Question

Let R = {(1, 3), (2, 2), (3, 2)} and S = {(2, 1), (3, 2), (2, 3)} be two relations on set A = {1, 2, 3}. Then $SoR$ =?
(a) {(1, 3), (2, 2), (3, 2), (2, 1), (2, 3)}
(b) {(3, 2), (1, 3)}
(c) {(2, 3), (3, 2), (2, 2)}
(d) {(3, 2), (2, 1), (2, 3)}

Answer 1

Write the relation $SoR=S\left( R\left( x \right) \right)$, where x is the domain of R. Find the corresponding value of $R\left( x \right)$ for each of the values of x taken from the set A = {1, 2, 3}. Now, consider $R\left( x \right)$ as the domain of S and find the corresponding values of S for each of the values of $R\left( x \right)$. Pair the values according to the obtained relation to get the answer.

Complete step-by-step solution:
Here we have been provided with two relations R = {(1, 3), (2, 2), (3, 2)} and S = {(2, 1), (3, 2), (2, 3)} be two relations on set A = {1, 2, 3}. We are asked to find the relation $SoR$. We can write $SoR=S\left( R\left( x \right) \right)$ where x is the domain of R whose values are taken from set A.
Now, in the relation R = {(1, 3), (2, 2), (3, 2)} we can conclude the following results: -
$\begin{aligned} & \Rightarrow R\left( 1 \right)=3 \\\ & \Rightarrow R\left( 2 \right)=2 \\\ & \Rightarrow R\left( 3 \right)=2 \\\ \end{aligned}$
Similarly, in the relation S = {(2, 1), (3, 2), (2, 3)} we can conclude the following results: -
$\begin{aligned} & \Rightarrow S\left( 2 \right)=1 \\\ & \Rightarrow S\left( 3 \right)=2 \\\ & \Rightarrow S\left( 2 \right)=3 \\\ \end{aligned}$
Now, in the relation $S\left( R\left( x \right) \right)$ the values of $R\left( x \right)$ will be the domain of $S\left( R\left( x \right) \right)$ and we have to find the corresponding values of S for each of the values of $R\left( x \right)$.
(i) At x = 1 we have $R\left( x \right)=3$ and for x = 3 we have $S\left( x \right)=2$. So we get,
$\begin{aligned} & \Rightarrow S\left( R\left( 1 \right) \right)=S\left( 3 \right) \\\ & \Rightarrow S\left( 3 \right)=2 \\\ \end{aligned}$
Therefore, the pair of values is (3, 2).
(ii) At x = 2 we have $R\left( x \right)=2$ and for x = 2 we have $S\left( x \right)=1$ or $3$. So we get,
$\Rightarrow S\left( R\left( 2 \right) \right)=S\left( 2 \right)$
$\Rightarrow S\left( 2 \right)=1$ or $3$
Therefore, the pair of values is (2, 1) or (2, 3).
(iii) At x = 3 we have again $R\left( x \right)=2$ and for x = 2 we have $S\left( x \right)=1$ or $3$, so here also we will get the same pairs of values as obtained in situation (ii). So we get,
Therefore, the pair of values is (2, 1) or (2, 3).
So, the relation $SoR$ can be given as $SoR$ = {(2, 1), (2, 3), ((3, 2)}. Hence, option (d) is the correct answer.

Note: Note that there is no value of x for which the value of $R\left( x \right)$ is 1, so we don’t have to consider the value 1 in the domain of $S\left( R\left( x \right) \right)$ and therefore we cannot find $S\left( 1 \right)$. Also, its value is not provided in the question. The relation of the type $S\left( R\left( x \right) \right)$ is called a composite relation.