# Naive Set Theory

Basic set theory, with ZF axioms:

• Paul Halmos. Naive Set Theory.

Countable and uncountable sets. A set is called countable if its elements can be enumerated; otherwise, it is called uncountable.

Clearly, all finite sets are countable. The set of integers and the set of rational numbers are also countable. However, the set of real numbers $$\mathbb{R}$$ is uncountable.

Subset and superset. $$A$$ is a subset of $$B$$ (or: $$B$$ is a superset of $$A$$), denoted as $$A \subseteq B$$ (or: $$B \supseteq A$$), if and only if for every $$x \in A$$, there is $$x \in B$$.

$$A$$ and $$B$$ are said to be equal, denoted as $$A = B$$, if and only if $$A \subseteq B$$ and $$B \subseteq A$$; otherwise, $$A$$ and $$B$$ are said to be unequal, denoted as $$A \neq B$$.

$$A$$ is a proper subset of $$B$$ (or: $$B$$ is a proper superset of $$A$$), denoted as $$A \subset B$$ (or: $$B \supset A$$), if and only if $$A \subseteq B$$ and $$A \neq B$$.

Union. $$A \cup B = \{ x : x \in A \lor x \in B \}$$.

Intersection. $$A \cap B = \{ x : x \in A \land x \in B \}$$.

Difference. $$A \setminus B = \{ x : x \not\in A \land x \in B \}$$.

Symmetric difference. $$A \triangle B = (A \setminus B) \cup (B \setminus A) = \{ x : x \in A \oplus x \in B \}$$.

Cartesian product. $$A \times B = \{ (x,y) : x \in A \land y \in B \}$$.

Power set. $$\mathcal{P}(A) = \{ X : X \subseteq A \}$$.

Empty set. The empty set is denoted as $$\varnothing$$. $$| \varnothing | = 0$$.

Disjoint sets. Two sets $$A$$ and $$B$$ are said to be disjoint, if and only if $$A \cap B = \varnothing$$.