Question

Let $R$ be a relation on the set $N$. defined by $\left\{\left(x,y\right):2x-y=10\right\}$; then $R$ is:

• A. Reflexive
• B. Symmetric
• C. Transitive
• D. None of the above

Answer 1

The correct option is A

Reflexive

Explanation for the correct answer:

Option $\left(A\right)$: Reflexive

A relation $R$ on set $N$ is said to be Reflexive when $\left(a,a\right)\in R$

The relation $R$of set $N$ is $\left\{\left(x,y\right):2x-y=10\right\}$

If $x=y$ in $\left\{\left(x,y\right):2x-y=10\right\}$

$\Rightarrow 2x-x=10$

$\Rightarrow x=10$

$\Rightarrow y=10$

Thus, the point $\left(10,10\right)$ $\in R$ is Reflexive

Explanation for the incorrect option:

Option $\left(B\right)$: A relation $R$ on set $N$is said to be symmetric when $\left(a,b\right)\in R\left(b,a\right)\in R$

The relation $R$of set $N$ is $\left\{\left(x,y\right):2x-y=10\right\}$

If $x=6,y=2$ in $\left\{\left(x,y\right):2x-y=10\right\}$

$\left(6,2\right)\in R$ but $\left(2,6\right)\notin R$

Thus , $R$ is not symmetric.

Option $\left(C\right)$: A relation $R$ on set $N$is said to be transitive when $\left(a,b\right)\in R\left(b,c\right)\in R\left(a,c\right)\in R$

The relation $R$of set $N$ is $\left\{\left(x,y\right):2x-y=10\right\}$

If $x=4,y=-2,z=6$ in $\left\{\left(x,y\right):2x-y=10\right\}$

$\left(4,-2\right)\in R\left(-2,6\right)\in R\left(4,6\right)\notin R$

Thus, $R$ is not transitive

Therefore, option $\left(A\right)$ is correct answer.