|
|
|
Viewing Version
3
of
'supremum'
|
[ view 'supremum'
|
back to history
]
| Title of object: |
supremum |
| Canonical Name: |
Supremum |
| Type: |
Definition |
| Created on: |
2001-10-18 22:53:56-04 |
| Modified on: |
2001-11-17 01:52:24-05 |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}
|
Content:
The supremum of a set $S$ is the least upper bound of $S$ and is denoted $\sup(S)$.
\medskip
Let $A$ be a set with an order $\leq$, and let $S \subseteq A$. For any $x \in A$, $x$ is an upper bound of $S$ if $y \leq x$ for any $y \in S$. $\sup(S)$ is the least such upper bound; that is, if $b$ is an upper bound of $S$, then $\sup(S) \leq b$.
\medskip
Note that it is not necessarily the case that $\sup(S) \in S$. Suppose $S = (0, 1)$; then $\sup(S) = 1$, but $1 \not\in S$. |
|
|
|
|
|
