Question

Let $A=\left\{2,4,6,8\right\}$. A relation $R$ on $A$is defined by $R=\left\{\left(2,4\right),\left(4,2\right),\left(4,6\right),\left(6,4\right)\right\}$. Then $R$ is

• A. Antisymmetric
• B. Reflexive
• C. Symmetric
• D. Transitive

Answer 1

The correct option is C

Symmetric

Explanation for the correct answer:

Option (C): Symmetric

Given,$A=\left\{2,4,6,8\right\}$,$R=\left\{\left(2,4\right),\left(4,2\right),\left(4,6\right),\left(6,4\right)\right\}$

A relation is said to be symmetric, if $(a,b)\in R$, then$(b,a)\in R.$

$R=\left\{\left(2,4\right),\left(4,2\right),\left(4,6\right),\left(6,4\right)\right\}$

$$\begin{gathered} \left(a,b\right)=\left\{\left(2,4\right),\left(4,2\right),\left(4,6\right),\left(6,4\right)\right\}\in R \\ \Rightarrow \left(b,a\right)=\left\{\left(4,2\right),\left(2,4\right),\left(6,4\right),\left(4,6\right)\right\}\in R \end{gathered}$$

Therefore, $R$ is symmetric

Explanation for the wrong answer:

Option (A): Antisymmetric

A relation R is not antisymmetric if there exists such that $(a,b)\in R$ and $(b,a)\in R$ but $a\ne b$.

$R=\left\{\left(2,4\right),\left(4,2\right),\left(4,6\right),\left(6,4\right)\right\}$

$$\begin{gathered} \left(a,b\right)=\left\{\left(2,4\right),\left(4,2\right),\left(4,6\right),\left(6,4\right)\right\}\in R \\ \Rightarrow \left(b,a\right)=\left\{\left(4,2\right),\left(2,4\right),\left(6,4\right),\left(4,6\right)\right\}\in R \end{gathered}$$

Therefore, $R$ is not antisymmetric

Option (B): Reflexive

A relation is said to be reflexive, if $(a,a)\in R$, for every$a\in A.$

$R=\left\{\left(2,4\right),\left(4,2\right),\left(4,6\right),\left(6,4\right)\right\}$

Let $2\in A\mathrm{but}\left(2,2\right)\notin R$

Therefore, $R$ is not reflexive

Option (D): Transitive

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

$R=\left\{\left(2,4\right),\left(4,2\right),\left(4,6\right),\left(6,4\right)\right\}$

For $(a,b)=(2,4)\in R$ and $(b,c)=\left(4,6\right)\in R$ then $(a,c)=\left(2,6\right)\notin R$

Therefore, $R$ is not Transitive.

Hence, option (C) is the correct answer.