Equivalence class
Definition and notation

An equivalence relation on a set X {\displaystyle X} is a binary relation ∼ {\displaystyle \,\sim \,} on X {\displaystyle X} satisfying the three properties:

The equivalence class of an element a {\displaystyle a} is defined as

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 {\displaystyle X} with respect to an equivalence relation R {\displaystyle R} is denoted as X / R, {\displaystyle X/R,} and is called X {\displaystyle X} modulo R {\displaystyle R} (or the quotient set of X {\displaystyle X} by R {\displaystyle R}). The surjective map x ↦ {\displaystyle x\mapsto } from X {\displaystyle X} onto X / R, {\displaystyle X/R,} which maps each element to its equivalence class, is called the canonical surjection, 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 {\displaystyle X/R} to X. Since its composition with the canonical surjection is the identity of X / R, {\displaystyle 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 canonical representatives. For example, in modular arithmetic, for every integer m greater than 1, the congruence modulo m is an equivalence relation on the integers, for which two integers a and b are equivalent—in this case, one says congruent —if m divides a − b; {\displaystyle a-b;} this is denoted a ≡ b (mod m). {\textstyle a\equiv b{\pmod {m}}.} Each class contains a unique non-negative integer smaller than m, {\displaystyle 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 mod m, {\displaystyle a{\bmod {m}},} and produces the remainder of the Euclidean division of a by m.
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