Question

If $n\left(U\right)=700,n\left(A\right)=200,n\left(B\right)=300$ and $n(A\cap B)=100,$ then $n(A^{\prime }\cap B^{\prime })$ = ____.

• A. 400
• B. 240
• C. 300
• D. 500

Answer 1

The correct option is C

300

By de-Morgan's law of complementation for sets A and B, we have
$(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }.$
The portion in green represents this set.
Thus, $n(A^{\prime }\cap B^{\prime })=n\left(\right(A\cup B{)}^{\prime }).$

But, $n\left(\right(A\cup B{)}^{\prime })=n(U)-n(A\cup B),$ where $U$ is the universal set.

$\text{Also,}n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$

$=200+300-100$

$=400$

$\therefore n(A\cup B{)}^{\prime }=n(U)-n(A\cup B)$

$=700-400$

$=300$

$\therefore n(A^{\prime }\cap B^{\prime })=n\left(\right(A\cup B{)}^{\prime })=300$