Editing Equality (mathematics) (section)

=== Equivalence relation ===
{{Main|Equivalence relation}}
[[File:Equivalentie.svg|thumb|upright=0.8|Graph of an example equivalence with 7 classes]]
An [[equivalence relation]] is a [[mathematical relation]] that generalizes the idea of similarity or sameness. It is defined on a [[Set (mathematics)|set]] <math>X</math> as a [[binary relation]] <math>\sim</math> that satisfies the three properties: [[Reflexive relation|reflexivity]], [[Symmetric relation|symmetry]], and [[Transitive relation|transitivity]]. Reflexivity means that every element in <math>X</math> is equivalent to itself (<math>a \sim a</math> for all <math>a \in X</math>). Symmetry requires that if one element is equivalent to another, the reverse also holds (<math>a \sim b \implies b \sim a</math>). Transitivity ensures that if one element is equivalent to a second, and the second to a third, then the first is equivalent to the third (<math>a \sim b</math> and <math>b \sim c \implies a \sim c</math>).{{Snf|Stoll|1963|p=29}} These properties are enough to [[Partition of a set|partition a set]] into disjoint [[equivalence class]]es. Conversely, every partition defines an equivalence class.{{Sfn|Stoll|1963|p=31}}

The equivalence relation of equality is a special case, as, if restricted to a given set <math>S,</math> it is the strictest possible equivalence relation on <math>S</math>; specifically, equality partitions a set into equivalence classes consisting of all [[singleton set]]s.{{Sfn|Stoll|1963|p=31}} Other equivalence relations, since they're less restrictive, generalize equality by identifying elements based on shared properties or transformations, such as [[Modular arithmetic#Congruence|congruence in modular arithmetic]] or [[Similarity (geometry)|similarity in geometry]].<ref>{{Cite book |last=Stark |first=Harold M. |url=https://mitpress.mit.edu/9780262690607/an-introduction-to-number-theory/ |title=An Introduction to Number Theory |date=May 30, 1978 |publisher=MIT Press |isbn=978-0-262-69060-7 |edition= |location=Cambridge, Massachusetts |pages=51–54}}</ref><ref>{{Cite web |date=2020-02-10 |title=2.2.1: Similarity |url=https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_II_(Illustrative_Mathematics_-_Grade_8)/02:_Dilations_Similarity_and_Introducing_Slope/2.02:_New_Page/2.2.1:_Similarity |access-date=2025-03-24 |website=Mathematics LibreTexts}}</ref>

==== Congruence relation ====
{{Main|Congruence relation}}
In [[abstract algebra]], a [[congruence relation]] extends the idea of an equivalence relation to include the [[Equality (mathematics)#Basic properties|operation-application property]]. That is, given a set <math>X,</math> and a set of operations on <math>X,</math> then a congruence relation <math>\sim</math> has the property that <math>a \sim b \implies f(a) \sim f(b)</math> for all operations <math>f</math> (here, written as unary to avoid cumbersome notation, but <math>f</math> may be of any [[arity]]). A congruence relation on an [[algebraic structure]] such as a [[Group (mathematics)|group]], [[Ring (mathematics)|ring]], or [[Module (mathematics)|module]] is an equivalence relation that respects the operations defined on that structure.<ref>{{Cite book |last=Hungerford |first=Thomas W. |date=1974 |title=Algebra |publisher=Springer |location=New York |series=Graduate Texts in Mathematics |volume=73 |doi=10.1007/978-1-4612-6101-8 |isbn=978-1-4612-6103-2 |issn=0072-5285}}</ref>