File:Bijection between natural and integer numbers.pdf
Summary
| Description |
English: Shows the bijective function f: ℤ → ℕ with f(x) = if x≥0 then 2x else -2x-1 fi, its inverse being f-1: ℕ→ℤ with f-1(y) = if y is even then y/2 else -(y+1)/2 fi, thus establishing that the infinite set ℤ of all integer numbers has the same cardinality as its proper subset ℕ of natural numbers. |
| Date | |
| Source | Own work |
| Author | Jochen Burghardt |
| Other versions | File:Bijection between natural and integer numbers svg.svg |
LaTeX source Code |
|---|
\documentclass{article}
\usepackage[pdftex]{color}
\usepackage[paperwidth=165mm,paperheight=30mm]{geometry}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{165mm}
\setlength{\textheight}{30mm}
\setlength{\oddsidemargin}{-27mm}
\setlength{\unitlength}{1mm}
\setlength{\parindent}{0mm}
\sloppy
\definecolor{cNat} {rgb}{0.99,0.00,0.00}
\definecolor{cInt} {rgb}{0.00,0.00,0.99}
\definecolor{cPMap} {rgb}{0.50,0.50,0.50}
\definecolor{cNMap} {rgb}{0.70,0.70,0.70}
\pagestyle{empty}
\begin{document}
\begin{picture}(160,25)
% pos map N-->Z
\textcolor{cPMap}{\put(80,15){\vector(0,-1){10}}}%
\textcolor{cPMap}{\put(90,15){\vector(0,-1){10}}}%
\textcolor{cPMap}{\put(100,15){\vector(0,-1){10}}}%
\textcolor{cPMap}{\put(110,15){\vector(0,-1){10}}}%
\textcolor{cPMap}{\put(120,15){\vector(0,-1){10}}}%
\textcolor{cPMap}{\put(130,15){\vector(0,-1){10}}}%
% pos map Z-->N
\textcolor{cPMap}{\put(80,5){\vector(0,1){10}}}%
\textcolor{cPMap}{\put(90,5){\vector(0,1){10}}}%
\textcolor{cPMap}{\put(100,5){\vector(0,1){10}}}%
\textcolor{cPMap}{\put(110,5){\vector(0,1){10}}}%
\textcolor{cPMap}{\put(120,5){\vector(0,1){10}}}%
\textcolor{cPMap}{\put(130,5){\vector(0,1){10}}}%
% neg map N-->Z
% 1 --> -1
\textcolor{cNMap}{\put(70,16){\vector(0,-1){11}}}%
\textcolor{cNMap}{\put(72,16){\oval(4,2)[tl]}}%
\textcolor{cNMap}{\put(72,17){\line(1,0){11}}}%
\textcolor{cNMap}{\put(83,16){\oval(4,2)[tr]}}%
\textcolor{cNMap}{\put(85,16){\vector(0,-1){1}}}%
% 3 --> -2
\textcolor{cNMap}{\put(60,17){\vector(0,-1){12}}}%
\textcolor{cNMap}{\put(62,17){\oval(4,2)[tl]}}%
\textcolor{cNMap}{\put(62,18){\line(1,0){31}}}%
\textcolor{cNMap}{\put(93,17){\oval(4,2)[tr]}}%
\textcolor{cNMap}{\put(95,17){\vector(0,-1){2}}}%
% 5 --> -3
\textcolor{cNMap}{\put(50,18){\vector(0,-1){13}}}%
\textcolor{cNMap}{\put(52,18){\oval(4,2)[tl]}}%
\textcolor{cNMap}{\put(52,19){\line(1,0){51}}}%
\textcolor{cNMap}{\put(103,18){\oval(4,2)[tr]}}%
\textcolor{cNMap}{\put(105,18){\vector(0,-1){3}}}%
% 7 --> -4
\textcolor{cNMap}{\put(40,19){\vector(0,-1){14}}}%
