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In mathematics, equivalent definitions are used in two somewhat different ways. First, within a particular mathematical theory (for example, Euclidean geometry), a notion (for example, ellipse or minimal surface) may have more than one definition.
Assuming the existence of an infinite set N consisting of all natural numbers and assuming the existence of the power set of any given set allows the definition of a sequence N, P(N), P(P(N)), P(P(P(N))), … of infinite sets where each set is the power set of the set preceding it. By Cantor's theorem, the cardinality of each set in this ...
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 ...
Given any set , an equivalence relation over the set [] of all functions can be obtained as follows. Two functions are deemed equivalent when their respective sets of fixpoints have the same cardinality, corresponding to cycles of length one in a permutation.
In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set that is not empty (where by definition, "is empty" means that the statement () is true) might not have an inhabitant (which is an such that ).
Define the two measures on the real line as = [,] () = [,] for all Borel sets. Then and are equivalent, since all sets outside of [,] have and measure zero, and a set inside [,] is a -null set or a -null set exactly when it is a null set with respect to Lebesgue measure.
A functor F : C → D yields an equivalence of categories if and only if it is simultaneously: . full, i.e. for any two objects c 1 and c 2 of C, the map Hom C (c 1,c 2) → Hom D (Fc 1,Fc 2) induced by F is surjective;
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...