Item 16 (005Q) — The Clowder Project

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

Interaction With Powersets and Groups. Let $X$ be a set.
1. The quadruple $\webleft (\mathcal{P}\webleft (X\webright ),\mathbin {\triangle },\emptyset ,\text{id}_{\mathcal{P}\webleft (X\webright )}\webright )$ is an abelian group.^[1]
2. Every element of $\mathcal{P}\webleft (X\webright )$ has order $2$ with respect to $\mathbin {\triangle }$, and thus $\mathcal{P}\webleft (X\webright )$ is a Boolean group (i.e. an abelian $2$-group).

Footnotes

[1] Here are some examples:

When $X=\emptyset $, we have an isomorphism of groups between $\mathcal{P}\webleft (\emptyset \webright )$ and the trivial group:
\[ \webleft (\mathcal{P}\webleft (\emptyset \webright ),\mathbin {\triangle },\emptyset ,\text{id}_{\mathcal{P}\webleft (\emptyset \webright )}\webright ) \cong \text{pt}. \]
When $X=\text{pt}$, we have an isomorphism of groups between $\mathcal{P}\webleft (\text{pt}\webright )$ and $\mathbb {Z}_{/2}$:
\[ \webleft (\mathcal{P}\webleft (\text{pt}\webright ),\mathbin {\triangle },\emptyset ,\text{id}_{\mathcal{P}\webleft (\text{pt}\webright )}\webright ) \cong \mathbb {Z}_{/2}. \]
When $X=\webleft\{ 0,1\webright\} $, we have an isomorphism of groups between $\mathcal{P}\webleft (\webleft\{ 0,1\webright\} \webright )$ and $\mathbb {Z}_{/2}\times \mathbb {Z}_{/2}$:
\[ \webleft (\mathcal{P}\webleft (\webleft\{ 0,1\webright\} \webright ),\mathbin {\triangle },\emptyset ,\text{id}_{\mathcal{P}\webleft (\webleft\{ 0,1\webright\} \webright )}\webright ) \cong \mathbb {Z}_{/2}\times \mathbb {Z}_{/2}. \]

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