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A Infimum is the Set Element of a Partially Ordered Set that is GreaterThanOrEqualTo all Set Elements of some Subset of the set.



    • In mathematics, particularly set theory, the infimum (plural infima) of a subset of some set is the greatest element (not necessarily in the subset) that is less than or equal to all elements of the subset. Consequently the term greatest lower bound (also abbreviated as glb or GLB) is also commonly used. Infima of real numbers are a common special case that is especially important in analysis. However, the general definition remains valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered.
    • Infima are in a precise sense dual to the concept of a supremum and thus additional information and examples are found in that article.
    • In analysis the infimum or greatest lower bound of a subset [math]\displaystyle{ S }[/math] of real numbers is denoted by inf(S) and is defined to be the biggest real number that is smaller than or equal to every number in S. If no such number exists (because [math]\displaystyle{ S }[/math] is not bounded below), then we define inf(S) = −∞. If [math]\displaystyle{ S }[/math] is empty, we define inf(S) = ∞ (see extended real number line).