In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) on a set
is a relation that is symmetric and transitive. In other words, it holds for all
that:
If is also reflexive, then
is an equivalence relation.
Properties and applications
In a set-theoretic context, there is a simple structure to the general PER on
: it is an equivalence relation on the subset
. (
is the subset of
such that in the complement of
(
) no element is related by
to any other.) By construction,
is reflexive on
and therefore an equivalence relation on
. Notice that
is actually only true on elements of
: if
, then
by symmetry, so
and
by transitivity. Conversely, given a subset Y of X, any equivalence relation on Y is automatically a PER on X.
PERs are therefore used mainly in computer science, type theory and constructive mathematics, particularly to define setoids, sometimes called partial setoids. The action of forming one from a type and a PER is analogous to the operations of subset and quotient in classical set-theoretic mathematics.
In mathematics, an equivalence relation is a binary relation that is at the same time a reflexive relation, a symmetric relation and a transitive relation. As a consequence of these properties an equivalence relation provides a partition of a set into equivalence classes.
Notation
Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R, the most common are "a ~ b" and "a ≡ b", which are used when R is the obvious relation being referenced, and variations of "a ~Rb", "a ≡Rb", or "aRb" otherwise.
Definition
A given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. That is, for all a, b and c in X:
X together with the relation ~ is called a setoid. The equivalence class of under ~, denoted
, is defined as
.
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson (1960) in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte (1973), drawing on the work in twelve-tone theory of Milton Babbitt. The concepts of set theory are very general and can be applied to tonal and atonal styles in any equally tempered tuning system, and to some extent more generally than that.
One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and can be related by musical operations such as transposition, inversion, and complementation. The methods of musical set theory are sometimes applied to the analysis of rhythm as well.
Mathematical set theory versus musical set theory
Although musical set theory is often thought to involve the application of mathematical set theory to music, there are numerous differences between the methods and terminology of the two. For example, musicians use the terms transposition and inversion where mathematicians would use translation and reflection. Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences (though mathematics does speak of ordered sets, and although these can be seen to include the musical kind in some sense, they are far more involved).