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# Set Notation in latex

* The empty set:
  * $$\exists A\ \[\ \forall x\ ( x \not\in A) \ ]$$
* Non-empty set:
  * $$\forall A\ \[\ \exists x\ ( x \in A) \ ]$$
* Subset:
  * $$\forall a\ \[\ a \in A \to a \in B \iff A \subseteq B \ ]$$
* Proper Subset:
  * $$\forall a\ \[ \ a \in A \to \ \[ \ a \in B \land \ (\exists b\ . \ b\in B \land b \not\in A) \ ] \ \iff A \subset B \ ]$$
* Equality
  * $$A \subseteq B \land B \subseteq A \iff A=B$$
* Axiom of regularity (foundation) states that every non-empty set $$x$$ contains a member $$y$$ such that $$x$$ and $$y$$ are disjoint sets.

$$
\forall x\[
\exists a(a\in x) \to
\exists y(y\in x  \land
\lnot \exists z (z \in y \land z \in x)
)
]
$$

$$
\forall x \exists y \ \[ \
y\in x \to \lnot \exists z (z \in y \land z \in x)
\ ]
$$

$$
\forall x \exists y \ \[ \
y\in x \to \lnot \exists z (z \in y \cap x)
\ ]
$$