\input{decls.tex}
\title{Sets}
\begin{document}
\maketitle
Set comprehensions can be written $\{ x | x \in \N \}$ or $\{ x : x \in \N \}$ - '$:$' or '$|$'
\begin{description}
	\item[Axiom of Extensionality / Set Equality] $A = B \iff \forall x. (x \in A \iff x \in B)$
	\item[$A \subseteq B$] \quad $\forall x \in A. x \in B$ \\
		Is transitive, reflexive, antisymmetric
	\item[$A \subset B$] \quad $(\forall x \in A.~x \in B) \land (\exists x \in B.~x \not\in A)$ \\
		Is transitive, antisymmetric
	\item[$\varnothing$] $\{\}$
	\item[$\cup$] Union
	\item[$\cap$] Intersection
	\item[$A \setminus B$] \quad $\{ x \in A : x \not\in B \}$
\end{description}
\end{document}