# Definitions for equivalence classequiv·a·lence class

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1. equivalence classnoun

Any one of the subsets into which an equivalence relation partitions a set, each of these subsets containing all the elements of the set that are equivalent under the equivalence relation.

### WikipediaRate this definition:0.0 / 0 votes

1. 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.

### WikidataRate this definition:0.0 / 0 votes

1. Equivalence class

In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element n in X is the subset of all elements in X which are equivalent to n. Equivalence classes among elements of a structure are often used to produce a smaller structure whose elements are the classes, distilling a relationship every element of the class shares with at least one other element of another class. This is known as modding out by the class. The class may assume the identity of one of the original elements, as when fractions are put in reduced form.

### Numerology

1. Chaldean Numerology

The numerical value of equivalence class in Chaldean Numerology is: 9

2. Pythagorean Numerology

The numerical value of equivalence class in Pythagorean Numerology is: 6

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"equivalence class." Definitions.net. STANDS4 LLC, 2024. Web. 13 Jun 2024. <https://www.definitions.net/definition/equivalence+class>.

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