Difference between revisions of "Least upper bound"
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| − | Given a [[subset]] <math>S</math> in some larger [[ordered set]] <math>R</math>, a '''least upper bound''' or '''supremum''' for <math>S</math> is an [[element]] <math>M \in R</math> such that <math>s \leq M</math> for every <math>s \in S</math> and there is no <math>m < M</math> with this same property. | + | Given a [[subset]] <math>S</math> in some larger [[ordered set]] <math>R</math>, a '''least upper bound''' or '''supremum''' for <math>S</math> is an [[element]] <math>\displaystyle M \in R</math> such that <math>s \leq M</math> for every <math>s \in S</math> and there is no <math>m < M</math> with this same property. |
If the least upper bound <math>M</math> of <math>S</math> is an element of <math>S</math>, it is also the [[maximum]] of <math>S</math>. If <math>M \not\in S</math>, then <math>S</math> has no maximum. | If the least upper bound <math>M</math> of <math>S</math> is an element of <math>S</math>, it is also the [[maximum]] of <math>S</math>. If <math>M \not\in S</math>, then <math>S</math> has no maximum. | ||
Revision as of 19:19, 3 November 2006
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Given a subset 
in some larger ordered set 
, a least upper bound or supremum for is an element 
such that 
for every 
and there is no 
with this same property.
If the least upper bound 
of is an element of , it is also the maximum of . If 
, then has no maximum.
