Show that if R is an equivalence relation on X then dom R=rngR =X.

Show that if R is an equivalence relation on X, then Dom R = Rng R = X.

Let R={(m,n):2 divides m+n} on Z. Show that R is an equivalence relation on Z.

If f:X to Y is a function. Define a relation R on X given by R={(a, b): f(a)=f(b)}. Show that R is an equivalence relation on X.

If R and S are two equivalence relation on the set then prove that RcapS is also an equivlaence relation on the set.

Let A={1,2,3,...,9) and R be the relation on A xx A defined by (a, b) R (c,d), if a+d=b+c for (a, b), (c,d) in A xx A. Prove that R is an equivalence relation and also obtain the equivalence class (2,5).

If a relation R: AtoA is an equivalence relation then prove that R^-1:A toA is also an equivalence relation.

Let R be the relation on the set R of real numbers such that aRb iff a-b is and integer. Test whether R is an equivalence relation. If so find the equivalence class of 1 "and" 1/2 wrt. This equivalence relation.

Show that the relation R on the set A of points in a plane given by R ={(P,Q): distance of the point P from the origin is same as the distance of the point Q from the origin) is an equivalence relation. Further, show that the set of all points related to a point P ne (0,0) is the circle passing through P with origin as centre.