Question

Define relation & show that if R be the selation

on the set z of all integers defined by xRy= x-y

is disable by n, then R is equivalence sedation​

Answer 1

In mathematics a relation or mainly a binary relation over two sets X and Y is a set of ordered pairs (x, y) consisting of elements x in X and y in Y. That is, it is a subset of the Cartesian product X × Y.

• Now R is the relation  on the set Z of all integers defined by xRy= x-y is divisible by n.
• Reflexive: xRx = x-x = 0. Now 0 is always divisible by n. So R is reflexive.
• Symmetric: xRy means n divides x-y. Now yRx = y-x = -(x-y) which is also divisible by n as n divides x-y. So R is symmetric.
• Transitive: xRy and yRz implies n divides both x-y and y-z. Now xRz = x-z = (x-y) + (y-z) which is also divisible by n as n divides both of x-y and y-z. So R is transitive.
• Now as R is Reflexive, Symmetric and Transitive then R is an Equivalence Relation.