\textcolor{cNMap}{\put(42,19){\oval(4,2)[tl]}}%
\textcolor{cNMap}{\put(42,20){\line(1,0){71}}}%
\textcolor{cNMap}{\put(113,19){\oval(4,2)[tr]}}%
\textcolor{cNMap}{\put(115,19){\vector(0,-1){4}}}%
% 9 --> -5
\textcolor{cNMap}{\put(30,20){\vector(0,-1){15}}}%
\textcolor{cNMap}{\put(32,20){\oval(4,2)[tl]}}%
\textcolor{cNMap}{\put(32,21){\line(1,0){91}}}%
\textcolor{cNMap}{\put(123,20){\oval(4,2)[tr]}}%
\textcolor{cNMap}{\put(125,20){\vector(0,-1){5}}}%
% 11 --> -6
\textcolor{cNMap}{\put(20,21){\vector(0,-1){16}}}%
\textcolor{cNMap}{\put(22,21){\oval(4,2)[tl]}}%
\textcolor{cNMap}{\put(22,22){\line(1,0){111}}}%
\textcolor{cNMap}{\put(133,21){\oval(4,2)[tr]}}%
\textcolor{cNMap}{\put(135,21){\vector(0,-1){6}}}%
% nat line
\thicklines
\textcolor{cNat}{\put(80,15){\line(1,0){60}}}%
\textcolor{cNat}{\put(141,15){\makebox(0,0)[l]{\bf\ldots}}}%
\textcolor{cNat}{\put(155,15){\makebox(0,0){$I\!\!N$}}}%
% nat points
\textcolor{cNat}{\multiput(80,15)(5,0){12}{\circle*{1}}}%
\textcolor{cNat}{\put(80,14){\makebox(0,0)[tl]{$0$}}}%
\textcolor{cNat}{\put(85,14){\makebox(0,0)[tl]{$1$}}}%
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\textcolor{cNat}{\put(105,14){\makebox(0,0)[tl]{$5$}}}%
\textcolor{cNat}{\put(110,14){\makebox(0,0)[tl]{$6$}}}%
\textcolor{cNat}{\put(115,14){\makebox(0,0)[tl]{$7$}}}%
\textcolor{cNat}{\put(120,14){\makebox(0,0)[tl]{$8$}}}%
\textcolor{cNat}{\put(125,14){\makebox(0,0)[tl]{$9$}}}%
\textcolor{cNat}{\put(130,14){\makebox(0,0)[tl]{$10$}}}%
\textcolor{cNat}{\put(135,14){\makebox(0,0)[tl]{$11$}}}%
% int line
\textcolor{cInt}{\put(9,5){\makebox(0,0)[r]{\bf\ldots}}}%
\textcolor{cInt}{\put(10,5){\line(1,0){130}}}%
\textcolor{cInt}{\put(141,5){\makebox(0,0)[l]{\bf\ldots}}}%
\textcolor{cInt}{\put(155,5){\makebox(0,0){$Z\!\!\!Z$}}}%
% int points
\textcolor{cInt}{\multiput(20,5)(10,0){12}{\circle*{1}}}%
\textcolor{cInt}{\put(20,4){\makebox(0,0)[t]{$-6$}}}%
\textcolor{cInt}{\put(30,4){\makebox(0,0)[t]{$-5$}}}%
\textcolor{cInt}{\put(40,4){\makebox(0,0)[t]{$-4$}}}%
\textcolor{cInt}{\put(50,4){\makebox(0,0)[t]{$-3$}}}%
\textcolor{cInt}{\put(60,4){\makebox(0,0)[t]{$-2$}}}%
\textcolor{cInt}{\put(70,4){\makebox(0,0)[t]{$-1$}}}%
\textcolor{cInt}{\put(80,4){\makebox(0,0)[tl]{$0$}}}%
\textcolor{cInt}{\put(90,4){\makebox(0,0)[tl]{$1$}}}%
\textcolor{cInt}{\put(100,4){\makebox(0,0)[tl]{$2$}}}%
\textcolor{cInt}{\put(110,4){\makebox(0,0)[tl]{$3$}}}%
\textcolor{cInt}{\put(120,4){\makebox(0,0)[tl]{$4$}}}%
\textcolor{cInt}{\put(130,4){\makebox(0,0)[tl]{$5$}}}%
\end{picture}
\end{document}
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