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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.
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 .
For example, in modular arithmetic, for every integer m greater than 1, the congruence modulo m is an equivalence relation on the integers, for which two integers a and b are equivalent—in this case, one says congruent—if m divides ; this is denoted ().
For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms. If is a group with operation , a congruence relation on is an equivalence relation on the elements of satisfying
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.
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]
For example, if the union of the equivalence relations has a property known as "Borel-boundedness" (which roughly means that any Borel assignment of functions : to points on the space can be "eventually bounded" by such a Borel assignment which is constant on equivalence classes), then it is hyperfinite. However, it is unknown whether every ...