Definition:Inverse of Subset

Monoid
Let $$\left({S, \circ}\right)$$ 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:


 * $$X^{-1} = \left\{{y \in S: x \in X, x \circ y = e_S}\right\}$$

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

Group
When $$\left({G, \circ}\right)$$ is a group, then $$C = G$$ as from the Cancellation Laws, all group elements are cancallable.

Thus $$X^{-1}$$ is then defined as:


 * $$X^{-1} = \left\{{x^{-1} \in G: x \in X}\right\}$$