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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.
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]
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 .
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 ...
Functional Requirements for Bibliographic Records (FRBR / ˈ f ɜːr b ər /) is a conceptual entity–relationship model developed by the International Federation of Library Associations and Institutions (IFLA) that relates user tasks of retrieval and access in online library catalogues and bibliographic databases from a user’s perspective.
An example is the relation "is equal to", because if a = b is true then b = a is also true. If R T represents the converse of R, then R is symmetric if and only if R = R T. [2] Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation. [1]
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 Z * (X) := Z[X] be the free abelian group on the algebraic cycles of X. Then an adequate equivalence relation is a family of equivalence relations, ~ X on Z * (X), one for each smooth projective variety X, satisfying the following three conditions: (Linearity) The equivalence relation is compatible with addition of cycles.