##### Wiktionary

**equivalence class**

n. (context set theory English) 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.

##### Wikipedia

**Equivalence class**

In mathematics, when the elements of some set*S* have a notion of equivalence (formalized as an equivalence relation) defined on them, 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 a and b are equivalent.

Formally, given a set and an equivalence relation on , the *equivalence class* of an element in is the set

{*x* ∈ *S* ∣ *x* ∼ *a*}

of elements which are equivalent to . It may be proven from the defining properties of "equivalence relations" that the equivalence classes form a partition of . This partition - the set of equivalence classes - is sometimes called the **quotient set** or the **quotient space** of by and is denoted by .

When the set has some structure (such as a group operation or a topology) and the equivalence relation ~ is defined in a manner suitably compatible with this structure, then 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.

**Equivalence class (music)**

*This article is about equivalency in music; for equivalency in mathematics see Equality (mathematics) and equivalence class.*

In music theory, **equivalence class** is an equality ( =) or equivalence between sets or twelve-tone rows. A relation rather than an operation, it may be contrasted with derivation. "It is not surprising that music theorists have different concepts of equivalence [from each other]..." "Indeed, an informal notion of equivalence has always been part of music theory and analysis. Pitch class set theory, however, has adhered to formal definitions of equivalence."

A definition of equivalence between two twelve-tone series that Schuijer describes as informal despite its air of mathematical precision, and that shows its writer considered equivalence and equality as synonymous:

Forte (1963, p. 76) similarly uses *equivalent* to mean *identical*, "considering two subsets as equivalent when they consisted of the same elements. In such a case, mathematical set theory speaks of the 'equality,' not the 'equivalence,' of sets."

Other equivalencies in music include:

- Enharmonic equivalency
- Inversional equivalency
- Octave equivalency
- Permutational equivalency
- Transpositional equivalency