Question:
If n(A ∩ B) = 10, n(B ∩ C) = 20 and n(A ∩ C) = 30, then the greatest possible value of n(A ∩ B ∩ C) is ____________.
Solution:
If n(A ∩ B) = 10
n(B ∩ C) = 20
n(A ∩ C) = 30
To find the greatest possible value of n(A ∩ B ∩ C)
Since A ∩ B ∩ C ≤ A ∩ B , A ∩ B ∩ C ≤ A ∩ C and A ∩ B ∩ C ≤ B ∩ C
⇒ n(A ∩ B ∩ C) ≤ n(A ∩ B), n(A ∩ B ∩ C) ≤ n(A ∩ C) and n(A ∩ B ∩ C) ≤ n(B ∩ C)
⇒ n(A ∩ B ∩ C) ≤ min{n(A ∩ B), n(A ∩ C), n(B ∩ C)}
≤ min {10, 20, 30} = 10
i.e maximum / greatest possible value of n(A ∩ B ∩ C) is 10.