Show that the relation R in the set Z of intergers given by
`R ={(a,b):2 ` divides a-b }
is an equivalence relation.

Show that the relation R in the set A = {1,2,3,4,5} given by R = {(a,b) : |a-b| is even}, is an equivalence relation. Show that all the elements of {1,3,5} are related to each other and all the elements of {2,4} are related to each other. But no element of {1,3,5} is related to any element of {2,4}.

Show that the relation R in the set R of real numbers, defined as R = {(a,b) : a le b ^(2)} is neither reflexive nor symmetric nor transitive.

Show that the relation R in 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.

In the set Z of all integers, which of the following relation R is not an equivalence relation

Let f: X to Y be a function. Define a relation R in X given by R = {(a,b):f (a) =f (b)}. Examine whether R is an equivalence relation or not.

Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a,b):a is sister of b} is the empty relation and R' ={(a,b): the difference between heights of a and b is less than 3 meters } is the universal relation.

Show that the relation R in the set {1,2,3} given by R = {(1,1) ,(2,2),(3,3) , (1,2) ., (2,3)} is reflexive but neither symmetric nor transitive.