File:Diagonal argument 01.pdf
Summary
| Description |
English: An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above. (Adapted from File:Diagonal argument 2.svg; see File:Diagonal argument powerset.pdf for a similar version for power sets.) |
| Date | |
| Source | Own work |
| Author | Jochen Burghardt |
| Other versions | File:Diagonal argument 01 svg.svg |
LaTeX source Code |
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\documentclass{article}
\usepackage[pdftex]{color}
\usepackage[paperwidth=50mm,paperheight=65mm]{geometry}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{50mm}
\setlength{\textheight}{65mm}
\setlength{\oddsidemargin}{-23mm}
\setlength{\parindent}{0mm}
\definecolor{cPos} {rgb}{0.99,0.00,0.00}
\definecolor{cNeg} {rgb}{0.00,0.00,0.99}
\pagestyle{empty}
\newcommand{\0}[1]{\textcolor{cPos}{#1}}
\newcommand{\1}[1]{\textcolor{cNeg}{#1}}
\begin{document}
$\begin{array}{|*{14}{@{\;}c}|}
\hline
s_1 & = & \00 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \ldots \\ % periodic 0
s_2 & = & 1 & \01 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \ldots \\ % periodic 1
s_3 & = & 0 & 1 & \00 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & \ldots \\ % periodic 0,1
s_4 & = & 1 & 0 & 1 & \00 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & \ldots \\ % periodic 1,0
s_5 & = & 1 & 1 & 0 & 1 & \00 & 1 & 1 & 0 & 1 & 0 & 1 & \ldots \\ % periodic 1,1,0,1,0
s_6 & = & 0 & 0 & 1 & 1 & 0 & \01 & 1 & 0 & 1 & 1 & 0 & \ldots \\ % 0, periodic 0,1,1
s_7 & = & 1 & 0 & 0 & 0 & 1 & 0 & \00 & 0 & 1 & 0 & 0 & \ldots \\ % periodic 1,0,0,0
s_8 & = & 0 & 0 & 1 & 1 & 0 & 0 & 1 & \01 & 0 & 0 & 1 & \ldots \\ % periodic 0,0,1,1
s_9 & = & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & \01 & 1 & 0 & \ldots \\ % periodic 1,1,0,0
s_{10} & = & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & \00 & 1 & \ldots \\ % binary numbers 1, 10, 11, 100, 101, ...)
s_{11} & = & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & \00 & \ldots \\ % 1 whenever new number starts in s_{10}
\vdots & & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\
\hline
\multicolumn{10}{c}{} \\
\hline
s & = & \11 & \10 & \11 & \11 & \11 & \10 & \11 & \10 & \10 & \11 & \11 & \ldots \\
\hline
\end{array}$
\end{document}
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