Question

If $A$and $B$are two equivalence relations defined on a set $C$, then

• A. $A\cap B$ is an equivalence relation
• B. $A\cap B$is not an equivalence relation
• C. $A\cup B$is an equivalence relation
• D. $A\cup B$is not an equivalence relation

Answer 1

The correct option is A

$A\cap B$ is an equivalence relation

Explanation for the correct answer:

Option$\left(A\right)$: $A\cap B$ is an equivalence relation

A relation $R$ on set $R$ is said to be equivalence relation if it is reflexive, symmetric and transitive.

Here, set $A$ and $B$ are in equivalence relation on set $C$, hence both must be reflexive, symmetric and transitive on the same set.

Then by the definition of set intersection: $A\cap B$ is the set of all pairs of elements in the set $A$ and $B.$

Therefore, $A\cap B$ is an equivalence relation

Hence, Option $\left(A\right)$ is the correct answer.