Question

Given, $\xi$ $=$ {Natural numbers between $25$ and $45$}; $A=$ {even numbers} and $B$ {multiples of $3$}. Find : $n(A$ $\cup$ $B$) $+n$($A$ $\cap$ $B$).

Answer 1

$\xi =$ { $26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44$ } as $x$ is a natural number between $25$ and $45$
$A=$ { $26,28,30,32,34,36,38,40,42,44$ } as $A$ is set of a multiple of $3$ within the limits of the universal set already defined as $\xi$.

$\Rightarrow n\left(A\right)=10$ as there are $10$ elements in the Set $A$.

And, $B=$ { $27,30,33,36,39,42$ } as $B$ is a set of multiple of $3$ within the limits of the universal set already defined as $\xi$

$\Rightarrow n\left(B\right)=6$ as there are $6$ elements in the Set $B$.

Using the formula, $n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$

So, $n(A\cup B)+n(A\cap B)=n\left(A\right)+n\left(B\right)=10+6=16$