Question

If the number of elements in each of universal set, set A and set B are represented as $n\left(U\right)$, $n\left(A\right)$, $n\left(B\right)$ respectively, then the number of elements that are not in set A and not in set B is given by ___.

• A. $n\left(AU{B}^{\prime }\right)$
• B. $n(A\cap B{)}^{\prime }$
• C. $n\left(U\right)-n(A\cap B)$
• D. $n\left(U\right)-n(A\cup B)$

Answer 1

The correct option is D

$n\left(U\right)-n(A\cup B)$

The elements that are not in set $A$ and not in set $B$ belong to the set $A^{\prime }\cap B^{\prime }.$

But by de-Morgan's law for complementation, we have
$A^{\prime }\cap B^{\prime }=(A\cup B{)}^{\prime }.$
Thus, $n(A^{\prime }\cap B^{\prime })=n\left(\right(A\cup B{)}^{\prime })=n(U)-n(A\cup B).$