Question

For any two sets A and B, if n(A) =15, n(B) = 12, A ∩ B ≠ ϕ and B ⊄ A, then the maximum and and minimum possible values of n(A ∆ B) are _______ and ___________ respectively.

Answer 1

If n(A) =15
n(B) = 12
A ∩ B ≠ ϕ
B ⊄ A
Then maximum and possible values of n(A ∆ B) = ?
Since A ∩ B ⊆ A and A ∩ B ⊆ B
⇒ n(A ∩ B) ≤ n(A) and n(A ∩ B) ≤ n(B)
⇒ n(A ∩ B ≤ min {n(A), n(B)} = 12
⇒ –n (A ∩ B) ≥ – 12
i.e n(A ∩ B) ≤ 12
also A ⊆ A ⋃ B, B ⊆ A ⋃ B
i.e n(A) ≤ n(A ⋃ B) and n(B) ≤ n(A ⋃ B)
⇒ n(A ⋃ B) ≥ max {n(A), n(B)} = 15
i.e. n(A ⋃ B) ≥ 15
⇒ n(A ∆ B) = n(A ⋃ B) – n(A ∩ B) ≥ 15 – 12 = 3
i.e n(A ∆ B) ≥ 3
i.e maximum value of n(A ∆ B) = 3