What is Equivalence relation?
In [[mathematics]], an '''equivalence relation''' <math>R</math> on a set is a [[relation (mathematics)|mathematical relation]] that is [[symmetry|symmetric]], [[transitivity|transitive]] and [[reflexive]]. For a given element <math>a</math> on that set, the set of all elements related to <math>a</math> (in the sense of <math>R</math>) is called the equivalence class of <math>a</math>, and written as <math>[a]</math>.<ref>{{Cite web|date=2020-03-25|title=Comprehensive List of Algebra Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/|access-date=2020-08-30|website=Math Vault|language=en-US}}</ref><ref>{{Cite web|date=2017-09-20|title=7.3: Equivalence Classes|url=https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/7%3A_Equivalence_Relations/7.3%3A_Equivalence_Classes|access-date=2020-08-30|website=Mathematics LibreTexts|language=en}}</ref> With an equivalence relation, it is possible to partition a [[set]] into distinct equivalence classes.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Equivalence Class|url=https://mathworld.wolfram.com/EquivalenceClass.html|access-date=2020-08-30|website=mathworld.wolfram.com|language=en}}</ref> As an example, consider the set of all animals on a farm and define the following relation: two animals are related if they belong to the same [[species]]. Under this relation, a [[cow]] is related to an [[ox]], but not to a [[chicken]]. In fact, this relation is an equivalence relation because: *It is ''reflexive'': each animal is of the same species as itself *It is ''symmetric'': if a first animal is in the same species as a second animal, then the second animal is also in the same species as the first animal. *It is ''transitive'': if a first animal is in the same species as a second animal, and the second animal is in the same species as a third animal, then the first animal is in the same species as the third animal. In this example, the set of all animals related to a pa