Question

If A and B are two sets such that $n\left( A \right)$ = 27, $n\left( B \right)$ = 35 and $n$$\left( {A \cup B} \right)$ = 50, find $n$ $\left( {A \cap B} \right)$.

Answer 1

Hint: Here in this type of question union sets and intersection sets are mentioned so we should know the formula of union and intersection sets i.e. $n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)$.

Complete Step-by-Step solution:
As we are given that
$n\left( A \right)$ = 27
$n\left( B \right)$ = 35
$n$$\left( {A \cup B} \right)$ = 50
Now we will use the formula- $n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)$
So by substituting the values
we get 50= 27+35-$n$ $\left( {A \cap B} \right)$
$ \Rightarrow $ 50 = 62-$n$ $\left( {A \cap B} \right)$.
 $ \Rightarrow $ $n$ $\left( {A \cap B} \right)$=12

Note: We must know the formula of the sets theory in order to solve this type of question. Along with the formula one must be clear about the difference between union sets and intersection sets. This concept is very easy if the difference between union sets, complement sets and intersection sets is understood properly.