In mathematics, particularly in order theory, an upper bound or majorant[1] of a subset S of some preordered set (K, ≤) is an element of K that is greater than or equal to every element of S.[2][3] Dually, a lower bound or minorant of S is defined to be an element of K that is less than or equal to every element of S. A set with an upper (respectively, lower) bound is said to be bounded from above or majorized[1] (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.[4]

A set with upper bounds and its least upper bound

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