Difference between revisions of "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. | 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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Revision as of 22:00, 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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