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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.
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.
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 .
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]
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 ...
For example, "being isomorphic" is an equivalence relation on groups, and the equivalence classes, called isomorphism classes, are not sets. The set of all equivalence classes in X {\displaystyle X} with respect to an equivalence relation R {\displaystyle R} is denoted as X / R , {\displaystyle X/R,} and is called X {\displaystyle X} modulo R ...
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.