Difference between revisions of "Maximum"

 
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Given an [[ordered set]] <math>S</math>, the '''maximum''' [[element]] of <math>S</math>, if it exists, is some <math>M \in S</math> such that for all <math>n \in S</math>, <math>n \leq M</math>.
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Given a [[partially ordered set]] <math>S</math>, the '''maximum''' [[element]] of <math>S</math>, if it exists, is some <math>M \in S</math> such that for all <math>n \in S</math>, <math>n \leq M</math>.
  
 
For example, the maximum element of the [[set]] <math>S_1 = \{0, e, \pi, 4\}</math> of [[real number]]s is <math>4</math>, since it is larger than every other element of the set.
 
For example, the maximum element of the [[set]] <math>S_1 = \{0, e, \pi, 4\}</math> of [[real number]]s is <math>4</math>, since it is larger than every other element of the set.
  
Every finite set has a maximum.  However, many infinite sets do not.  The [[integer]]s, <math>\mathbb Z</math> have no maximum, since for any <math>n \in \mathbb Z</math> we can find <math>m \in \mathbb Z</math> such that <math>m > n</math>.  (Taking <math>m = n + 1</math> works nicely.)
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Every [[finite]] [[subset]] of a [[totally ordered set]] such as the reals has a maximum.  However, many [[infinite]] sets do not.  The [[integer]]s, <math>\mathbb Z</math> have no maximum, since for any <math>n \in \mathbb Z</math> we can find <math>m \in \mathbb Z</math> such that <math>m > n</math>.  (Taking <math>m = n + 1</math> works nicely.)
  
A more subtle example of this phenomenon is the set <math>K = \{0, \frac 12, \frac 23, \frac 34, \frac 45, \ldots\} = \{1 - \frac 1n \mid n \in \mathbb{Z}_{> 0}\}</math>.  While this set has a [[least upper bound]] 1, it has no maximum.
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A more subtle example of this phenomenon is the set <math>K = \left\{0, \frac 12, \frac 23, \frac 34, \frac 45, \ldots\right\} = \left\{1 - \frac 1n \mid n \in \mathbb{Z}_{> 0}\right\}</math>.  While this set has a [[least upper bound]] 1, it has no maximum.
  
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The previous example suggests the following formulation: if <math>S</math> is a set contained in some larger ordered set <math>R</math> with the [[least upper bound property]], then <math>S</math> has a maximum if and only if the least upper bound of <math>S</math> is a member of <math>S</math>.
  
The previous example suggests the following formulation: if <math>S</math> is a set contained in some larger ordered set <math>R</math> with the [[least upper bound property]], then <math>S</math> has a maximum if and only if the least upper bound of <math>S</math> is a member of <math>S</math>.
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[[Category:Definition]]
 
[[Category:Definition]]
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Latest revision as of 12:55, 9 February 2007

Given a partially ordered set $S$, the maximum element of $S$, if it exists, is some $M \in S$ such that for all $n \in S$, $n \leq M$.

For example, the maximum element of the set $S_1 = \{0, e, \pi, 4\}$ of real numbers is $4$, since it is larger than every other element of the set.

Every finite subset of a totally ordered set such as the reals has a maximum. However, many infinite sets do not. The integers, $\mathbb Z$ have no maximum, since for any $n \in \mathbb Z$ we can find $m \in \mathbb Z$ such that $m > n$. (Taking $m = n + 1$ works nicely.)

A more subtle example of this phenomenon is the set $K = \left\{0, \frac 12, \frac 23, \frac 34, \frac 45, \ldots\right\} = \left\{1 - \frac 1n \mid n \in \mathbb{Z}_{> 0}\right\}$. While this set has a least upper bound 1, it has no maximum.

The previous example suggests the following formulation: if $S$ is a set contained in some larger ordered set $R$ with the least upper bound property, then $S$ has a maximum if and only if the least upper bound of $S$ is a member of $S$.

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