Question

If $n\left(A\right)=30,n\left(B\right)=40,n\left(A\cup B\right)=45$, find $n\left(A\cap B\right)$.

Short answer

$n\left(A\cap B\right)=25$.

Step-by-step solution

Step 1. Given Information.

The given values are

$n\left(A\right)=30,n\left(B\right)=40,n\left(A\cup B\right)=45$.

Step 2. Value of nA∩B.

Using the Counting principle, we get

$\begin{array}{rcl}n\left(A\cup B\right)& =& n\left(A\right)+n\left(B\right)-n\left(A\cap B\right)\\ 45& =& 30+40-n\left(A\cap B\right)\\ 45& =& 70-n\left(A\cap B\right)\\ -25& =& -n\left(A\cap B\right)\\ n\left(A\cap B\right)& =& 25\end{array}$.