Question:
If $A$ and $B$ are two sets such that $n(A)=23, n(b)=37$ and $n(A-B)=8$ then find $n(A \cup B)$
$\operatorname{Hint} n(A)=n(A-B)+n(A \cap B) n(A \cap B)=(23-8)=15 .$
Solution:
Given: $n(A)=23, n(B)=37, n(A-B)=8$
Using the hint
$n(A)=n(A-B)+n(A \cap B)$
$\Rightarrow 23=8+n(A \cap B)$
$\Rightarrow n(A \cap B)=23-8$
$\Rightarrow n(A \cap B)=15$
Visualizing the hint given,
We know that $n(A \cup B)=n(A)+n(B)-n(A \cap B)$
$\Rightarrow n(A \cup B)=23+37-15$
$\Rightarrow \mathrm{n}(\mathrm{A} \cup \mathrm{B})=45$
Hence $n(A \cup B)=45$