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In combinatorial number theory, the Lambek–Moser theorem is a generalization of Beatty's theorem that defines a partition of the positive integers into two subsets from any monotonic integer-valued function. Conversely, any partition of the positive integers into two subsets may be defined from a monotonic function in this way. The theorem was discovered by Leo Moser and Joachim Lambek. provides a visual proof of the result.〔For another proof, see 〕 ==Statement of the theorem== The theorem applies to any non-decreasing and unbounded function ''f'' that maps positive integers to non-negative integers. From any such function ''f'', define ''f'' * to be the integer-valued function that is as close as possible to the inverse function of ''f'', in the sense that, for all ''n'', :''f''(''f'' *(''n'')) < ''n'' ≤ ''f''(''f'' *(''n'') + 1). It follows from this definition that ''f'' * * = ''f''. Further, define :''F''(''n'') = ''f''(''n'') + ''n'' and ''G''(''n'') = ''f'' *(''n'') + ''n''. Then the result states that ''F'' and ''G'' are strictly increasing and that the ranges of ''F'' and ''G'' form a partition of the positive integers. 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「Lambek–Moser theorem」の詳細全文を読む スポンサード リンク
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