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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. A simpler example is equality. Any number is equal to itself (reflexive).
In the category of sets, the coequalizer of two functions f, g : X → Y is the quotient of Y by the smallest equivalence relation ~ such that for every x ∈ X, we have f(x) ~ g(x). [1] In particular, if R is an equivalence relation on a set Y, and r 1, r 2 are the natural projections (R ⊂ Y × Y) → Y then the coequalizer of r 1 and r 2 is ...
A major area of research in descriptive set theory is the classification of Borel equivalence relations, and in particular those which are countable. Among these, finite equivalence relations are considered to be the simplest (for instance, they admit Borel transversals). Therefore, it is natural to ask whether certain equivalence relations ...
In his controversial 1878 paper, Cantor explicitly defined the notion of "power" of sets and used it to prove that the set of all natural numbers and the set of all rational numbers are equinumerous (an example where a proper subset of an infinite set is equinumerous to the original set), and that the Cartesian product of even a ...
Similarly, R on X is left Euclidean if, and only if, R ′ := R| dom(R) is an equivalence and for each x in X\dom(R), all elements that are related to x under R are equivalent under R ′. A left Euclidean relation is left-unique if, and only if, it is antisymmetric. Similarly, a right Euclidean relation is right unique if, and only if, it is ...
In mathematics, when the elements of some set have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set into equivalence classes. These equivalence classes are constructed so that elements a {\displaystyle a} and b {\displaystyle b} belong to the same equivalence class if, and only if , they are ...
In mathematics, given a category C, a quotient of an object X by an equivalence relation: is a coequalizer for the pair of maps , =,, where R is an object in C and "f is an equivalence relation" means that, for any object T in C, the image (which is a set) of : = (,) () is an equivalence relation; that is, a reflexive, symmetric and transitive relation.
Elementary equivalence; Equals sign; Equality (mathematics) Equality operator; Equipollence (geometry) Equivalence (measure theory) Equivalence class; Equivalence of categories; Equivalence of metrics; Equivalence relation; Equivalence test; Equivalent definitions of mathematical structures; Equivalent infinitesimal; Equivalent latitude ...
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