Question

Let $A,B\subseteq U$ be a universal set.

If $n\left(U\right)=60,n\left(A\right)=21,n\left(B\right)=43$, then the greatest value of $n(A\cup B)$ and least value of $n(A\cup B)$ are

• A. $60,43$
• B. $50,36$
• C. $70,44$
• D. $60,38$

Answer 1

The correct option is A $60,43$

$A,B\subset U$

$\Rightarrow n\left(A\right),n\left(B\right)\le n\left(U\right)$

Also, $A,B\subset U$

$\Rightarrow A\cup B\subset U$

$\Rightarrow n(A\cup B)\le n\left(U\right)$

$\Rightarrow n\left(A\right)+n\left(B\right)-n(A\cap B)\le n\left(U\right)=60$ and $A\cap B\ne \varphi$

Value of $n(A\cup B)$ is greatest when $n(A\cup B)=n\left(U\right)=60$

Value of $n(A\cup B)$ will be least value when $A\subset B$

Also, $n(A\cap B)=21$ occurs when $A\subset B$
$\Rightarrow n(A\cup B)=21+43-21=43$ is the least value

Thus, $43\le n(A\cup B)\le 60$

Hence, option $A$ is correct.