If n(A ∩ B) = 10, n(B ∩ C) = 20 and n(A ∩ C) = 30,
Question:

If n(A ∩ B) = 10, n(B ∩ C) = 20 and n(A ∩ C) = 30, then the greatest possible value of n(A ∩ ∩ 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 ∩ ∩ C)

Since A ∩ B ∩ ≤ A ∩ B , A ∩ B ∩ ≤  A ∩ and A ∩ ∩ ≤ 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 ∩ ∩ C) is 10.

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