An identity element for a binary operation * as described by Definition 3.12 is sometimes referred to as "a two-sided identity element." Using complete sentences, give analogous definitions for
a. A left identity element e_l for*, 

b. A right identity element e_R for*·

Theorem 3.13 shows that if a two-sided identity element for* exists, it is unique. Is the same true for a one-sided identity element you just defined? If so, prove it. If not, give a counterexample ( S, *) for a finite set S and find the first place where the proof of Theorem 3.13 breaks down.

Data from Definition 3.12

Definition: Let (S, *) be a binary structure. An element e of S is an identity element for * if e * s = s * e = s for all s ∈ S.