Question

Let $A$ be the set of first ten natural numbers and let $R$ be a relation on $A$ defined by $(x,y)\in R^{\prime }$ $x+2y=10$, i.e. $R=\{(x,y);x\in A,y\in A\text{and}x+2y=10\}$. Express $R$ and $R^{-1}$ as sets of ordered pairs. Find the ranges of $R^{-1}$.

• A. $\{2,4,6,8\}$
• B. $\{1,2,3,4\}$
• C. $\{1,3,5,7\}$
• D. $\{0,2,4,6,8\}$

Answer 1

Answer: (A)

$\{2,4,6,8\}$

$y=\frac{10-x}{2}$
where both $x$ and $y$ are natural numbers between $1$ and $10.$
Clearly for $x=2,4,6,8$ we get values of $y$ as $4,3,2,1.$
$\therefore R=\{(2,4),(4,3),(6,2),(8,1)\}$
$\therefore R^{-1}=\{(4,2),(3,4)(2,6),(1,8)\}$
Domain of $R$ is $\{2,4,6,8\}=$ Range of $R^{-1}$
Domain of $\{4,3,2,1\}=$ Range of $R$