Inverse of Inverse of Subset of Group

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

$\blacksquare$