Inverse Subset Group Product

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Theorem

Let $\left({G, \circ}\right)$ be a group.

Let $X, Y \subseteq G$.

Then:

$\left({X \circ Y}\right)^{-1} = Y^{-1} \circ X^{-1}$

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

First, note that.

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $		$\displaystyle x \in X, y \in Y$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle x^{-1} \in X^{-1}, y^{-1} \in Y^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Inverse Subset
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle y^{-1} \circ x^{-1} \in Y^{-1} \circ X^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Subset Product

Now:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x \circ y$	$\in$	$\displaystyle X \circ Y$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Subset Product
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle \left({x \circ y}\right)^{-1}$	$\in$	$\displaystyle \left({X \circ Y}\right)^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Inverse Subset
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle y^{-1} \circ x^{-1}$	$\in$	$\displaystyle \left({X \circ Y}\right)^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Inverse of Group Product
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle Y^{-1} \circ X^{-1}$	$\subseteq$	$\displaystyle \left({X \circ Y}\right)^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Subset

By a similar argument we see that $\left({X \circ Y}\right)^{-1} \subseteq Y^{-1} \circ X^{-1}$.

Hence the result.

$\blacksquare$

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

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