Show that the relation on , defined by is an equivalent relation.

Solution:

If is Reflexive, Symmetric and Transitive, then is an equivalence relation.

Reflexivity:
Suppose and be an arbitrary element of

Therefore, is reflexive.

Symmetric:
Suppose and such that

Therefore, is symmetric.

Transitivity:
Suppose such that and
and
On adding both the equations we obtain

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