Inverse of Product of Subsets of Group/Proof 1

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Theorem

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

Let $X, Y \subseteq G$.


Then:

$\paren {X \circ Y}^{-1} = Y^{-1} \circ X^{-1}$

where $X^{-1}$ is the inverse of $X$.


Proof

First, note that.

\(\displaystyle \) \(\) \(\displaystyle x \in X, y \in Y\)
\(\displaystyle \) \(\leadsto\) \(\displaystyle x^{-1} \in X^{-1}, y^{-1} \in Y^{-1}\) Definition of Inverse of Subset of Group
\(\displaystyle \) \(\leadsto\) \(\displaystyle y^{-1} \circ x^{-1} \in Y^{-1} \circ X^{-1}\) Definition of Subset Product


Now:

\(\displaystyle x \circ y\) \(\in\) \(\displaystyle X \circ Y\) Definition of Subset Product
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {x \circ y}^{-1}\) \(\in\) \(\displaystyle \paren {X \circ Y}^{-1}\) Definition of Inverse of Subset of Group
\(\displaystyle \leadsto \ \ \) \(\displaystyle y^{-1} \circ x^{-1}\) \(\in\) \(\displaystyle \paren {X \circ Y}^{-1}\) Inverse of Group Product
\(\displaystyle \leadsto \ \ \) \(\displaystyle Y^{-1} \circ X^{-1}\) \(\subseteq\) \(\displaystyle \paren {X \circ Y}^{-1}\) Definition of Subset


By a similar argument we see that $\paren {X \circ Y}^{-1} \subseteq Y^{-1} \circ X^{-1}$.


Hence the result.

$\blacksquare$