A relation `R = {(x, y): x-y` is divisible by `5, x, y in Z}` is defined on set of integers (Z). Prove that R is an equivalence relation.

A relation R = {(x, y): x-y is divisible by 4, x, y in Z} is defined on set of integers (Z). Prove that R is an equivalence relation.

A relation R = { (x, y) : x-y is divisible' by 6, x, y in Z} is defined on set of integers (Z). Prove that R is an equivalent relation.

If relation R defined on set A is an equivalence relation, then R is

Check whether relation R = {(x, y): x le y^2, x, y in R} , defined on set of real numbers R, is reflexive, symmetric and transitive.

Let Z be the set of all integers and R be the relation on Z defined as R = (a,b) : a,b in Z and a-b is divisible by 5) Prove that R is an equivalence relation.

Determine whether each of the following relations are reflexive, symmetric and transitive : (i) Relation R in the set A = {1, 2, 3,......, 13, 14} defined as R={(x,y): 3x-y=0} (ii) Relation R in the set N of natural numbers defined as R={(x,y): y=x+5 " and " x lt 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} defined as R = {(x, y): y is divisible by x}. iv) Relation R in the set Z, of all integers defined as R = {(x, y) : x -y is an integer}.

Prove that the relation R defined on the set Z of integers as R= {(a, b) : 5 divides abs(a-b)} is an equivalence relation.

Prove that the relation R defined on the set Z of integers as R= {(a, b) : 4 divides abs(a-b)} is an equivalence relation.

Prove that the relation R defined on the set Z of integers as R= {(a, b) : 3 divides abs(a-b)} is an equivalence relation.

Let R be a relation on the set A of ordered pairs of positive integers defined by R, (x, y) R (u, v) , if and only if xv = yu . Show that R is an equivalence relation.