Question

If R is a relation from \[\left\{ 11,12,13 \right\}\] to \[\left\{ 8,10,12 \right\}\] defined by \[y=x-3\] . Then \[{{R}^{-1}}\] is equal to
A) \[\left\{ \left( 8,11 \right),\left( 10,13 \right) \right\}\]
B) \[\left\{ \left( 11,18 \right),\left( 13,10 \right) \right\}\]
C) \[\left\{ \left( 10,13 \right),\left( 8,11 \right),\left( 8,10 \right) \right\}\]
D) None of these

Answer 1

Hint: Here we have been given two sets whose relation is defined by an equation and we have to find the inverse of that relation. Firstly by using the equation we will find the ordered pair contained in the relation given. Then we will use the concept to find the inverse of the relation obtained by interchanging the values inside the ordered pair and hence get our desired answer.

Complete answer: It is given that $R$ is a relation from \[\left\{ 11,12,13 \right\}\] to \[\left\{ 8,10,12 \right\}\] defined as \[y=x-3\] .
Let the two set given be denoted as,
\[X=\left\{ 11,12,13 \right\}\] , \[Y=\left\{ 8,10,12 \right\}\]….$\left( 1 \right)$
Now as we know that as the relation is defined on the equation given the ordered pair in the equation should satisfy the equation.
The equation is given as:
\[y=x-3\]
On rearranging the terms we get,
$\Rightarrow x-y=3$
So this means that the ordered pair in the relation will satisfy above value that is,
$R=\left\{ \left( x,y \right):x\in X,y\in Y,\,x-y=3 \right\}$…..$\left( 2 \right)$
So from equation (1) we get the number of ordered pair as follows:
$\left( 11,8 \right),\left( 11,10 \right),\left( 11,12 \right),\left( 12,8 \right),\left( 12,10 \right),\left( 12,12 \right),\left( 13,8 \right),\left( 13,10 \right),\left( 13,12 \right)$
Using equation (2) only below values satisfies the condition,
$R=\left\{ \left( 11,8 \right),\left( 13,10 \right) \right\}$……$\left( 3 \right)$
We got our relation now we will find its inverse.
 The inverse of the relation as,
${{R}^{-1}}=\left\{ \left( y,x \right):\left( x,y \right)\in R \right\}$
Using above concept in equation (3) we get,
${{R}^{-1}}=\left\{ \left( 8,11 \right),\left( 10,11 \right) \right\}$
Hence the correct option is (A).

Note:
 In this type of questions we firstly need to know all the ordered pairs in the relation satisfying the given condition and then we interchange the two values in each pair to get the inverse of the relation. For finding the ordered pair, take each value from the first set with each value in the second set and check whether they satisfy the equation. Relation is a subset of the Cartesian product or we can simply say a bunch of points (ordered pairs). Relation defines a relationship between two different sets of information.