Difference between revisions of "Maximum"
I_like_pie (talk | contribs) |
I_like_pie (talk | contribs) |
||
| Line 7: | Line 7: | ||
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>. | ||
| − | + | {{stub}} | |
[[Category:Definition]] | [[Category:Definition]] | ||
| − | |||
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 .
This article is a stub. Help us out by expanding it.
