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 |
$\blacksquare$