Let R be a equivalence relation on set A. Show that R R = R. Solution given that

Question

Let R be a equivalence relation on set A. Show that R R = R.

Explanation / Answer

given that R is an equivalence relation. (1) R is reflexive. so, we have xR = x consider x( RoR) = x( R(R))= xR(R) = x ( R)= xR = x ==> x(RoR)= x so, RoR is reflexive. (2) R is symmetric. so, we have if xR = y then yR = x such that x and y are not equal. keeping this in view, suppose x(RoR)= y ==> xR = yR consider y(RoR)= y(R(R)) = yR(R)= = xR = yR = x by hypothesis. so, whenever, x(RoR)= y, we get y(RoR)=x and x not equal to y. therefore, RoR is symmetric. (3) given R is transitive. i.e. , x R = y and yR = z ==> x R = z keeping this in view, suppose x(RoR)=y and y(RoR)=z so, we can use xR=y and yR= z consider x(RoR) = xR(R)=yR=z so, x(RoR)= z therefore, RoR is transitive. (1),(2),(3) ==> RoR is also an equivalence relation

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