ThmDex – An index of mathematical definitions, results, and conjectures.
▼ Set of symbols
▼ Alphabet
▼ Deduction system
▼ Theory
▼ Zermelo-Fraenkel set theory
▼ Set
▼ Binary cartesian set product
▼ Binary relation
▼ Map
▼ Bijective map
▼ Set of bijections
Definition D16
Countable set
Formulation 0
Let $\mathbb{N}$ be the D225: Set of natural numbers.
A D11: Set $X$ is a countable set if and only if \begin{equation} \exists \, E \subseteq \mathbb{N} : \text{Bij}(E \to X) \neq \emptyset \end{equation}
Formulation 1
Let $\mathbb{N}$ be the D225: Set of natural numbers.
A D11: Set $X$ is a countable set if and only if there exists a D18: Map $f : E \to X$ such that
(1) $E \subseteq \mathbb{N}$ is a D78: Subset of $\mathbb{N}$
(2) $f$ is a D468: Bijective map from $E$ to $X$
Children
▶ D1881: Cocountable set
▶ D17: Finite set
Results
▶ R4627
▶ R263: Binary cartesian product of countable sets is countable
▶ R3507: Binary union of countable sets is countable
▶ R371: Countable set iff injection to natural numbers
▶ R262: Countable union of countable sets is countable
▶ R508: Finite cartesian product of countable sets is countable
▶ R3506: Finite union of countable sets is countable
▶ R511: Natural number plane is countable