First slide

Operations on Sets

Question

Suppose A₁, A₂, ….., A₃₀ are thirty sets each having 5 elements and B₁, B_{2 ………}, B_n are n sets each with 3 elements, let $\cup_{i = 1}^{30} A_{i} = \cup_{j = 1}^{n} B_{j} = S$
and each element of S belongs to exactly 10 of the A_i ’s and exactly 9 of the B_j ’s. Then n is equal to

Moderate

A

15
B

3
C

45
D

none of these

Solution

$S = \cup_{i = 1}^{30} A_{i}, so n (S) = \frac{1}{10} (5 \times 30) = 15$ (since element in the union S belongs to exactly 10 of the $A_{i}$ ’s).
Again , $S = \cup_{i = 1}^{n} B_{i}$ so
$n (S) = 1 / 9 (3 \times n) = n / 3 = 15 \Rightarrow n = 45$

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