Extensional set

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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a setoid (''X'', ~) is a set (or type) ''X'' equipped with an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
~. A setoid may also be called E-set,
Bishop A bishop is an ordained clergy member who is entrusted with a position of authority and oversight in a religious institution. In Christianity, bishops are normally responsible for the governance of dioceses. The role or office of bishop is ...
set, or extensional set. Setoids are studied especially in
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding parts, ...
and in type-theoretic
foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathem ...
. Often in mathematics, when one defines an equivalence relation on a set, one immediately forms the quotient set (turning equivalence into equality). In contrast, setoids may be used when a difference between identity and equivalence must be maintained, often with an interpretation of
intension In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — an intension is any property or quality connoted by a word, phrase, ...
al equality (the equality on the original set) and extensional equality (the equivalence relation, or the equality on the quotient set).

# Proof theory

In proof theory, particularly the proof theory of
constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...
based on the Curry–Howard correspondence, one often identifies a mathematical
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
with its set of proofs (if any). A given proposition may have many proofs, of course; according to the principle of proof irrelevance, normally only the truth of the proposition matters, not which proof was used. However, the Curry–Howard correspondence can turn proofs into
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s, and differences between algorithms are often important. So proof theorists may prefer to identify a proposition with a ''setoid'' of proofs, considering proofs equivalent if they can be converted into one another through beta conversion or the like.

# Type theory

In type-theoretic foundations of mathematics, setoids may be used in a type theory that lacks quotient types to model general mathematical sets. For example, in
Per Martin-Löf Per Erik Rutger Martin-Löf (; ; born 8 May 1942) is a Swedish logician, philosopher, and mathematical statistician. He is internationally renowned for his work on the foundations of probability, statistics, mathematical logic, and computer sc ...
's intuitionistic type theory, there is no type of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s, only a type of regular Cauchy sequences of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
s. To do
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include con ...
in Martin-Löf's framework, therefore, one must work with a ''setoid'' of real numbers, the type of regular Cauchy sequences equipped with the usual notion of equivalence. Predicates and functions of real numbers need to be defined for regular Cauchy sequences and proven to be compatible with the equivalence relation. Typically (although it depends on the type theory used), the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
will hold for functions between types (intensional functions), but not for functions between setoids (extensional functions). The term "set" is variously used either as a synonym of "type" or as a synonym of "setoid".

# Constructive mathematics

In
constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...
, one often takes a setoid with an
apartness relation In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be more basic than equality. It is often written as \# (⧣ in unicode) to distinguish from the negation of equality (the ''denial inequa ...
instead of an equivalence relation, called a constructive setoid. One sometimes also considers a partial setoid using a partial equivalence relation or partial apartness. (see e.g. Barthe ''et al.'', section 1)

*
Groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial func ...

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