Let `Z` be the set of integers. Show that the relation `R={(a ,\ b): a ,\ b in Z` and `a+b` is even} is an equivalence relation on `Zdot`

(i) Prove that the relation ""_(x)R_(y) iff 3 is a factor of (x-y) , defined on the set of integers is an equivalence relation. (ii) If x is the set of real numbers, then prove that the relation R={(a,b): a in x, b in x and a=b} is an equivalence relation.

Show that the relation R defined by R={(a , b):a-b is divisible by 3; a , bZ} is an equivalence relation.

Show that the relation R defined by R={(a , b):a-b is divisible by 3; a , bZ} is an equivalence relation.

Show that the relation R defined by R={(a , b):a-b is divisible by 3; a , bZ} is an equivalence relation.

Prove that the relation R on Z defined by (a ,\ b) in RhArr a-b is divisible by 5 is an equivalence relation on Z .

Prove that the relation R in set A = {1, 2, 3, 4, 5} given by R = {(a,b): |a-b| is even} is an equivalence relation .

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.

Let m be a given fixed positive integer. Let R={(a.b) : a,b in Z and (a-b) is divisible by m} . Show that R is an equivalence relation on Z .

Let Z be the set of all integers. A relation R is defined on Z by xRy to mean x-y is divisible by 5. Show that R is an equivalence relation on Z.

Statement-1: On the set Z of all odd integers relation R defined by (a, b) in R iff a-b is even for all a, b in Z is an equivalence relation. Statement-2: If a relation R on a set A is symmetric and transitive, then it is reflexive and hence an equivalence relation, because (a, b) in Rimplies(b, a)in R" [By symmetry]" (a, b)in R and (b, a) in Rimplies (a,a)in R " [By transitivity]"