$A$ and $B$ are two sets such that $n (A - B) = 32 + x, n (B - A) = 5 x$ and $n (A \cap B) = x$ Illustrate the information by means of a Venn diagram. Given that $n (A) = n (B)$ .
Calculate (i) The value of $x$ (ii) $n (A \cup B)$ .

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Solution

$n (A - B) = 32 + x$
$n (B - A) = 5 x$
$n (A \cap B) = x$
$n (A) = n (B)$
$(1) n (A) = n (B)$
$32 + x + x = 5 x + x$
$32 + 2 x = 6 x$
$6 x - 2 x = 32$
$4 x = 32$
$x = 8$

$(2) n (A) = 32 + x + x$
$= 32 + 2 x$
$= 32 + 2 \times 8$
$x = 48$

$n (B) = 5 x + x$
$= 6 x$
$= 6 \times 8$
$= 48$
$n (A \cup B) = n (A) + n (B) - n (A \cap B)$
$= 48 + 48 - 8$
$= 96 - 8$
$= 88$
so, the value of $x = 8$ and $n (A \cup B) = 88$

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