Exercise 2.3

Let \((M, *)\) be a monoid, and let \(e, e' \in M\). Assume that \(e\) and \(e'\) are both identity elements; that is,

\[ \begin{align}\begin{aligned}\forall x \in M.\; e * x = x * e = x\\\forall x \in M.\; e' * x = x * e' = x.\end{aligned}\end{align} \]

Now what is the result of \(e * e'\)? Since \(e\) is an identity, we must have that \(e * e' = e'\). Additionally, since \(e'\) is an identity, we must have that \(e * e' = e\). The only way that \(e * e'\) can be equal to both of these two things at once is if they are the same, so we conclude that \(e = e'\), i.e. any monoid has exactly one identity element.