 18 enjoyed mixed drinks and ice tea, and 16 enjoyed icetea and wine, and 4 passengers enjoyed them all.
a) How many passengers want only iced tea with theirmeals?
b) How many passengers do not like any of them?
a) 36 passengers want only iced tea.
b) 10 passengers do not like any of them(= 160 – 24 – 52 – 36 – 12 – 8 – 14 – 4)
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