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Question

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

A
RS, RS are both equivalence relations
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B
RS is an equivalence relation
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C
RS is an equivalence relation
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D
Neither RS nor RS is an equivalence relation
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Solution

The correct option is C RS is an equivalence relation
RS is an equivalence relation as can be seen from proof given below.
Let xϵA(x,x)ϵR and (x,x)ϵS (since R and S are reflexive)
(x,x)ϵRS also RS is reflexive
Now, (x,y)ϵRS
(x,y)ϵR and (x,y)ϵS
(y,x)ϵR and (y,x)ϵS
(Since R and S are symmetric)
(y,x)ϵRS
(x,y)ϵRS
(y,x)ϵRS
RS is therefore symmetric
Now consider
(x,y) and (y,z)ϵRS
(x,y) and (y,z)ϵR
and (x,y) and (y,z)ϵS
(x,z)ϵR and (x,z)ϵS
(Since R and S are transitive)
(x,z)ϵRS
RS is transitive also. Since RS is reflexive, symmetric and transitive.
RS is equivalence relation.
Note: A similar argument cannot be made from RS

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