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| Title of object: |
infinite |
| Canonical Name: |
Infinite |
| Type: |
Definition |
| Created on: |
2001-11-16 00:44:25 |
| Modified on: |
2004-04-23 12:45:15 |
| Classification: |
msc:03-00 |
| Synonyms: |
infinite=infinite set |
Preamble:
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Content:
A set $S$ is infinite if it is not finite; that is, there is no $n \in \mathbb{N}$ for which there is a bijection between $n$ and $S$. Hence an infinite set has a cardinality greater than any natural number:
$$ \vert S\vert \ge \aleph_0 $$
Infinite sets can be divided into countable and uncountable. For countably infinite sets $S$, there is a bijection between $S$ and $\mathbb{N}$. This is not the case for uncountably infinite sets (like the reals and any non-trivial real interval).
Some examples of finite sets:
\begin{itemize}
\item The empty set: $\{\}$.
\item $\{0, 1\}$
\item $\{1, 2, 3, 4 , 5\}$
\item $\{1,1.5, e, \pi\}$
\end{itemize}
Some examples of infinite sets:
\begin{itemize}
\item $\{1, 2, 3, 4, \ldots\}$ (countable)
\item The primes: $\{2, 3, 5, 7, 11, \ldots\}$ (countable)
\item An interval of the reals: $(0, 1)$ (uncountable)
\item The rational numbers: $\mathbb{Q}$ (countable)
\end{itemize} |
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