Identity Element

• Definition: An identity element of a Set with a binary operation is an Element that leaves all other elements unchanged when combined with them.
• Formally: For a set $S$ with operation $\star$, an element $e\in S$ is an identity if: $\forall a\in S,a\star e=e\star a=a$
• Examples:
  • 0 is the identity for addition on $\mathbb{Z}$.
  • 1 is the identity for multiplication on $\mathbb{R}$.
  • $\epsilon$ is the identity for string concatenation.