# 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

 $\ds \paren {X^{-1} }^{-1}$ $=$ $\ds \set {x^{-1}: x \in X^{-1} }$ Definition of Inverse of Subset of Group $\ds$ $=$ $\ds \set {\paren {x^{-1} }^{-1}: x \in X}$ Definition of Inverse of Subset of Group $\ds$ $=$ $\ds \set {x: x \in X}$ Inverse of Group Inverse $\ds$ $=$ $\ds X$ Definition of Set Definition by Predicate

$\blacksquare$