Difference between revisions of "Least upper bound"

Revision as of 00:37, 6 March 2008

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Given a subset $S$ in some larger partially ordered set $R$ , a least upper bound or supremum, for $S$ is an element $M \in R$ such that $s \leq M$ for every $s \in S$ and there is no $m < M$ with this same property.

If the least upper bound $M$ of $S$ is an element of $S$ , it is also the maximum of $S$ . If $M \not\in S$ , then $S$ has no maximum.

Completeness: This is one of the fundamental axioms of real analysis.

A set $S$ is said to be complete if any nonempty subset of $S$ that is bounded above has a supremum.

The fact that $\mathbb{R}$ is complete is something intuitively clear but impossible to prove using only the field and order properties of $\mathbb{R}$

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-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.
+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>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.
-'''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.
+'''Completeness''': This is one of the fundamental axioms of real [[analysis]].
+A set <math>S</math> is said to be '''complete''' if any [[empty set | nonempty]] [[set|subset]] of <math>S</math> that is [[bounded]] above has a supremum.
+The fact that <math>\mathbb{R}</math> is complete is something intuitively clear but impossible to prove using only the field and order properties of <math>\mathbb{R}</math>
 [[Category:Definition]]

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Difference between revisions of "Least upper bound"

Revision as of 00:37, 6 March 2008