*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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