Give an example to show that the union of two equivalence relations on

1.	Give an example to show that the union of two equivalence relations on a set A need not be an equivalence relation on A.
Answer» Let R₁= {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} and R₂ = {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)} are two equivalence relations on set A = {1, 2, 3}. Now, R₁ ∪ R₂ = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)}. Since, (2, 1) ∊ R₁ ∪ R₂ and (1, 3) ∊ R₁ ∪ R₂. But (2, 3) ∉ R₁ ∪ R₂. Hence, relation R₁ ∪ R₂ is not transitive relation and hence, not equivalence relation. Thus, R₁ ∪ R₂ is not an equivalence relation even when relation R₁ and R₂ are equivalence relation.

Give an example to show that the union of two equivalence relations on a set A need not be an equivalence relation on A.

Answer»

Let R₁= {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} and R₂ = {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)} are two equivalence relations on set A = {1, 2, 3}.

Now, R₁ ∪ R₂ = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)}.

Since, (2, 1) ∊ R₁ ∪ R₂ and (1, 3) ∊ R₁ ∪ R₂.

But (2, 3) ∉ R₁ ∪ R₂.

Hence, relation R₁ ∪ R₂ is not transitive relation and hence, not equivalence relation.

Thus, R₁ ∪ R₂ is not an equivalence relation even when relation R₁ and R₂ are equivalence relation.

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Give an example to show that the union of two equivalence relations on a set A need not be an equivalence relation on A.

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