Question

Which one of the following relations on R is an equivalence relation?

• A. $a{R}_{1}b\iff \left|a\right|=\left|b\right|$
• B. $a{R}_{2}b\iff a\ge b$
• C. ${\mathrm{aR}}_{3}b\iff \text{a divides}b$
• D. $a{R}_{4}b\iff a<b$

Answer 1

The correct option is A

$a{R}_{1}b\iff \left|a\right|=\left|b\right|$

Explanation for the correct option

For option (a)

$a{R}_{1}b\iff \left|a\right|=\left|b\right|$

Reflexive: Let $a\in R_1$

$\left|a\right|=\left|a\right|$

Hence, $R_1$ is a reflexive relation.

Symmetric: Let $a,b\in R_1$

$$\begin{gathered} \Rightarrow \left|a\right|=\left|b\right| \\ \Rightarrow \left|b\right|=\left|a\right| \\ \therefore b{R}_{1}a \end{gathered}$$

Hence, $R_1$ is a symmetric relation.

Transitive: Let $a,b,c\in R_1$

$\Rightarrow \left|a\right|=\left|b\right|$ and $\left|b\right|=\left|c\right|$

$\Rightarrow \left|a\right|=\left|c\right|$

$\because a{R}_{1}c$

Hence, $R_1$ is a transitive relation.

Therefore, $R_1$ is an equivalence relation.

Explanation for incorrect options

For option (b)

$a{R}_{2}b\iff a\ge b$

For symmetric relation,

$a{R}_{2}b\iff a\ge b$ then it does not imply $b\ge a$

Hence, $R_2$ is not a symmetric relation

Therefore, $R_2$ is not a equivalence relation.

For option (c)

${\mathrm{aR}}_{3}b\iff \text{a divides}b$

${\mathrm{aR}}_{3}b\iff \text{a divides}b$ then it does not imply $\text{b divides a}$

Hence, $R_3$ is not a symmetric relation

Therefore, $R_3$ is not a equivalence relation.

For option (d)

$a{R}_{4}b\iff a<b$

$a{R}_{4}b\iff a<b$ then it does not imply $a<b$

Hence, $R_4$ is not a symmetrirelation

Therefore, $R_4$ is not a equivalence relation.

Hence the correct option is option (A).