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Question

Let R and S be two equivalence relations on set A. Prove that RS is an equivalence relation.

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Solution

Equivalence relation mean reflexive, symmetric and transitive.
Let an element aA. Since R and S are equivalence relations they are reflexive.
Therefore (a,a)R and (a,a)S
So (a,a)R and (a,a)S
So (a,a)RS
RS is reflexive ......(1)
Let (a,b)RS
(a,b)R and (a,b)S
(b,a)R and (b,a)S
( Since R and S are symmetric),
(b,a)RS
RS is symmetric ....(2)
Let (a,b),(b,c)RS
(a,b),(b,c)R(a,c)R
(a,b),(b,c)S(a,c)S
Since R and S are transitive,
(a,c)RS
RS is transitive ....(3)
From (1), (2) and (3), RS is an equivalence relation.

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