| Version current |
Version 13 |
| \PMlinkescapeword{between} |
\PMlinkescapeword{between} |
| \PMlinkescapeword{finite} |
\PMlinkescapeword{finite} |
| \PMlinkescapeword{equivalent} |
\PMlinkescapeword{equivalent} |
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| A set $S$ is \emph{infinite} if it is not \PMlinkname{finite}{Finite}; that is, there is no $n \in \mathbb{N}$ for which there is a bijection between $n$ and $S$. |
A set $S$ is \emph{infinite} if it is not \PMlinkname{finite}{Finite}; that is, there is no $n \in \mathbb{N}$ for which there is a bijection between $n$ and $S$. |
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| Assuming the \PMlinkname{Axiom of Choice}{AxiomOfChoice} (or the Axiom of Countable Choice), this definition of infinite sets is equivalent to that of \PMlinkname{Dedekind-infinite sets}{DedekindInfinite}. |
Assuming the \PMlinkname{Axiom of Choice}{AxiomOfChoice} (or the Axiom of Countable Choice), this definition of infinite sets is equivalent to that of \PMlinkname{Dedekind-infinite sets}{DedekindInfinite}. |
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| Some examples of finite sets: |
Some examples of finite sets: |
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| \begin{itemize} |
\begin{itemize} |
| \item The empty set: $\{\}$. |
\item The empty set: $\{\}$. |
| \item $\{0, 1\}$ |
\item $\{0, 1\}$ |
| \item $\{1, 2, 3, 4 , 5\}$ |
\item $\{1, 2, 3, 4 , 5\}$ |
| \item $\{1,1.5, e, \pi\}$ |
\item $\{1,1.5, e, \pi\}$ |
| \end{itemize} |
\end{itemize} |
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| Some examples of infinite sets: |
Some examples of infinite sets: |
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| \begin{itemize} |
\begin{itemize} |
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\item $\{1, 2, 3, 4, \ldots\}$.
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\item $\{1, 2, 3, 4, \ldots\}$ (countable)
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\item The primes: $\{2, 3, 5, 7, 11, \ldots\}$.
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\item The primes: $\{2, 3, 5, 7, 11, \ldots\}$ (countable)
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\item The rational numbers: $\mathbb{Q}$.
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\item An interval of the reals: $(0, 1)$ (uncountable)
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\item An interval of the reals: $(0, 1)$.
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\item The rational numbers: $\mathbb{Q}$ (countable)
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| \end{itemize} |
\end{itemize} |
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| The first three examples are countable, but the last is uncountable. |
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