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Mathematical construction of a set with an equivalence relation
In mathematics, a setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional
Setoid
Mathematical ways to group elements of a set
equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory. A partition of a set X is a
Partition_of_a_set
Category where every morphism is invertible; generalization of a group
A → C {\displaystyle h\circ g:A\rightarrow C} . Special cases include: Setoid: a set that comes with an equivalence relation, G-set: a set equipped with
Groupoid
Mathematical concept
equivalence relation – Generalization of equivalence classes to scheme theory Setoid – Mathematical construction of a set with an equivalence relation Transversal
Equivalence_class
Data type in type theory
types, setoids (sets explicitly equipped with an equivalence relation) are often used instead of quotient types. However, unlike with setoids, many type
Quotient_type
Additional mathematical object
(geometries), orders, graphs, events, differential structures, categories, setoids, and equivalence relations. Sometimes, a set is endowed with more than
Mathematical_structure
Mathematical concept for comparing objects
{\displaystyle X} together with the relation ∼ {\displaystyle \,\sim \,} is called a setoid. The equivalence class of a {\displaystyle a} under ∼ , {\displaystyle \
Equivalence_relation
Mathematical set with an ordering
given two elements. This definition is equivalent to a partial order on a setoid, where equality is taken to be a defined equivalence relation rather than
Partially_ordered_set
t_{y}(\Gamma (f)(i))=A(f,i)(t_{x}(i))} . Other models of type theory include the setoid model,[citation needed] the groupoid model, the simplicial set model, the
Semantics_of_type_theory
Mathematical concept for comparing objects
more commonly used, particularly to define setoids, sometimes called partial setoids. Forming a partial setoid from a type and a PER is analogous to forming
Partial_equivalence_relation
Basic notion of sameness in mathematics
Inequality Logical equality Logical equivalence Relational operator § Equality Setoid Theory of pure equality Uniqueness quantification "Equality (n.), sense
Equality_(mathematics)
apartness relation is known as a constructive setoid. A function f : A → B {\displaystyle f:A\to B} between such setoids A {\displaystyle A} and B {\displaystyle
Apartness_relation
Abstract data type for storing distinct values
may be modeled by refinement types, and quotient sets may be replaced by setoids.) The characteristic function F {\displaystyle F} of a set S {\displaystyle
Set_(abstract_data_type)
Alternative foundation of mathematics
is somewhat more cumbersome, since intensional reasoning requires using setoids or similar constructions. There are many common mathematical objects that
Intuitionistic_type_theory
Axiomatic set theories based on the principles of mathematical constructivism
Variants of the functional predicate definition using apartness relations on setoids have been defined as well. A subset of a function is still a function and
Constructive_set_theory
Logic principle
foundations of mathematics are generally not extensional in this sense, and setoids are commonly used to maintain a difference between intensional equality
Extensionality
If C is cartesian closed or locally cartesian closed, then so is Cex. setoid Menni, Matias (2000). "Exact completion and toposes" (PDF). Retrieved 18
Exact_completion
Genus of fungi
Bovista sclerocystis is the only species in the genus with mycosclereids (setoid elements) in the peridium. Spores are brown to purple-brown, roughly spherical
Bovista
(A,B)\simeq \operatorname {Hom} (B,f(A))^{*}} for any objects A, B. setoid A setoid is an object in the free exact completion of the category of sets.
Glossary_of_category_theory
SETOID
SETOID
SETOID
SETOID
Girl/Female
Maori
Industrious, from Amelia.
Boy/Male
Indian, Punjabi, Sikh
Acting to Attain the Spirit
Biblical
the praise of the Lord
Boy/Male
Hindu, Indian, Traditional
At the Feet of the Lord
Boy/Male
Arabic
Servant of the merciful.
Surname or Lastname
English
English : metronymic from Goody.
Girl/Female
British, English
Many
Boy/Male
Indian
Prince, The Moon
Male
German
Old High German name derived from the word wari(n), WARIN means "cover, shelter," from warnôn "to be careful, watchful."Â
Girl/Female
Italian Latin Greek
Golden.
SETOID
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