Equivalence class

https://en.wikipedia.org/wiki/Equivalence_classes

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.