The Uniqueness of The Identity

The identity element is unique. One of the first proofs I ever did. Still one of my favorites. It goes like this.

Theorem: Let \(S\) be a non-empty set closed under binary operation \(*: S \times S \rightarrow S\). If \(e \in S\) is an identity element of the binary algebraic structure \(\langle S, ∗\rangle\), then \(e\) is unique.

Proof: Supose \(e\) and \(\bar{e}\) are both identity elements of \(S\). Consider the element \(e*\bar{e}\in S\). Since \(e\) is an identity element, \(e*\bar{e}=\bar{e}\). And since \(\bar{e}\) is an identity element, \(e*\bar{e}=e\). So \(e=e*\bar{e}=\bar{e}\) and therefore \(e=\bar{e}\). \(\square\)

What I really like about this proof is how simple it is. It doesn't even assume that \(\langle S, ∗\rangle\) is a group. The uniqueness of the identity is an imediate consequence of its existence.