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Supremum

The supremum is the least upper bound of a set , defined as a quantity such that no member of the set exceeds , but if is any positive quantity, however small, there is a member that exceeds (Jeffreys and Jeffreys 1988). When it exists (which is not required by this definition, e.g., does not exist), it is denoted (or sometimes simply for short). The supremum is implemented in the Wolfram Language as MaxValue[f, constr, vars].More formally, the supremum for a (nonempty) subset of the affinely extended real numbers is the smallest value such that for all we have . Using this definition, always exists and, in particular, .Whenever a supremum exists, its value is unique. On the real line, the supremum of a set is the same as the supremum of its set closure.Consider the real numbers with their usual order. Then for any set , the supremum exists (in ) if and only if is bounded from above and nonempty...

Infimum

The infimum is the greatest lower bound of a set , defined as a quantity such that no member of the set is less than , but if is any positive quantity, however small, there is always one member that is less than (Jeffreys and Jeffreys 1988). When it exists (which is not required by this definition, e.g., does not exist), the infimum is denoted or . The infimum is implemented in the Wolfram Language as MinValue[f, constr, vars].Consider the real numbers with their usual order. Then for any set , the infimum exists (in ) if and only if is bounded from below and nonempty.More formally, the infimum for a (nonempty) subset of the affinely extended real numbers is the largest value such that for all we have . Using this definition, always exists and, in particular, .Whenever an infimum exists, its value is unique...

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