Question

If $A$ and $B$ have some elements in common, then $n(A\cup B)$ is:

• A. $=n\left(A\right)+n\left(B\right)$
• B. $>n\left(A\right)+n\left(B\right)$
• C. $<n\left(A\right)+n\left(B\right)$
• D. $\ge n\left(A\right)+n\left(B\right)$

Answer 1

The correct option is C $<n\left(A\right)+n\left(B\right)$

We know that $n(A\cup B)+n(A\cap B)=n\left(A\right)+n\left(B\right)$

But $n(A\cap B)>0$ always

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

Hence, $n(A\cup B)<n\left(A\right)+n\left(B\right)$.