Question

If A and B are two sets such that n(A)= 115, n(B)= 326, n(A-B)= 47, then write $n\left( A\cup B \right)$.

Answer 1

Hint: In this question we are given the number of elements in two sets and the number of elements belonging only to one set and not to the other is given we have to find the number of elements in their union. So, first, we should understand the definition of the union of two sets and use the formula for finding the number of elements in a set to find the answer to this question.

Complete step-by-step answer:

In set theory, a set is a well-defined collection of objects. Therefore, each set will contain some elements. The union of two sets is defined as the collection of all the elements which belong to either A or B. Mathematically, we can write

$A\cup B=\left\{ x:x\in A\text{ or }x\in B \right\}............(1.1)$

Where the symbol $\cup $ represents union and the symbol $\in $ stands for belongs to.

The expression $A-B$ is defined as the collection of all the elements which belong to A but do not belong to B. Mathematically, we can write

$A-B=\left\{ x:x\in A\text{ and }x\notin B \right\}............(1.2)$

The expression $A\cap B$ is defined as the collection of all the elements which belong to both A and B. Mathematically, we can write

$A\cap B=\left\{ x:x\in A\text{ and }x\in B \right\}............(1.3)$

Now, as any element belonging to A has to either belong to B or not belong to B, we should have

$n(A)=n\left( A-B \right)+n\left( A\cap B \right)..........(1.4)$

Where n(A) denotes the number of terms in a set A.

As it is given that n(A)=115 and n(A-B)=47, therefore, using these values in equation (1.4), we have

$115=47+n\left( A\cap B \right)$

$\Rightarrow n\left( A\cap B \right)=115-47=68..........(1.5)$

Also, from set theory, we have the formula for the number of elements in the union of sets A and B as

$n(A\cup B)=n(A)+n(B)-n\left( A\cap B \right)........(1.6)$

Using the values of n(A) =115, n(B)=326 as given in the question and $n(A\cap B)=68$ from equation (1.5) in equation (1.6), we get

$n(A\cup B)=115+326-68=373$

Thus, the answer to this question should be 373.

Note: Note that we could also have solved this question by writing $n(A)-n\left( A\cap B \right)=n\left( A-B \right)$ from equation (1.4) and using it in equation (1.6) to obtain $n(A\cup B)=n(B)+n(A)-n\left( A\cap B \right)=n(B)+n(A-B)$ and as the values of n(B) and n(A-B) are given, we could have obtained the answer without finding the value of $n(A\cap B)$. However, the answer in both methods would remain the same.