Question

In order that a relation $R$ defined on a non–empty set $A$ is an equivalence relation, it is sufficient, if $R$:

• A. Is reflexive
• B. Is symmetric
• C. Is transitive
• D. Possesses all the above three properties

Answer 1

The correct option is D

Possesses all the above three properties

Explanation For Correct Option:

Illustrating the definition of equivalence relation

A relation $R$ defined on a set $A$is said to be an equivalence relation if and only if $R$ is

• Reflexive - $R$ is reflexive if$(a,a)\in R$ for all $a\in A$
• Symmetric - $R$ is symmetric if and only if $(a,b)\in R\Rightarrow (b,a)\in R$ for all $a,b\in A$
• Transitive - $R$ is transitive if and only if $(a,b)\in Rand(b,c)\in R\Rightarrow (a,c)\in R$ for all $a,b,c\in A$

For Example:

Consider a relation “ $=$” is an equivalence relation on a set of numbers $A$.

So, for all elements $a,b,c\in A,$ we have

• It is reflexive because $a=a$ for all $a\in A$
• It is symmetric because$a=b\Rightarrow b=a$ for all $a,b\in A$
• It is transitive because $a=b,b=c\Rightarrow a=c$ for all $a,b,c\in A$

Hence, option D is the correct answer.