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. $a{R}_{3}b\iff a\text{divides}b$
• 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 correct option:

Let $R$ be a relation

Reflexive relation: A homogeneous binary relation $R$ on a set X is reflexive if it relates every element of $X$ to itself.

Symmetric relation: a relation R is symmetric only if $(b,a)\in R$ is true when $(a,b)\in R$.

Transitive relation: In transitive relation $aRbandbRc\Rightarrow aRc\forall a,b,c\in A$ i.e., $(a,b)\in R$ and $(b,c)\in R$ then it implies $(a,c)\in R$

Equivalence relation: if a relation satisfies all the conditions of reflexive, symmetric, and transitive relation then we say that the relation has an equivalence relation.

For option (a)

For reflexive it is always true that if $(a,a)\in R_1\iff \left|a\right|=\left|a\right|$

For symmetric relation

let $(a,b)\in R_1\iff \left|a\right|=\left|b\right|$

$\iff \left|a\right|=\left|b\right|$

$\iff \left|b\right|=\left|a\right|$

$\iff (b,a)\in R_1$

This relation satisfies symmetric relation.

For transitive relation

let $(a,b)\in R_1\text{and}(b,c)\in R_1\iff \left|a\right|=\left|b\right|\text{and}\left|b\right|=\left|c\right|$

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

$\iff (a,c)\in R_1$

This relation satisfies the transitive relation.

$a{R}_{1}b\iff \left|a\right|=\left|b\right|$ satisfies reflexive relation, symmetric relation, and transitive relation.

Hence this expression satisfies the equivalence relation.

Therefore, option (a) is the correct option.

Explanation for incorrect options:

Option (B)

If $a\ge b$, then $a\nleqq b$.

So $R_2$ does not satisfy the symmetric property.

Hence $R_2$ is not an equivalence relation.

Option (C)

$a\text{divides}b$ does not imply $b\text{divides}a$.

So $R_3$ does not satisfy the symmetric property.

Hence $R_3$ is not an equivalence relation.

Option (D)

If $a>b$, then it does not imply $a<b$.

So $R_4$ does not satisfy the symmetric property.

Hence $R_4$ is not an equivalence relation.

Therefore option (A) is correct.