Let be the set of all real numbers. Show that the relation is symmetric but neither reflexive nor transitive.

Solution:

and (As given)

Non-Reflexivity:
Assume be an arbitrary element of
, which is not equal to 1

Therefore, is not reflexive.

Symmetric:
Assume and , such that
(As addition is commutative)
Therefore, is symmetric.

Non-Transitivity:
Assume and , such that and and
On adding both the equation, we obtain

, which is not equal to 1

Therefore, is not transitive.
As a result, is symmetric but neither transitive nor reflexive.