Difference between revisions of "Maximum"
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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 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.) | ||
− | 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. |
Revision as of 21:00, 3 November 2006
Given an 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 set 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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