# Naive Set Theory

Mort YaoBasic set theory, with ZF axioms:

- Paul Halmos.
*Naive Set Theory.*

**Finite sets and infinite sets**. A set is called *finite* if it contains finitely many elements; otherwise, it is called *infinite*.

**Countable set 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 natural numbers \(\mathbb{N}\), the set of integers \(\mathbb{Z}\) and the set of rational numbers \(\mathbb{Q}\) 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 (cross 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 \(A \cap B = \varnothing\).