A= { x belongs to Z:0 <= x <=12}

A= { 0,1,2,3,4,5,6,7,8,9,10,11,12}

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

For a belongs to A and (a,a) belongs to R as |a-a|=0 which is a multiple of 4.

Hence it is reflexive.

Let (a,b) belongs to R and |a-b| is a multiple of 4

then |b-a| is also a multiple of 4.

Hence it is symmetric.

Let (a,b) belongs to R and |a-b| is a multiple of 4 and (b,c) belongs to R and |b-c| is also a multiple of 4,

then (a-c) = (a-b) + (b-c) is also a multiple of 4

This implies that (a-c) is also a multiple of 4

and also (a,c) belongs to R

Hence it is transitive.

This three criteria implies that the relation is a equivalence relation.

Hope this helps you.

All the best !!