equivalence classes - meaning and definition. What is equivalence classes
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What (who) is equivalence classes - definition

MATHEMATICAL CONCEPT
Quotient set; Equivalence classes; Factor space; Canonical projection; Canonical projection map; Equivalence class representative; Equivalence Class Representative; Equivalence Class Of Y; Class representative (mathematics); Quotient sets; Equivalence set; Representative (mathematics); Canonical surjection
  • Congruence]] is an example of an equivalence relation. The leftmost two triangles are congruent, while the third and fourth triangles are not congruent to any other triangle shown here. Thus, the first two triangles are in the same equivalence class, while the third and fourth triangles are each in their own equivalence class.
  • Graph of an example equivalence with 7 classes

equivalence class         
<mathematics> An equivalence class is a subset whose elements are related to each other by an equivalence relation. The equivalence classes of a set under some relation form a partition of that set (i.e. any two are either equal or disjoint and every element of the set is in some class). (1996-05-13)
Equivalence class         
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.
equivalence class         
¦ noun Mathematics & Logic the class of all members of a set that are in a given equivalence relation.

Wikipedia

Equivalence class

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

Formally, given a set S {\displaystyle S} and an equivalence relation {\displaystyle \,\sim \,} on S , {\displaystyle S,} the equivalence class of an element a {\displaystyle a} in S , {\displaystyle S,} denoted by [ a ] , {\displaystyle [a],} is the set

of elements which are equivalent to a . {\displaystyle a.} It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S . {\displaystyle S.} This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of S {\displaystyle S} by , {\displaystyle \,\sim \,,} and is denoted by S / {\displaystyle S/{\sim }} .

When the set S {\displaystyle S} has some structure (such as a group operation or a topology) and the equivalence relation {\displaystyle \,\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.