Question

The relation ‘is less than’ on a set of natural numbers is

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

Answer 1

The correct option is C

only transitive

Explanation for correct answer:

Option (C): only transitive

Let $N$ be the set of natural numbers. Then relation$R=\left\{(a,b):a<b,a,b\in N\right\}$

Reflexivity: A natural number is not less than itself

$\Rightarrow (a,a)\notin R$ where $a\in N$

$\Rightarrow R$ is not reflexive

Symmetric: $\forall a,b\in N,(a,b)\in R$

$\Rightarrow a<b$but $b$ can not be less than $a$

$\Rightarrow (b,a)\notin R$

$\Rightarrow R$is not symmetric.

Transitivity: $\forall a,b,c\in N,(a,b)\in R$ and $(b,c)\in R$

$\Rightarrow a<bandb<c$

$\Rightarrow a<c(a,c)\in R$

$\Rightarrow R$is transitive.

Hence, option (c ) is the correct answer.