Question

Let A, B, C be finite sets. Suppose that n (A) = 10, n (B) = 15, n (C) = 20, n (A ∩ B) = 8 and n (B ∩ C) = 9. Then the possible value of n (A ∪ B ∪ C) is ______.

विकल्प

• 26

• 27

• 28

• Any of the three values 26, 27, 28 is possible

Answer 1

Let A, B, C be finite sets. Suppose that n (A) = 10, n (B) = 15, n (C) = 20, n (A ∩ B) = 8 and n (B ∩ C) = 9. Then the possible value of n (A ∪ B ∪ C) is any of the three values 26, 27, 28 is possible.

Explanation:

We have

n (A ∪ B ∪ C) = n (A) + n (B) + n (C) – n (A ∩ B) – n (B ∩ C) – n (C ∩ A) + n (A ∩ B ∩ C)

= 10 + 15 + 20 – 8 – 9 – n (C ∩ A) + n (A ∩ B ∩ C)

= 28 – {n(C ∩ A) – n (A ∩ B ∩ C)}  ...(i)

Since n (C ∩ A) ≥ n (A ∩ B ∩ C)

We have n (C ∩ A) – n (A ∩ B ∩ C) ≥ 0  ...(ii)

From (i) and (ii): n (A ∪ B ∪ C) ≤ 28  ...(iii)

Now, n (A ∪ B) = n (A) + n (B) – n (A ∩ B)

= 10 + 15 – 8

= 17

and n (B ∪ C) = n (B) + n (C) – n (B ∩ C)

= 15 + 20 – 9

= 26

Since, n (A ∪ B ∪ C) ≥ n (A ∪ C) and

n (A ∪ B ∪ C) ≥ n (B ∪ C), we have

n (A ∪ B ∪ C) ≥ 17 and n (A ∪ B ∪ C) ≥ 26

Hence n (A ∪ B ∪ C) ≥ 26  ...(iv)

From (iii) and (iv) we obtain

26 ≤ n (A ∪ B ∪ C) ≤ 28

Also n (A ∪ B ∪ C) is a positive integer

∴ n(A ∪ B ∪ C) = 26 or 27 or 28