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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). If , then (symmetric).
An equivalence relation on a set is a binary relation on satisfying the three properties: [1] for all (reflexivity), implies for all (symmetry), if and then for all (transitivity). The equivalence class of an element is defined as [2] The word "class" in the term "equivalence class" may generally be considered as a synonym of "set", although ...
Equivalence of categories. In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves ...
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 .
Category theory. Schematic representation of a category with objects X, Y, Z and morphisms f, g, g ∘ f. (The category's three identity morphisms 1 X, 1 Y and 1 Z, if explicitly represented, would appear as three arrows, from the letters X, Y, and Z to themselves, respectively.) Category theory is a general theory of mathematical structures ...
In mathematics, a binary relation on a set is reflexive if it relates every element of to itself. [1][2] An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity.
Partial equivalence relation. In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation[1]) is a homogeneous binary relation that is symmetric and transitive. If the relation is also reflexive, then the relation is an equivalence relation.
In mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalence relation. A ternary equivalence relation is symmetric, reflexive, and transitive, where those terms are meant in the sense defined below. The classic example is the relation of collinearity among three points in Euclidean space.