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...

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...

foundations of mathematics are generally not extensional in this sense, and setoids are commonly used to maintain a difference between intensional equality...

{\displaystyle X} together with the relation ∼ {\displaystyle \,\sim \,} is called a setoid. The equivalence class of a {\displaystyle a} under ∼ , {\displaystyle \...

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...

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...

types, setoids (sets explicitly equipped with an equivalence relation) are often used instead of quotient types. However, unlike with setoids, many type...

is somewhat more cumbersome, since intensional reasoning requires using setoids or similar constructions. There are many common mathematical objects that...