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In mathematics, two sets ''A'' and ''B'' are equinumerous if there exists a one-to-one correspondence (a bijection) between them, i.e. if there exists a function from ''A'' to ''B'' such that for every element ''y'' of ''B'' there is exactly one element ''x'' of ''A'' with ''f''(''x'') = ''y''.〔 This definition can be applied to both finite and infinite sets and allows one to state whether two sets have the same size even if they are infinite. The study of cardinality is often called equinumerosity (''equalness-of-number''). The terms equipollence (''equalness-of-strength'') and equipotence (''equalness-of-power'') are sometimes used instead. The statement that two sets ''A'' and ''B'' are equinumerous is usually denoted : or , or Georg Cantor, the inventor of set theory, showed in 1874 that there is more than one kind of infinity, specifically that the collection of all natural numbers and the collection of all real numbers, while both infinite, are not equinumerous (see Cantor's first uncountability proof). In a controversial 1878 paper, Cantor explicitly defined the notion of "power" of sets and used it to prove that the set of all natural numbers and the set of all rational numbers are equinumerous, and that the Cartesian product of even a countably infinite number of copies of the real numbers is equinumerous to a single copy of the real numbers. Cantor's theorem from 1891 implies that no set is equinumerous to its power set.〔 This allows the definition of greater and greater infinite sets starting from a single infinite set. Equinumerous finite sets have the same number of elements. Equinumerosity has the characteristic properties of an equivalence relation.〔 Equinumerous sets are said to have the same cardinality, and the cardinal number of a set is the equivalence class of all sets equinumerous to it.〔 The statement that any two sets are either equinumerous or one has a smaller cardinality than the other is equivalent to the axiom of choice.〔 Unlike finite sets, some infinite sets are equinumerous to proper subsets of themselves.〔 == Cardinality == Equinumerous sets are said to have the same cardinality. The cardinality of a set ''X'' is a measure of the "number of elements of the set" and can be defined as the equivalence class of all sets equinumerous to ''X''.〔 This is possible because equinumerosity has the characteristic properties of an equivalence relation (reflexivity, symmetry, and transitivity): ;Reflexivity: Given a set ''A'', the identity function on ''A'' is a bijection from ''A'' to itself, showing that every set ''A'' is equinumerous to itself: . ;Symmetry: For every bijection between two sets ''A'' and ''B'' there exists an inverse function which is a bijection between ''B'' and ''A'', implying that if a set ''A'' is equinumerous to a set ''B'' then ''B'' is also equinumerous to ''A'': implies . ;Transitivity: Given three sets ''A'', ''B'' and ''C'' with two bijections and , the composition of these bijections is a bijection from ''A'' to ''C'', so if ''A'' and ''B'' are equinumerous and ''B'' and ''C'' are equinumerous then ''A'' and ''C'' are equinumerous: and together imply . The definition of the cardinality of a set as the equivalence class of all sets equinumerous to it is problematic in Zermelo–Fraenkel set theory, the standard form of axiomatic set theory, because the equivalence class of a non-empty set is too large to be a set: it is a proper class. Within the framework of Zermelo–Fraenkel set theory relations are by definition restricted to sets (a binary relation on a set ''A'' is a subset of the Cartesian product ), and there is no set of all sets in Zermelo–Fraenkel set theory. In Zermelo–Fraenkel set theory, instead of defining the cardinality of a set as the equivalence class of all sets equinumerous to it one tries to assign a representative set to each equivalence class (cardinal assignment). In some other systems of axiomatic set theory, e.g. Von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory, relations are extended to classes. A set ''A'' is said to have cardinality smaller than or equal to the cardinality of a set ''B'' if there exists a one-to-one function (an injection) from ''A'' into ''B''. This is denoted |''A''| ≤ |''B''|. If ''A'' and ''B'' are not equinumerous, then the cardinality of ''A'' is said to be strictly smaller than the cardinality of ''B''. This is denoted |''A''| < |''B''|. The law of trichotomy holds for cardinal numbers, so that any two sets are either equinumerous, or one has a strictly smaller cardinality than the other.〔 The law of trichotomy for cardinal numbers is equivalent to the historically highly controversial axiom of choice. The Schröder-Bernstein theorem states that any two sets ''A'' and ''B'' for which there exist two one-to-one functions and are equinumerous: if |''A''| ≤ |''B''| and |''B''| ≤ |''A''|, then |''A''| = |''B''|.〔〔 This theorem does not rely on the axiom of choice. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Equinumerosity」の詳細全文を読む スポンサード リンク
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