    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 R∩S is an equivalence relationR∩S 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)ϵR∩S also ∴R∩S is reflexive Now, (x,y)ϵR∩S ⇒(x,y)ϵR and (x,y)ϵS ⇒(y,x)ϵR and (y,x)ϵS (Since R and S are symmetric) ⇒(y,x)ϵR∩S ∴(x,y)ϵR∩S ⇒(y,x)ϵR∩S R∩S is therefore symmetric Now consider (x,y) and (y,z)ϵR∩S ⇒(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)ϵR∩S ∴R∩S is transitive also. Since R∩S is reflexive, symmetric and transitive. ∴R∩S is equivalence relation. Note: A similar argument cannot be made from R∪S  Suggest Corrections  0      Similar questions
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