Editing Cardinality (section)

==== Equivalence ====
[[File:Example for a composition of two functions.svg|thumb|Example for a composition of two functions.|282x282px]]
A fundamental result necessary in developing a theory of cardinality is showing it is an [[equivalence relation]]. A binary [[Relation (mathematics)|relation]] is an equivalence relation if it satisfies the three basic properties of equality: [[Reflexive relation|reflexivity]], [[Symmetric relation|symmetry]], and [[Transitive relation|transitivity]]. A relation <math>R</math> is reflexive if, for any <math>a,</math> <math>aRa</math> (read: <math>a</math> is <math>R</math>-related to <math>a</math>); symmetric if, for any <math>a</math> and <math>b,</math> if <math>aRb,</math> then <math>bRa</math> (read: if <math>a</math> is related to <math>b,</math> then <math>b</math> is related to <math>a</math>); and transitive if, for any <math>a,</math> <math>b,</math> and <math>c,</math> if <math>aRb</math> and <math>bRc,</math> then <math>aRc.</math>

Given any set <math>A,</math> there is a bijection from <math>A</math> to itself by the [[identity function]], therefore cardinality is reflexive. Given any sets <math>A</math> and <math>B,</math> such that there is a bijection <math>f</math> from <math>A</math> to <math>B,</math> then there is an [[inverse function]] <math>f^{-1}</math> from <math>B</math> to <math>A,</math> which is also bijective, therefore cardinality is symmetric. Finally, given any sets <math>A,</math> <math>B,</math> and <math>C</math> such that there is a bijection <math>f</math>  from <math>A</math> to <math>B,</math> and <math>g</math> from <math>B</math> to <math>C,</math> then their [[Function composition|composition]] <math>g \circ f</math> (read: <math>g</math> after <math>f</math>) is a bijection from <math>A</math> to <math>C,</math> and so cardinality is transitive. Thus, cardinality forms an equivalence relation. This means that cardinality [[Partition of a set|partitions sets]] into [[equivalence classes]], and one may assign a representative to denote this class. This motivates the notion of a [[Cardinality#Cardinal numbers|cardinal number]].

Somewhat more formally, a relation must be a certain set of [[ordered pairs]]. Since there is no [[set of all sets]] in standard set theory (see: ''{{section link||Cantor's paradox}}''), cardinality is not a relation in the usual sense, but a [[Predicate (logic)|predicate]] or a relation over [[Class (set theory)|classes]].