Show that the relation R defined on the set A = (1, 2, 3, 4, 5), given by
R = {(a, b) : |a – b| is even} is an equivalence relation.

Solution:

and is even where
(As given)
If is Reflexive, Symmetric and Transitive, therefore equivalence relation.

Reflexivity:
Suppose be an arbitrary element of A

As 0 is even

Therefore, is reflexive.

Symmetric:
Suppose and , such that

Therefore, is symmetric.

Transitivity:
Suppose and , such that and
is even and is even
This is only possible when and both are even or odd and and both are even or odd.
Two cases arise:

Case I: If is an even
Suppose and
is even and is even
As is even
a is even and is even
is even

Case II: If b is an odd
Suppose and
is even and is even
As is odd
is odd and is odd
As difference of two odd numbers is even
is even

Therefore, R is transitive.
As a result, R is an equivalence relation.