Let $X = {n \in N : 1 \leq n \leq 50}$ . If $A = {n \in X : n is a multiple of 2}$ and $B = {n \in X : n is a multiple of 7}$ , then the number of elements in the smallest subset of $X$ containing both $A$ and $B$ is

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Solution

$X = {x : 1 \leq x \leq 50, x \in N}$
$A = {x : x is a multiple of 2} = {2, 4, 6, 8, \dots, 50}$
$B = {x : x is a multiple of 7} = {7, 14, 21, \dots, 49}$

Smallest subset of $X$ which contains elements of both $A$ and $B$ is a set with multiple of $2$ or $7$ less than $50$ .
$n (A \cup B) = n (A) + n (B) - n (A \cap B)$
$= 25 + 7 - 3$
$= 29$

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