# Equivalence class When elements of a set `S` have a notion of equivalence defined on them (and formalized as an equivalence relation), then the set `S` may be naturally split into *equivalence classes*. For `a,b ∈ S`, if `a ~ b` (a is equivalent to b) then `a` and `b` belong to the same equivalence class. Formally, given a set `S` and an equivalence relation `~` on `S`, the equivalence class of an element `a` in `S`, denoted by `[a]` is the set `{ x ∈ S | x ~ a }` of elements which are equivalent to `a`. It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of `S`. This partition (the set of equivalence classes) is called the *quotient set* of `S` by `~` and denoted by `S\~`. When a set $$S$$ has some structure (e.g. group operation) and the equivalence relation $$\sim$$ is compatible with this structure, the quotient set often inherits a similar structure from its parent set. ## Examples If $$X$$ is the set of all cars, and $$\sim$$ is the equivalence relation **has-the-same-color-as**, then one particular *equivalence class* consists of *all green cars*. $$X/\sim$$ could be naturally identified with the set of all car colors. Let $$X$$ be the set of all rectangles in a plane, and $$\sim$$ the equivalence relation *has the same area as*. For each positive real number $$A$$ there will be an equivalence class of all the rectangles that have area $$A$$. Consider the modulo 2 equivalence relation on the set of integers: $$x \sim y$$ iff their difference $$x − y$$ is an even number. This relation gives rise to exactly *2 equivalence classes*: one class consisting of all even numbers, and the other consisting of all odd numbers. Under this relation $$\[7]$$, $$\[9]$$, and $$\[1]$$ all represent the same element of $$Z/\sim$$. Let X be the set of ordered pairs of integers (a,b) with b not zero, and define an equivalence relation \~ on X according to which (a,b) \~ (c,d) if and only if ad = bc. Then the equivalence class of the pair (a,b) can be identified with the rational number a/b, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers.\[3] The same construction can be generalized to the field of fractions of any integral domain. If X consists of all the lines in, say the Euclidean plane, and L \~ M means that L and M are parallel lines, then the set of lines that are parallel to each other form an equivalence class as long as a line is considered parallel to itself. In this situation, each equivalence class determines a point at infinity. ## Formal definition An equivalence relation on a set $$X$$ is a binary relation $$\sim$$ on $$X$$ satisfying these 3 properties. $$\forall a, b, c \in X$$: * reflexivity: $$\forall a : a \sim a$$ * symmetry: $$\forall a,b : (a \sim b) \to (b \sim a)$$ * transitivity, $$\forall a,b,c :(a \sim b \land b \sim c) \to (a \sim c)$$ The equivalence class of an element $$a$$ is denoted $$\[a]$$ or $$\[a]\_{\sim}$$, and is defined as the set $${x\in X\mid a\sim x}$$ of elements that are related to $$a$$ by $$\sim$$. The word "class" in the term "equivalence class" does not refer to a class of set theory, however equivalence classes do often turn out to be proper classes. --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/220-relation-theory/terms/equivalence-class.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.