Definition:Inverse of Subset

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 $$X$$.

When $$\left({G, \circ}\right)$$ is a group, then $$C = G$$ as All Group Elements are Cancellable.

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

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