Identity Element
Let \(A\) be a set , and \(\ast:A \times A \rightarrow A\) be a binary operation on \(A\). We define an identity element of \(A\) to be an element \(e \in A\) such that:
\[ e \ast a=a \ast e=a \]for every \(a \in A\).
(sidenote: We remark that when \(A\) is a group , then the identity element is unique. )