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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
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
Mathematical concept
equivalence relation – Generalization of equivalence classes to scheme theory Setoid – Mathematical construction of a set with an equivalence relation Transversal
Equivalence_class
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 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
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
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
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
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
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
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
SETOID
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Boy/Male
Gujarati, Hindu, Indian, Kannada, Sanskrit
Strifes to Triumph
Boy/Male
Hindu, Indian
King of Love
Surname or Lastname
English
English : patronymic from the medieval personal name Jan (see Jayne).
Female
French
Feminine form of French Frédéric, FRÉDÉRIQUE means "peaceful ruler."
Boy/Male
Hindu
Beautiful slender girl
Surname or Lastname
English
English : habitational name from a place in North Yorkshire, named in Old English ‘farmstead (Old English tūn) of a man called Ælfwine or a woman called Ælfwynn’. This is now a very rare name in England.
Boy/Male
Tamil
Shark
Boy/Male
Indian, Punjabi, Sikh
Love for Flowers
Boy/Male
Muslim
Name of a companion
Female
Russian
 Contracted form of Russian Ekaterina, KARINA means "pure." Compare with other forms of Karina.
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