Question

The relation ‘less than’ in the set of natural numbers is

• A. only symmetric
• B. only transitive
• C. only reflexive
• D. equivalence relation

Answer 1

The correct option is B

only transitive

Explanation for correct option

Option (B): only transitive

$\bullet$In a transitive relation, if $\mathrm{aRb}$ and $\mathrm{bRc}$ then $\mathrm{aRc}$

Here, relation is ‘less than’, so $\mathrm{a}<\mathrm{b}$and $\mathrm{b}<\mathrm{c}$ then $\mathrm{a}<\mathrm{c}$

For example: Let $\mathrm{a}=2,\mathrm{b}=3$ and $\mathrm{c}=10$

Here, $2<3$ and $3<10$, so $2<10$, which is true.

Thus option(B) is correct.

Explanation for incorrect options

Option (A): only symmetric

In a symmetric relation, if $\mathrm{aRb}$ then $\mathrm{bRa}$

Here, relation is ‘less than’, so $\mathrm{a}<\mathrm{b}$ then $\mathrm{b}<\mathrm{a}$

For example: Let $\mathrm{a}=2,$ and $\mathrm{b}=3$

Here, $2<3$ but $3<2$ is not true.

Thus option(A) is incorrect.

Option (C): only reflexive

In a reflexive relation, $\mathrm{aRa}$

Here, relation is ‘less than’, so $\mathrm{a}<\mathrm{a}$

For example: Let $\mathrm{a}=2,$ so $2<2$ which is not true.

Thus option(C) is incorrect.

Option (D): equivalence relation

A relation on a set is said to be equivalent if and only if the relation is reflexive, symmetric and transitive.

As the given relation is not symmetric and reflexive, so it is not an equivalence relation.

Thus option(D) is incorrect.

Hence, the correct option is option(B) i.e. only transitive