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Equivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number.
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 ...
Graph of an example equivalence with 7 classes An equivalence relation is a mathematical relation that generalizes the idea of similarity or sameness. It is defined on a set X {\displaystyle X} as a binary relation ∼ {\displaystyle \sim } that satisfies the three properties: reflexivity , symmetry , and transitivity .
Hence, given the information that the identity functors form an equivalence of categories, in this example one still can choose between two natural isomorphisms for each direction. The category of sets and partial functions is equivalent to but not isomorphic with the category of pointed sets and point-preserving maps. [2]
An example of a left quasi-reflexive relation is a left Euclidean relation, which is always left quasi-reflexive but not necessarily right quasi-reflexive, and thus not necessarily quasi-reflexive. An example of a coreflexive relation is the relation on integers in which each odd number is related to itself and there are no other relations. The ...
Equivalence relationships exist between exact copies of the same manifestation of a work or between an original item and reproductions of it, so long as the intellectual content and authorship are preserved. Examples include reproductions such as copies, issues, facsimiles and reprints, photocopies, and microfilms.
The examples "is greater than", "is at least as great as", and "is equal to" are transitive relations on various sets. As are the set of real numbers or the set of natural numbers: whenever x > y and y > z, then also x > z whenever x ≥ y and y ≥ z, then also x ≥ z whenever x = y and y = z, then also x = z. More examples of transitive ...