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In mathematics, the infimum (abbreviated inf; plural infima) of a subset ''S'' of a partially ordered set ''T'' is the greatest element in ''T'' that is less than or equal to all elements of ''S'', if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ''GLB'') is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset ''S'' of a partially ordered set ''T'' is the least element in ''T'' that is greater than or equal to all elements of ''S'', if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ''LUB''). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. If the supremum of a subset ''S'' exists, it is unique. If ''S'' contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to ''S'' (or does not exist). Likewise, if the infimum exists, it is unique. If ''S'' contains a least element, then that element is the infimum; otherwise, the infimum does not belong to ''S'' (or does not exist). The concepts of supremum and infimum are similar to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have ''no minimum or maximum''. For instance, the positive real numbers ℝ+ * does not have a minimum, because any given element of ℝ+ * could simply be divided in half resulting in a smaller number that is still in ℝ+ *. There is, however, exactly one infimum of the positive real numbers: 0, which is smaller than all the positive real numbers and greater than any other number which could be used as a lower bound. Note that 0 ∉ ℝ+ *. == Formal definition == A ''lower bound'' of a subset ''S'' of a partially ordered set (''P'', ≤) is an element ''a'' of ''P'' such that * ''a'' ≤ ''x'' for all ''x'' in ''S''. A lower bound ''a'' of ''S'' is called an ''infimum'' (or ''greatest lower bound'', or ''meet'') of ''S'' if * for all lower bounds ''y'' of ''S'' in ''P'', ''y'' ≤ ''a'' (''a'' is larger than any other lower bound). Similarly, an ''upper bound'' of a subset ''S'' of a partially ordered set (''P'', ≤) is an element ''b'' of ''P'' such that * ''b'' ≥ ''x'' for all ''x'' in ''S''. An upper bound ''b'' of ''S'' is called a ''supremum'' (or ''least upper bound'', or ''join'') of ''S'' if * for all upper bounds ''z'' of ''S'' in ''P'', ''z'' ≥ ''b'' (''b'' is less than any other upper bound). 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「Infimum and supremum」の詳細全文を読む スポンサード リンク
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