Let A and B are two sets containing 23 and 60 elements respectively. Find the maximum and minimum number of elements in $A\cup B$

$n\left(A\right)=23,n\left(B\right)=60$

$n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$

$=23+60-n(A\cap B)=83-n(A\cap B)$

$n(A\cup B)$ will be maximum for minimum value of $n(A\cap B)$ which is zero

$\Rightarrow n{(A\cup B)}_{\max }=83$

Similarly $n(A\cap B)$ is minimum when $n(A\cap B)$ is maximum

(when $A\subset B$)

$\Rightarrow n{(A\cup B)}_{\max }=83-n{(A\cap B)}_{\max }$

$=83-23=60$.