Subset Product is Subset of Generator
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Theorem
Let $\struct {G, \circ}$ be a group.
Let $X, Y \subseteq \struct {G, \circ}$.
Then $X \circ Y \subseteq \gen {X, Y}$ where:
- $X \circ Y$ is the Subset Product of $X$ and $Y$ in $G$.
- $\gen {X, Y}$ is the subgroup of $G$ generated by $X$ and $Y$.
Proof
It is clear from Set of Words Generates Group that $\map W {\hat X \cup \hat Y} = \gen {X, Y}$.
It is equally clear that $X \circ Y \subseteq \map W {\hat X \cup \hat Y}$.
$\blacksquare$