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As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum.
The maximum of a subset of a preordered set is an element of which is greater than or equal to any other element of , and the minimum of is again defined dually. In the particular case of a partially ordered set , while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements.
If it does, it is a minimum or least element of . Similarly, if the supremum of belongs to , it is a maximum or greatest element of . For example, consider the set of negative real numbers (excluding zero).
The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value.
In a totally ordered set the maximal element and the greatest element coincide; and it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum. [1] The dual terms are minimum and absolute minimum. Together they are called the absolute extrema. Similar conclusions ...
Six's maximum and minimum thermometer is a registered thermometer that can record the maximum and minimum temperatures reached over a period of time, for example 24 hours. It is used to record the extremes of temperature at a location, for instance in meteorology and horticulture .
The maximum monthly Social Security benefit is $5,108 but it’s very difficult to get it. You must earn the maximum taxable wage for 35 years or longer. You also must not claim Social Security ...
The minimum and the maximum value are the first and last order statistics (often denoted X (1) and X (n) respectively, for a sample size of n). If the sample has outliers, they necessarily include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low. However, the sample maximum and minimum need not ...