Question

In order that a relation defined on a non- empty set A is an equivalence relation, it is sufficient, if R
1. Is reflexive
2. Is symmetric
3. Is transitive
4. Possesses all the above properties

Answer 1

Hint: For a non- empty set $A$ , If $R$ is a relation defined on it, then we have to tell what is the sufficient condition for it to be an equivalence relation.
We know, a relation is said to be an equivalence relation, if it is reflexive, symmetric as well as transitive.
And for that we must know the definitions of reflexive, symmetric and transitive relations, which we will see in the following steps.

Complete step-by-step answer:
Given a non-empty set $A$ and a relation $R$ is defined on it.
To find the sufficient condition, for which $R$ is an equivalence relation.
Now, we know, a relation $R$ is said to be a reflexive relation on a non- empty set $A$ , if for every $a$ in $A$ , $\left( {a,a} \right) \in R$ .
Whereas, a relation $R$ is said to be a symmetric relation on a non- empty set $A$ , if for every $a,b$ in $A$ , if $\left( {a,b} \right) \in R$ , then, $\left( {b,a} \right) \in R$ .
And finally, a relation $R$ is said to be a transitive relation on a non- empty set $A$ , if for every $a,b,c$ in $A$ , if $\left( {a,b} \right) \in R$ and $\left( {b,c} \right) \in R$ are given, then, $\left( {a,c} \right) \in R$ .
And we know, if a relation satisfies all these properties i.e., if a relation is reflexive, symmetric as well as transitive, then it is known as an equivalence relation.
Hence, option $(4)$ i.e., “possesses all above properties” is the correct option.

So, the correct answer is “Option 4”.

Note: A relation $R$ is said to be an equivalence relation if and only if it satisfies all the three properties and if even one the properties are kept unsatisfied then, that relation will not be an equivalence relation.
An equivalence relation defines a partition of the set into disjoint equivalence classes.
A relation can be denoted by ‘ $ \sim $ ‘ or simply by $R$ , as if we say that $a$ is in relation to $b$ , then we can write it as $a \sim b$ or $aRb$ .