[[Naïve set theory MOC]]
# Cardinality
The **cardinality**^[Mächtigkeit] of a set is a [[Cardinal]] uniquely corresponding to the set's [[Isomorphism class]] within [[Category of sets]]. #m/def/set

- $\abs{A} = \abs{B} \iff A \cong B$ iff there exists a [[Surjectivity, injectivity, and bijectivity|bijection]] between sets $A$ and $B$.
  $A$ and $B$ are thence said to be **equinumerous**.
- $\abs{A} \leq \abs{B}$  if and only if there exists an [[Surjectivity, injectivity, and bijectivity|injection]] $f : A \rightarrowtail B$, or equivalently iff $A$ is equinumerous with some $C \sube B$.

For finite sets, cardinality is given by the number of elements in the set.
A set $A$ with $\abs A \leq \abs{\mathbb{N}} = \aleph_{0}$ is called [[Countability|countable]].
The naturals have the smallest transfinite cardinality.

[^Mächtigket]: German _Mächtigkeit_ oder _Kardinalität_



## Properties

- [[Upper bound on the cardinality of an arbitrary union]]


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