"E-set" redirects here. For the technique in fertility medicine, see e-SET.
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 set.[1]
Setoids are studied especially in proof theory and in type-theoretic foundations of mathematics. 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 intensional equality (the equality on the original set) and extensional equality (the equivalence relation, or the equality on the quotient set).
^Alexandre Buisse, Peter Dybjer, "The Interpretation of Intuitionistic Type Theory in Locally Cartesian Closed Categories—an Intuitionistic Perspective", Electronic Notes in Theoretical Computer Science 218 (2008) 21–32.
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...
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...
foundations of mathematics are generally not extensional in this sense, and setoids are commonly used to maintain a difference between intensional equality...
equivalence relation – Generalization of equivalence classes to scheme theory Setoid – Mathematical construction of a set with an equivalence relation Transversal...
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...
types, setoids (sets explicitly equipped with an equivalence relation) are often used instead of quotient types. However, unlike with setoids, many type...
A → C {\displaystyle h\circ g:A\rightarrow C} . Special cases include: Setoids: sets that come with an equivalence relation, G-sets: sets equipped with...
{\displaystyle X} together with the relation ∼ {\displaystyle \,\sim \,} is called a setoid. The equivalence class of a {\displaystyle a} under ∼ , {\displaystyle \...
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...
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...
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...
is somewhat more cumbersome, since intensional reasoning requires using setoids or similar constructions. There are many common mathematical objects that...
Bovista sclerocystis is the only species in the genus with mycosclereids (setoid elements) in the peridium. Spores are brown to purple-brown, roughly spherical...
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