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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク König's theorem (set theory) ： ウィキペディア英語版 König's theorem (set theory) In set theory, König's theorem states that if the axiom of choice holds, ''I'' is a set, ''mi'' and ''ni'' are cardinal numbers for every ''i'' in ''I'', and for every ''i'' in ''I'' then : The ''sum'' here is the cardinality of the disjoint union of the sets ''mi'' and the product is the cardinality of the Cartesian product. However, without the use of the axiom of choice, the sum and the product cannot be defined as cardinal numbers, and the meaning of the inequality sign would need to be clarified. König's theorem was introduced by in the slightly weaker form that the sum of a strictly increasing sequence of nonzero cardinal numbers is less than their product. == Details == The precise statement of the result: if ''I'' is a set, ''Ai'' and ''Bi'' are sets for every ''i'' in ''I'', and : where < means ''strictly less than in cardinality,'' i.e. there is an injective function from ''Ai'' to ''Bi,'' but not one going the other way. The union involved need not be disjoint (a non-disjoint union can't be any bigger than the disjoint version, also assuming the axiom of choice). In this formulation, König's theorem is equivalent to the Axiom of Choice. (Of course, König's theorem is trivial if the cardinal numbers ''mi'' and ''ni'' are finite and the index set ''I'' is finite. If ''I'' is empty, then the left sum is the empty sum and therefore 0, while the right hand product is the empty product and therefore 1). König's theorem is remarkable because of the strict inequality in the conclusion. There are many easy rules for the arithmetic of infinite sums and products of cardinals in which one can only conclude a weak inequality ≤, for example: if for all ''i'' in ''I'', then one can only conclude :$\backslash sum\_\; m\_i\; \backslash le\; \backslash sum\_\; n\_i$ since, for example, setting $m\_i\; =\; 1$ & $n\_i\; =\; 2$ where the index set ''I'' is the natural numbers, yields the sum $\backslash aleph\_0$ for both sides and we have a strict equality. 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「König's theorem (set theory)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.