Editing Upper and lower bounds (section)

{{short description|Majorant and minorant in mathematics}}
{{About|precise bounds|asymptotic bounds|Big O notation}}
[[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 (mathematics)|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. 
The terms '''bounded above''' ('''bounded below''') are also used in the mathematical literature for sets that have upper (respectively lower) bounds.<ref>{{Cite web|url=http://mathworld.wolfram.com/UpperBound.html|title=Upper Bound|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-03}}</ref>