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.

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

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

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