Inverse of Inverse of Subset of Group

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Theorem

Let $\struct {G, \circ}$ be a group.

Let $X \subseteq G$.


Then:

$\paren {X^{-1} }^{-1} = X$.

where $X^{-1}$ denotes the inverse of $X$.


Proof

\(\displaystyle \paren {X^{-1} }^{-1}\) \(=\) \(\displaystyle \set {x^{-1}: x \in X^{-1} }\) Definition of Inverse of Subset of Group
\(\displaystyle \) \(=\) \(\displaystyle \set {\paren {x^{-1} }^{-1}: x \in X}\) Definition of Inverse of Subset of Group
\(\displaystyle \) \(=\) \(\displaystyle \set {x: x \in X}\) Inverse of Group Inverse
\(\displaystyle \) \(=\) \(\displaystyle X\) Axiom of Extension

$\blacksquare$