The set of pairs

is known a the diagonal from the table of ordered pairs.

There are

pairs in that table so

Therefore there are

off diagonal pairs.

Any reflexive relation on

must contain

.

So any reflexive relation is the union of

with any subset of off diagonal pairs.

How many of those are there?

Any symmetric relation on

,

, has the property that

.

There are

pairs either on the diagonal or above it.

Any subset of those pairs corresponds to a symmetric relation.

Just take that subset and unite it with its inverse.

How many are there?