Question

Show that the relation R on the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by R = {(a, b) : |a – b| is a multiple of 4} is an equivalence relation.

Answer 1

We have the given relation,

R = {(a,b) : |a-b| is a multiple of 4},

where a,b ∈ A and 

A = {x ∈ Z : 0 ≤ x ≤ 12}

= {0,1,2,....,12}

We discuss the following properties of relation R on set A.

Reflexivity :

For any a ∈ A we have

|a - a| = 0, which is multiple of 4

(a,a) ∈ R for all a ∈ R.

So, R is reflexive.

Symmetry :

Let (a,b) ∈ R.

 

So, R is Symmetric.

Transitivity :

Let a,b,c ∈ A such that (a,b) ∈ R and (b,c) ∈ R

So, R is transitive.

Hence, R is an equivalence ralation.