# Definition:Inverse of Subset

## Definition

### Monoid

Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$.

Let $C \subseteq S$ be the set of cancellable elements of $S$.

Let $X \subseteq C$.

Then the inverse of the subset $X$ is defined as:

$X^{-1} = \set {y \in S: \exists x \in X: x \circ y = e_S}$

That is, it is the set of all the inverses of all the elements of $X$.

### Group

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

Let $X \subseteq G$.

Then the inverse of the subset $X$ is defined as:

$X^{-1} = \set {x \in G: x^{-1} \in X}$

or equivalently:

$X^{-1} = \set {x^{-1}: x \in X}$