Does the Operation in Set Theory Imply a Singleton Set?

[set]

Homework Statement

Let S be a set with an operation * which assigns an element a*b of S for any a,b in S. Let us assume that the following two rules hold:
1. If a, b are any objects in S, then a*b = a
2. If a, b are any objects in S, then a*b = b*a
(Herstein, Abstract Algebra, 2ed)

Homework Equations

Is it safe to assume that the symmetry, transitivity, and reflexibility hold?

The Attempt at a Solution

a = a*b = b*a = b
But I am not sure if this is sufficient as it is my first course (in fact, first problem!) in abstract algebra...
[Edit]
Or with the relation obtained from the axioms of S, shall I proceed with proof by contradiction?

 

[mfb]

a = a*b = b*a = b

This is true for all a,b in S, and shows that S just has one single element. That is a strange problem.
The validity of those axioms will follow from that, but where is the point?