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The equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. The canonical map ker : X^X → Con X, relates the monoid X^X of all functions on X and Con X. ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f : X → X to its ...
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.
Let be the set of ordered pairs of integers (,) with non-zero , and define an equivalence relation on such that (,) (,) if and only if =, then the equivalence class of the pair (,) can be identified with the rational number /, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of ...
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. In an abstract set, a ternary equivalence relation determines a collection of equivalence classes or pencils that form a linear ...
Conversely, every partition defines an equivalence class. The equivalence relation of equality is a special case, as, if restricted to a given set , it is the strictest possible equivalence relation on ; specifically, equality partitions a set into equivalence classes consisting of all singleton sets.
The equivalence relation corresponding to a partition defines two elements as being equivalent when they belong to the same partition subset as each other. Conversely, every equivalence relation corresponds to a partition into equivalence classes. [2] Therefore, the Bell numbers also count the equivalence relations.
An equivalent formulation in this context is the following: [4] A congruence relation on an algebra A is a subset of the direct product A × A that is both an equivalence relation on A and a subalgebra of A × A. The kernel of a homomorphism is always a congruence. Indeed, every congruence arises as a kernel.
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so too are its restrictions. However, the transitive closure of a restriction is a subset of the restriction of the ...