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Let R1 and R2 be two equivalence relations on a set. Is R1 ∪ R2 an equivalence relation?

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Let R1 and R2 be two equivalence relations on a set. Is R1 ∪ R2 an equivalence relation?

(a) an equivalence relation

(b) reflexive closure of relation

(c) not an equivalence relation

(d) partial equivalence relation

The question was asked in an internship interview.

This intriguing question comes from Closure on Relations topic in portion Relations of Discrete Mathematics

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Right option is (a) an equivalence relation

The explanation is: R1 union R2 is not equivalence relation because transitivity property of closure need not hold. For instance, (x, y) can be in R1 and (y, z) be in R2 and (x, z) not in either R1 or R2. However, R1 intersection R2 is an equivalence relation.

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