# GATE | GATE-CS-2005 | Question 42

Let R and S be any two equivalence relations on a non-empty set A. Which one of the following statements is TRUE?

(A)

R âˆª S, R âˆ© S are both equivalence relations

(B)

R âˆª S is an equivalence relation

(C)

R âˆ© S is an equivalence relation

(D)

Neither R âˆª S nor R âˆ© S is an equivalence relation

Explanation:

The correct answer is (C), R âˆ© S is an equivalence relation.
An equivalence relation is a relation that is reflexive, symmetric, and transitive. Reflexive means that every element is related to itself. Symmetric means that if two elements are related, then the reverse is also true. Transitive means that if two elements are related, and the second element is related to a third element, then the first element is also related to the third element.
The intersection of two equivalence relations is always an equivalence relation. This is because the intersection of two relations is always reflexive, symmetric, and transitive.

To see this, let R and S be two equivalence relations on a non-empty set A. Then, the intersection of R and S is the set of all elements that are related in both R and S.

Reflexivity: For every element x in A, x is related to itself in R and S, so x is also related to itself in the intersection of R and S.
Symmetry: If x is related to y in R and S, then y is related to x in R and S, so y is also related to x in the intersection of R and S.
Transitivity: If x is related to y in R and S, and y is related to z in R and S, then x is also related to z in R and S, so x is also related to z in the intersection of R and S.
Therefore, the intersection of two equivalence relations is always an equivalence relation.

The other options are incorrect. Option (A) is incorrect because the union of two equivalence relations is not always an equivalence relation. Option (B) is incorrect because the union of two equivalence relations can be an equivalence relation, but it is not always an equivalence relation. Option (D) is incorrect because the intersection of two equivalence relations is always an equivalence relation.

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