Question:
If A and B are two sets such than n(A) = 54, n(B) = 39 and n(B – A) = 13 then find n(A ∪ B).
Hint $n(B)=n(B-A)+n(A \cap B) \Rightarrow n(A \cap B)=(39-13)=26$
Solution:
Given: n(A) = 54, n(B) = 39, n(B – A) = 13
Using the hint
$n(B)=n(B-A)+n(A \cap B)$
$\Rightarrow 39=13+n(A \cap B)$
$\Rightarrow n(A \cap B)=39-13$
$\Rightarrow n(A \cap B)=26$
Visualizing the hint given
We know that $n(A \cup B)=n(A)+n(B)-n(A \cap B)$
$\Rightarrow \mathrm{n}(\mathrm{A} \cup \mathrm{B})=54+39-26$
$\Rightarrow \mathrm{n}(\mathrm{A} \cup \mathrm{B})=67$
Hence $n(A \cup B)=67$