Difference between revisions of "Maximum"
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− | Given | + | 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.) | + | 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. | + | 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. |
+ | 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 11:55, 9 February 2007
Given a partially ordered set , the maximum element of , if it exists, is some such that for all , .
For example, the maximum element of the set of real numbers is , 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, have no maximum, since for any we can find such that . (Taking works nicely.)
A more subtle example of this phenomenon is the set . While this set has a least upper bound 1, it has no maximum.
The previous example suggests the following formulation: if is a set contained in some larger ordered set with the least upper bound property, then has a maximum if and only if the least upper bound of is a member of .
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