Illustration -: If A and B be two sets containing 3 and 6 element respectively, what can be the minimum number of elements in A ∪ B? Find also, the maximum number of elements in A ∪ B.
Solution: We have, n ≤ A ∪ B) = n≤ A) + n≤ B) – n≤ A ∩ B)
This shows that n ≤ A ∪ B) is minimum or maximum according as
n ≤ A ∩ B) is maximum or minimum respectively.
Case 1: When n ≤ A ∩ B) is minimum, ie. n ≤ A ∩ B) = 0. This is possible only when A ∩ B = ϕ. In this case,
n≤ A ∪ B) = n ≤ A) + n ≤ B) – 0 = n≤ A) + n ≤ B) = 3 +6 = 9
n ≤ A ∪ B)max = 9
Case 2: When n ≤ A ∩ B) is maximum
This is possible only when A ⊆ B.
In this case n ≤ A ∩ B) = 3
\ n ≤ A∪B) = n≤ A) + n≤ B) – n ≤ A ∩B) = ≤ 3+6-3)=6
n ≤ A ∪ B)min = 6.
Illustration -: In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Hindi only? How many can speak Bengali? How many can speak both Hindi and Bengali?
Solution: Total number of people = 1000
n ≤ H) = 750
n ≤ B) = 400
n ≤ H ∪ B) = n ≤ H) + n ≤ B) – n ≤ H ∩ B)
n ≤ H ∩ B) = 750 + 400 – 1000
= 150 speaking Hindi and Bengali both.
People speaking only Hindi = n ≤ H) – n ≤ H ∩ B) = 750 – 150 = 600
People speaking only Bengali = n ≤ B) – n ≤ H ∩ B) = 400 – 150 = 250.
Illustration -: A survey shows that 63% of the Americans like cheese whereas 76% like apples. If x% of the Americans like both cheese and apples, find the value of x.
Solution: Let A denote the set of Americans who like cheese and let B denote those who like apples. Let the population ofAmerica be 100. Then,
n≤ A) = 63, n≤ B) = 76
Now, n≤ A ∪ B) = n≤ A) + n≤ B) – n≤ A ∩ B)
⇒ n≤ A∪B) = 63+76-n≤ A ∩ B)
⇒ n ≤ A ∩ B) = 139 – n≤ A ∪ B)
But n≤ A∪B) ≤ 100 ⇒ n ≤ A ∩ B ) ³ 39 …≤ i)
Now, A ∩ B ⊆ A and A ∩ B ⊆ B
⇒n≤ A ∩ B ) ≤ n ≤ A) and n ≤ A ∩ B) ≤ n ≤ B)
⇒n ≤ A ∩ B) ≤ 63 …≤ ii)
From i) and ii), we have 39 ≤ n ≤ A ∩B ) ≤ 63 ⇒ 39 ≤ x ≤ 63.
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