Item 15 (005P) — The Clowder Project

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Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
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*To be replaced with Linus Romer's Elemaints when it is released.

Bijectivity. Given $A,B\subset \mathcal{P}\webleft (X\webright )$, the maps
\begin{align*} A\mathbin {\triangle }- & \colon \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ),\\ -\mathbin {\triangle }B & \colon \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ) \end{align*}

are bijections with inverses given by

\begin{align*} \webleft (A\mathbin {\triangle }-\webright )^{-1} & = -\cup \webleft (A\cap -\webright ),\\ \webleft (-\mathbin {\triangle }B\webright )^{-1} & = -\cup \webleft (B\cap -\webright ). \end{align*}

Moreover, the map

\[ C\mapsto C\mathbin {\triangle }\webleft (A\mathbin {\triangle }B\webright ) \]

is a bijection of $\mathcal{P}\webleft (X\webright )$ onto itself sending $A$ to $B$ and $B$ to $A$.

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