# equivalence relation

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*or equivalence class*)A relation which is transitive, symmetric, and reflexive divides its field into exclusive classes of things. Within each class everything bears the relation to everything else, and nothing bears it to anything in a different class. ‘Of the same height as’ or ‘having the same number of members as’ are equivalence relations that divide the field of people into classes of equally tall people, and sets into different classes depending on how many members they have. Defining an equivalence class of things at the same point on the scale is the fundamental operation in establishing any kind of measure of a property or quantity. In mathematical logic an early definition of a cardinal number ( Frege's ) was as an equivalence class of equinumerous classes.

*Philosophy dictionary.
Academic.
2011.*

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**Equivalence relation**— In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. Let a , b , and c be arbitrary elements of some set X . Then a b or a ≡ b denotes that a is… … Wikipedia**equivalence relation**— Math. a relation that is reflexive, symmetrical, and transitive, as equality. [1945 50] * * * In mathematics, a generalization of the idea of equality between elements of a set. All equivalence relations (e.g., that symbolized by the equals sign) … Universalium**equivalence relation**— noun A binary relation that is reflexive, symmetric and transitive. See Also: equivalence class, setoid, ≡, equivalence relation … Wiktionary**equivalence relation**— /əˈkwɪvələns rəˌleɪʃən/ (say uh kwivuhluhns ruh.layshuhn) noun Mathematics, Logic a reflexive, symmetric and transitive relation, that establishes any two elements in the set as equivalent or non equivalent. Also, equivalence … Australian English dictionary**equivalence relation**— noun Mathematics & Logic a relation between elements of a set which is reflexive, symmetric, and transitive and which defines exclusive classes whose members bear the relation to each other and not to those in other classes … English new terms dictionary**equivalence relation**— noun Date: circa 1949 a relation (as equality) between elements of a set (as the real numbers) that is symmetric, reflexive, and transitive and for any two elements either holds or does not hold … New Collegiate Dictionary**equivalence relation**— noun : a relation (as equality) between elements of a set (as the real numbers) that is symmetric, reflexive, and transitive and for any two elements either holds or does not hold … Useful english dictionary**Adequate equivalence relation**— In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well working theory of such cycles, and in particular, well defined… … Wikipedia**Partial equivalence relation**— In mathematics, a partial equivalence relation (often abbreviated as PER) R on a set X is a relation which is symmetric and transitive . In other words, it holds for all a, b and c in X that:# (Symmetry) if a R b then b R a # (Transitivity) if a… … Wikipedia**Borel equivalence relation**— In mathematics, a Borel equivalence relation on a Polish space X is an equivalence relation on X that is a Borel subset of X times; X (in the product topology).Given Borel equivalence relations E and F on Polish spaces X and Y respectively, one… … Wikipedia