Question

If $R$ is 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 these

Answer 1

The correct option is D

None of these

Explanation for the incorrect options:

Check for reflexive relation:

Since the relation is defined on the set of natural numbers $N$.

The relation $R$ can be written as $R=\{(6,2),(7,4),(8,6),(9,8),(10,10)......\}$

Therefore, the domain of $R=\{6,7,8,9,10,......\}$ and the range is $R=\{2,4,6,8,10,......\}$

A relation is reflexive if $(a,a)\in R$ for every $a\in N$,

Let $12\in N$, but $(12,12)\notin R$, so, the relation is not reflexive.

Thus, option (A) is incorrect.

Check for symmetric relation:

A relation is symmetric if $(a,b)\in R$ then $(b,a)\in R$

Here, $(6,2)\in R$ but $(2,6)\notin R$, so the relation is not symmetric.

Thus, option (B) is incorrect.

Check for transitive relation:

A relation is transitive if $(a,b)\in R$ and $(b,c)\in R$ then $(a,c)\in R$.

Here $(8,6)\in R$ and $(6,2)\in R$ but $(8,2)\notin R$. so the relation is not transitive.

Thus, option (C) is incorrect.

Since the relation is neither reflexive, symmetric nor transitive.

Hence, the correct option is (D).