Question

Let $R=\{x|x\in N,x$ is a multiple of $3,x\le 100\}$, $S=\{x|x\in N,x$ is a multiple of $5\text{,}x\le 100\}$. What is the number of elements in $\left(R\times S\right)\cap \left(S\times R\right)$.

• A. $36$
• B. $33$
• C. $20$
• D. $6$

Answer 1

The correct option is A

$36$

Explanation for the correct answer :

Finding number of elements in $\left(R\times S\right)\cap \left(S\times R\right)$.

Given, $R=\{x|x\in N,x$ is a multiple of $3,x\le 100\}$,

$S=\{x|x\in N,x$ is a multiple of $5\text{,}x\le 100\}$.

So, $$\begin{gathered} R=\left\{3,6,9,12,15,\dots \dots ,99\right\} \\ S=\left\{5,10,15,20,25,\dots \dots ,100\right\} \end{gathered}$$

Using property of set operation,
$\left(a\times b\right)\cap \left(c\times d\right)=(a\cap b)\times \left(c\cap d\right)$

Thus, $\left(R\times S\right)\cap \left(S\times R\right)=(R\cap S)\times \left(S\cap R\right)\dots \dots \left(1\right)$

Now finding, $\left(R\cap S\right)=\left\{15,30,45,60,75,90\right\}=6$ elements

$\left(S\cap R\right)=\left\{15,30,45,60,75,90\right\}=6$

Substitute the value in equation $1$ , we get

$\left(R\times S\right)\cap \left(S\times R\right)=6\times 6=36$ elements

Thus, the total number of elements is $36$.

Hence, the correct option is (A)