Question

If A and B are two sets such that $n\left( A \right)=20,n\left( B \right)=25,n\left( A\cup B \right)=40$, then write $n\left( A\cap B \right)$.

Answer 1

Hint:To solve this question, we require the knowledge of the most basic formula of two sets of an event in set, that is, $n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$, where $n\left( A \right)$ represents the number of elements in the set A, $n\left( B \right)$ represents the number of elements in the set B, $n\left( A\cup B \right)$ represents the total number of elements participating in the event and $n\left( A\cap B \right)$ represents the number of elements present in both.

Complete step-by-step answer:
In this question, we are asked to find the value of $n\left( A\cap B \right)$ and we have been given that $n\left( A \right)=20,n\left( B \right)=25,n\left( A\cup B \right)=40$. To solve this question, we will need to know the formula of two sets, that is, $n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$, where $n\left( A \right)$ represents the number of elements in the set A, $n\left( B \right)$ represents the number of elements in the set B, $n\left( A\cup B \right)$ represents the total number of elements participating in the event and $n\left( A\cap B \right)$ represents the number of elements present in both the events. Here, we are given, $n\left( A \right)=20,n\left( B \right)=25,n\left( A\cup B \right)=40$. Now, we will use these values in the above formula. So, we get,
$40=20+25-n\left( A\cap B \right)$
We will now simplify it further to find the value of $n\left( A\cap B \right)$. So, we get,
$40=45-n\left( A\cap B \right)$
Now, we will take $n\left( A\cap B \right)$ on the left hand side and the numerical values on the right hand side. So, we get,
$\begin{align}
  & n\left( A\cap B \right)=45-40 \\
 & \Rightarrow n\left( A\cap B \right)=5 \\
\end{align}$
Hence, we get the value of $n\left( A\cap B \right)$ as 5, when $n\left( A \right)=20,n\left( B \right)=25,n\left( A\cup B \right)=40$.

Note: The possible mistakes one can make while solving this question can be the calculation mistakes. Also, one can write $-n\left( A\cap B \right)$ as $+n\left( A\cap B \right)$ in the formula in a hurry, which will give a wrong answer. So, one should always remember to apply the formula of, $n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$ whenever there are two sets.