number 11 Let R be the relation in the set of natural number

number 11

Let R be the relation in the set of natural numbers N defined by the sentence (x - y) is divisible by 2\', that is R = {(x, y)| x N, y N, 2|(x - y)}} Show that R is an equivalence relation. Let X = {-2, -1, 0, 1, 2}. Let the function g: X rightarrow R be defined by the formula g(x) = x^3 = 1. Find the image of g. Is the function one-to one? If so, find inverse.

Solution

For the relation to be an equivalence relation, it must be reflexive, symmetric and transitive

Relation is said to be reflexive if (x,x) belongs to N

Since (x-x) = 0, which is divisible by 2, hence the relation is reflexive in nature

Relation is said to be symmetric if (x,y) belong to N, then (y,x) must also belong to N

Since (x-y) is divisible by 2, hence we can (x-y) = 2k

(y-x) = -(x-y) = -2k = 2(-k) = 2k1, hence (y-x) also divides 2. therefore the relation is symmetrix in nature

Relation is said to be transitive if (x,y) and (y,z) belongs to N, then it implies (x,z) belongs to R

(x-y) = 2k1

(y-z) = 2k2

(x-z) = (x-y) + (y-z) = 2k1 + 2k2 = 2(k1+k2) = 2k3

Hence (x-z) is divisible by 2

Therefore relation is transitive in nature

Since relation is reflexive, symmetric and transitive, it is an equivalence relation