Inverse Element

• Definition: Given a set with a binary operation $\star$ and an Identity Element $e$, an inverse element of $a\in S$ is an Element $b\in S$ such that: $a\star b=b\star a=e$
• Examples:
  • In $(\mathbb{Z},+)$, the inverse of $a$ is $-a$ since $a+(-a)=0$
  • In $(\mathbb{R}\setminus \{0\},\times )$, the inverse of $a$ is $1/a$, since $a\cdot (1/a)=1$.
  • For strings under concatenation, most elements have no inverse (since nothing undoes concatenation to return $\epsilon$, except $\epsilon$ itself).