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>\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. | + | Given a [[subset]] <math>S</math> in some larger [[partially 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. | ||
| − | '''The Least Upper Bound Axiom''': This is one of the fundamental axioms of real analysis. According to it, any nonempty set of real | + | |
| + | '''The Least Upper Bound Axiom''': This is one of the fundamental axioms of real [[analysis]]. According to it, any [[empty set | nonempty]] [[set]] of [[real number]]s that is bounded above has a supremum. This is something intuitively clear but impossible to prove using only the field properties, order properties and completeness property of the set of real numbers. | ||
[[Category:Definition]] | [[Category:Definition]] | ||
Revision as of 13:07, 9 February 2007
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Given a subset 
in some larger partially 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.
The Least Upper Bound Axiom: This is one of the fundamental axioms of real analysis. According to it, any nonempty set of real numbers that is bounded above has a supremum. This is something intuitively clear but impossible to prove using only the field properties, order properties and completeness property of the set of real numbers.
