Let R be a reflexive and transitive relation on X Let R1 be

Let R be a reflexive and transitive relation on X. Let R1 be the relation on X defined by the following: for x, y X, (x, y) R1 if and only if (y, x) R. Show that R R1 is an equivalence relation on X.

Solution

The relation is said to be an equivalence relation if it is reflexive, symmetric and transitive

1) Relation is said to be reflexive if (a,a) belongs to R

2) Relation is said to be symmetric if (a,b) belongs to R, then (b,a) must also belong to R

3) Relation is said to be transitive if (a,b) & (b,c) belongs to R, then (a,c) will also belong to R

It is given that relation R is reflexive and transitive, it implies first and third condition are also met

R^(-1) implies that (x,y) will belong to R if (y,x) belong to R, Therefore, for every (x,y) we will have (y,x) belonging in R

Hence the only members which will belong to the set will be (a,a) since if (a,b) belongs to R, then (b,a) would also have belong to R

Hence R (int) R\' = ({a,a},{b,b},....}

so the members of R (int) R\' will only be of the form (x,x), hence this relation will be an equivalence relation