Canonical Projection Map

In mathematics, when the elements of some set $S$ have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set $S$ into equivalence classes. These equivalence classes are constructed so that elements $a$ and $b$ belong to the same equivalence class if, and only if, they are equivalent.

Formally, given a set $S$ and an equivalence relation $\,\sim\,$ on $S,$ the of an element $a$ in $S,$ denoted by is the set
$\$
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 sometimes called the quotient set or the quotient space of $S$ by $\,\sim\,,$ and is denoted by $S / \sim.$

When the set $S$ has some structure (such as a group operation or a topology) and the equivalence relation $\,\sim\,$ is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.

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 would consist of all green cars, and $X / \sim$ 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", then 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, $\Z,$ such that $x \sim y$ if and only if their difference $x - y$ is an even number. This relation gives rise to exactly two equivalence classes: one class consists of all even numbers, and the other class consists of all odd numbers. Using square brackets around one member of the class to denote an equivalence class under this relation, and /math> all represent the same element of $\Z / \sim.$
* Let $X$ be the set of ordered pairs of integers $(a, b)$ with non-zero $b,$ and define an equivalence relation $\,\sim\,$ on $X$ such that $(a, b) \sim (c, d)$ if and only if $a d = b c,$ 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. 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 $L \sim M$ 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.

Definition and notation

An equivalence relation on a set $X$ is a binary relation $\,\sim\,$ on $X$ satisfying the three properties:
* $a \sim a$ for all $a \in X$ ( reflexivity),
* $a \sim b$ implies $b \sim a$ for all $a, b \in X$ ( symmetry),
* if $a \sim b$ and $b \sim c$ then $a \sim c$ for all $a, b, c \in X$ ( transitivity).

The equivalence class of an element $a$ is often denoted /math> or , and is defined as the set $\$ of elements that are related to $a$ by $\,\sim.$ The word "class" in the term "equivalence class" may generally be considered as a synonym of " set", although some equivalence classes are not sets but proper classes. For example, "being isomorphic" is an equivalence relation on groups, and the equivalence classes, called isomorphism classes, are not sets.

The set of all equivalence classes in $X$ with respect to an equivalence relation $R$ is denoted as $X / R,$ and is called $X$ modulo $R$ (or the of $X$ by $R$). The surjective map x \mapsto /math> from $X$ onto $X / R,$ which maps each element to its equivalence class, is called the , or the canonical projection.

Every element of an equivalence class characterizes the class, and may be used to ''represent'' it. When such an element is chosen, it is called a representative of the class. The choice of a representative in each class defines an injection from $X / R$ to . Since its composition with the canonical surjection is the identity of $X / R,$ such an injection is called a section, when using the terminology of category theory.

Sometimes, there is a section that is more "natural" than the other ones. In this case, the representatives are called . For example, in modular arithmetic, for every integer greater than , the congruence modulo is an equivalence relation on the integers, for which two integers and are equivalent—in this case, one says ''congruent'' —if divides $a-b;$ this is denoted $a\equiv b \pmod m.$ Each class contains a unique non-negative integer smaller than $m,$ and these integers are the canonical representatives.

The use of representatives for representing classes allows avoiding to consider explicitly classes as sets. In this case, the canonical surjection that maps an element to its class is replaced by the function that maps an element to the representative of its class. In the preceding example, this function is denoted $a \bmod m,$ and produces the remainder of the Euclidean division of by .

Properties

Every element $x$ of $X$ is a member of the equivalence class Every two equivalence classes /math> and /math> are either equal or disjoint