Question

Let $A,B,C$ be subsets of the universal set $U$. If $n\left(U\right)=692$, $n\left(B\right)=230$, $n\left(C\right)=370$, $n(B\cap C)=20$, $n(A\cap B^{\prime }\cap C^{\prime })=10$, find $n(A\cap B^{\prime }\cap C^{\prime })$.

Answer 1

$n(A^{\prime }\cap B^{\prime }\cap C^{\prime })=n\{(B^{\prime }\cap C^{\prime })\cap \left(A^{\prime }\right)\}$

$=n\left\{(B^{\prime }\cap C^{\prime })\right\}-n\{(B^{\prime }\cap C^{\prime })\cap A\}$

$=n{(B\cup C)}^{{}^{\prime }}-n(A\cap B^{\prime }\cap C^{\prime })$

$=n\left(U\right)-n(B\cup C)-n(A\cap B^{\prime }\cap C^{\prime })$

$=n\left(U\right)-n\left(B\right)-n\left(C\right)+n(B\cap C)-n(A\cap B^{\prime }\cap C^{\prime })$

$=692-230-370+90-10=172.$