Solve , If we have $n\left( A \right)=15,n\left( A\cup B \right)=29,n\left( A\cap B \right)=7$ find $n\left( B \right)$.

Hint: Use the data given in the question , $n\left( A \right)=15,n\left( A\cup B \right)=29$ and $n\left( A\cap B \right)=7$ and substitute in the formula , \[n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)\] and thus find $n\left( B \right)$ .

Complete step-by-step solution -
In the question we are provided with values of $n\left( A \right),n\left( A\cup B \right)$ AND $n\left( A\cap B \right)$ which is 15, 29, 7 respectively and we have to find value of $n\left( B \right)$ .
Here in the question $n\left( A \right)$ represent number of elements of A, $n\left( A\cup B \right)$ represent number of elements of A and B collectively and lastly $n\left( A\cap B \right)$ represent number of elements common to both A and B .
Here to find $n\left( B \right)$ which means number of elements of B we can use the formula that is, $n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$ .
 So, on rearranging we can rewrite it as,
$n\left( B \right)=n\left( A\cup B \right)+n\left( A\cap B \right)-n\left( A \right)$ .
On substituting values $n\left( A\cup B \right)$ as 29, $n\left( A\cap B \right)$ as 7 and $n\left( A \right)=15$ we get, $n\left( B \right)=29+7-15$ .
=21.
So the value of $n\left( B \right)$ is 21.

Note: Students generally have confusion between the sign $\cup $ and $\cap $. The sign $\cup $ means union which means the $A\cup B$ then it contains all the elements of A and B collectively and $\cap $ means intersection like the $A\cap B$ contains common elements of respective two sets A & B. At the time of using these signs, students need to be careful with their meaning.