Question

If n(A) = 5 and n(B) = 10 , then find maximum and minimum possible values of n(A ∩ B) if
(a) n(U) = 8
(b) n(U) = 20
Also explain the answer.

Answer 1

Answer:

n(A∪B) = n(A) + (B) - n(A∩B)

now, n(A∪B) will be maximum when the minus term i.e. n (A∩B) will be minimum and n(A∪B) will be Minimum when the minus term i.e. n (A∩B) will be Maximum.

now, the minimum value of n (A∩B) will be when Sets A and B are disjoint sets, i.e. n(A∩B) = 0, and in this condition the value of n(A∪B) will be Maximum.

and, the maximum value of n (A∩B) will be min. (n(A), n(B)).

so, Minimum value of n (A∪B) = 10 +8 -8 = 10

Maximum value of n(A∪B) = 10 + 8 - 0 = 18.

thus, 10≤ n(A∪B) ≤ 18.

Answer 2

Answer:

n(A∪B) = n(A) + (B) - n(A∩B)

now, n(A∪B) will be maximum when the minus term i.e. n (A∩B) will be minimum and n(A∪B) will be Minimum when the minus term i.e. n (A∩B) will be Maximum.

now, the minimum value of n (A∩B) will be when Sets A and B are disjoint sets, i.e. n(A∩B) = 0, and in this condition the value of n(A∪B) will be Maximum.

and, the maximum value of n (A∩B) will be min. (n(A), n(B)).

so, Minimum value of n (A∪B) = 10 +8 -8 = 10

Maximum value of n(A∪B) = 10 + 8 - 0 = 18.

thus, 10≤ n(A∪B) ≤ 18.