Question

If $A$ and $B$ are subsets of $U$ such that $n\left(U\right)=700,n\left(A\right)=200,n\left(B\right)=300,n(A\cap B)=100,$ then find $n(A^{\prime }\cap B^{\prime })$

Answer 1

We know that ${(A\cup B)}^{\prime }=A^{\prime }\cap B^{\prime },$ so we need to find $(A\cup B)$ first.
$n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)=200+300-100=400$

$n{(A\cup B)}^{\prime }=n\left(U\right)-n(A\cup B)=700-400=300\Rightarrow n(A^{\prime }\cap B^{\prime })=300$