If I is the set of real numbers, then prove that the relation `R={(x,y) : x,y in I and x-y " is an integer "}` , then prove that R is an equivalence relations.

Prove that the relation R={(x,y): x, y in N and x-y " is divisible by 7 "} defined on positive integers N is an equivlance relations.

(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.

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

If R is the set of real numbers prove that a function f: R -> R,f(x)=e^x , x in R is one to one mapping.

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 L be the set of all lines in X Y -plane and R be the relation in L defined as R={(L_1, L_2): L_1 is parallel to L_2} . Show that R is an equivalence relation. Find the set of all lines related to the line y=2x+4 .

The ordered pair (5,2) belongs to the relation R={(x,y):y=x-5,x,y in Z}

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