What is "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.

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.

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