Question

For $n,mϵN,n/m$means that $n$ is a factor of $m$, the relation $/$is

• A. Reflexive and Symmetric
• B. Transitive and Symmetric
• C. Reflexive Transitive and Symmetric
• D. Reflexive Transitive and not Symmetric.

Answer 1

The correct option is D

Reflexive Transitive and not Symmetric.

Explanation for the correct option:

Step1. Prove that relation is reflexive:

For a relation to be reflexive $\text{'}n\text{'}$ relate to itself. $\left[UsingdefinitionofReflexive\forall x\in R,x=x\right]$

$\therefore n/n,\forall n\in N$represent reflexive relation.

$\mathrm{n}/\mathrm{n}\mathrm{means}\text{'}\mathrm{n}\text{'}\mathrm{is}\mathrm{a}\mathrm{factor}\mathrm{of}\text{'}\mathrm{n}\text{'},\forall \mathrm{n}\in \mathrm{N}$ , all-natural numbers are factors for themselves.

Example: $3$is a factor of $3$

All-natural numbers are factors for themselves.

Hence, a relation is reflexive.

Step2. Prove that relation is symmetric:

For a relation to be symmetric $\mathrm{n}/\mathrm{m}$then $\mathrm{m}/\mathrm{n},\forall \mathrm{n},\mathrm{m}\in \mathrm{N}$ to be true. $\mathrm{[}\mathrm{Using}\mathrm{}\mathrm{definition}\mathrm{}\mathrm{of}\mathrm{}\mathrm{symmetric}\mathrm{}\mathrm{\forall }\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }\mathrm{R}\mathrm{}\mathrm{if}\mathrm{}\mathrm{x}\mathrm{=}\mathrm{y}\mathrm{}\mathrm{then}\mathrm{}\mathrm{y}\mathrm{=}\mathrm{x}\mathrm{}\mathrm{]}$

$\mathrm{n}/\mathrm{m}$ means $\text{'}n\text{'}$is a factor of $\text{'}m\text{'}$

$\text{'}n\text{'}$is a factor of $\text{'}m\text{'}$does not imply $\text{'}m\text{'}$is a factor of $\text{'}n\text{'}$

$\text{'}m\text{'}$is a multiple of $\text{'}n\text{'}$

$\therefore$The relation is not symmetric.

Example: $3$ is a factor of $6$ but $6$ is not a factor of$3$.

Step3.Prove that relation is transitive:

For a relation to be transitive $\mathrm{n}/\mathrm{m},\mathrm{m}/\mathrm{t}\mathrm{then}\mathrm{n}/\mathrm{t},\forall \mathrm{n},\mathrm{m},\mathrm{t}\in \mathrm{N}$

$n/m$means $\text{'}n\text{'}$is a factor of $\text{'}m\text{'}$ $\mathrm{[}\mathrm{Using}\mathrm{}\mathrm{definition}\mathrm{}\mathrm{of}\mathrm{}\mathrm{transitive}\mathrm{}\mathrm{\forall }\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{,}\mathrm{z}\mathrm{\in }\mathrm{R}\mathrm{}\mathrm{if}\mathrm{}\mathrm{x}\mathrm{=}\mathrm{y}\mathrm{,}\mathrm{}\mathrm{y}\mathrm{=}\mathrm{z}\mathrm{}\mathrm{then}\mathrm{}\mathrm{x}\mathrm{=}\mathrm{z}\mathrm{]}$

$m/t$means $\text{'}m\text{'}$is a factor of $\text{'}t\text{'}$

$\Rightarrow \text{'}n\text{'}$ is a factor of $\text{'}t\text{'}$

$\Rightarrow n/t$ is true.

Example: $3$ is a factor of $6$ and $6$ is a factor of $18$

$\Rightarrow 3$ is a factor of $18$

$\therefore$The relation is transitive.

Hence, Option(D) is the correct answer.