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[[File:Illustration of supremum.svg|thumb|300px|A set with upper bounds and its least upper bound]]
In [[mathematics]], particularly in [[order theory]], an '''upper bound''' or '''majorant'''<ref name=schaefer/> of a [[subset]] {{mvar|S}} of some [[Preorder|preordered set]] {{math|(''K'', ≤)}} is an element of {{mvar|K}} that is {{nobr|[[greater than or equal to]]}} every element of {{mvar|S}}.<ref name="MacLane-Birkhoff" /><ref>{{Cite web|url=https://www.mathsisfun.com/definitions/upper-bound.html|title=Upper Bound Definition (Illustrated Mathematics Dictionary)|website=Math is Fun|access-date=2019-12-03}}</ref>
[[Duality (order theory)|Dually]], a '''lower bound''' or '''minorant''' of {{mvar|S}} is defined to be an element of {{mvar|K}} that is less than or equal to every element of {{mvar|S}}.
A set with an upper (respectively, lower) bound is said to be '''bounded from above''' or '''majorized'''<ref name=schaefer/> (respectively '''bounded from below''' or '''minorized''') by that bound.
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The set {{math|1=''S'' = {{mset|42}}}} has {{math|42}} as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that {{mvar|S}}.
Every subset of the [[natural number]]s has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above. An infinite subset of the
Every finite subset of a non-empty [[totally ordered set]] has both upper and lower bounds.
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