File:MapsEvenToNat.pdf
Summary
| Description |
English: Shows three mappings (each on an own page) from the set E of even numbers to the set N of natural numbers, with different properties w.r.t. surjectivity and injectivity ( denoting the truncating integer division. Since some surjective and injective mapping from E to N exists, both are of same cardinality by definition. — In contrast, for any finite sets A and B, if a surjective and injective mapping from A to B exists, then (both are of same cardinality again, and hence) each mapping from A to B is surjective iff it is injective. |
| Date | |
| Source | Own work |
| Author | Jochen Burghardt |
LaTeX source code |
|---|
\documentclass[12pt]{article}
\setlength{\unitlength}{1mm}
\usepackage[pdftex]{color}
\usepackage[paperwidth=125mm,paperheight=90mm]{geometry}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{125mm}
\setlength{\textheight}{90mm}
\setlength{\oddsidemargin}{-23mm}
\setlength{\parindent}{0cm}
\pagestyle{empty}
\newcommand{\tdiv}{/\!\!/} % truncating division
\newcommand{\Nat}{I\!\!N} % natural numbers
\newcommand{\Even}{I\!\!E} % even numbers
\begin{document}
% define command for picture parts to be used several times in different
% arrangements
\newcommand{\SiMapping}{%
\color{fSi}%
\put( 8,53){\vector(0,-1){40}}%
\put(20,53){\vector(1,-4){10}}%
\put(30,53){\vector(1,-2){20}}%
\put(40,53){\vector(3,-4){30}}%
\put(50,53){\vector(1,-1){40}}%
}
\newcommand{\sIMapping}{%
\color{fsI}%
\put( 10,53){\vector(0,-1){40}}%
\put( 20,53){\vector(-1,-4){10}}%
\put( 30,53){\vector(-1,-4){10}}%
\put( 40,53){\vector(-1,-2){20}}%
\put( 50,53){\vector(-1,-2){20}}%
\put( 60,53){\vector(-3,-4){30}}%
\put( 70,53){\vector(-3,-4){30}}%
\put( 80,53){\vector(-1,-1){40}}%
\put( 90,53){\vector(-1,-1){40}}%
\put(100,53){\line(-4,-3){40}}%...
%\put(60,23){\vector(-1,-1){10}}%
}
\newcommand{\siMapping}{%
\color{fsi}%
\put( 9,53){\vector(0,-1){40}}%
\put( 20,53){\vector(0,-1){40}}%
\put( 30,53){\vector(0,-1){40}}%
\put( 40,53){\vector(0,-1){40}}%
\put( 50,53){\vector(0,-1){40}}%
\put( 60,53){\vector(0,-1){40}}%
\put( 70,53){\vector(0,-1){40}}%
\put( 80,53){\vector(0,-1){40}}%
\put( 90,53){\vector(0,-1){40}}%
\put(100,53){\vector(0,-1){40}}%
}
\newcommand{\doPicture}[2]{%
\begin{picture}(120,85)%
%\put(0,0){\makebox(0,0){$+$}}%
%\put(120,85){\makebox(0,0){$+$}}%
#1 % lowlighted mapping arros
\color{black}%
\put(117,63){\makebox(0,0)[tr]{$\Even$}}%
\put(117,3){\makebox(0,0)[br]{$\Nat$}}%
\thicklines%
\put(60,56){\oval(120,20)}%
\put(9,56){\makebox(0,0){$ 0$}}%
\put(20,56){\makebox(0,0){$ 2$}}%
\put(30,56){\makebox(0,0){$ 4$}}%
\put(40,56){\makebox(0,0){$ 6$}}%
\put(50,56){\makebox(0,0){$ 8$}}%
\put(60,56){\makebox(0,0){$10$}}%
\put(70,56){\makebox(0,0){$12$}}%
\put(80,56){\makebox(0,0){$14$}}%
\put(90,56){\makebox(0,0){$16$}}%
\put(100,56){\makebox(0,0){$18$}}%
\put(110,56){\makebox(0,0){\ldots}}%
\put(60,10){\oval(120,20)}%
\put(9,10){\makebox(0,0){$ 0$}}%
\put(20,10){\makebox(0,0){$ 1$}}%
\put(30,10){\makebox(0,0){$ 2$}}%
\put(40,10){\makebox(0,0){$ 3$}}%
\put(50,10){\makebox(0,0){$ 4$}}%
\put(60,10){\makebox(0,0){$ 5$}}%
\put(70,10){\makebox(0,0){$ 6$}}%
\put(80,10){\makebox(0,0){$ 7$}}%
\put(90,10){\makebox(0,0){$ 8$}}%
\put(100,10){\makebox(0,0){$ 9$}}%
\put(110,10){\makebox(0,0){\ldots}}%
\put(30,83){\makebox(0,0)[r]{\textcolor{fsi}{surjective,}}}%
\put(60,83){\makebox(0,0)[r]{\textcolor{fsi}{injective:}}}%
\put(65,83){\makebox(0,0)[l]{\textcolor{fsi}{$n \mapsto n \tdiv 2$}}}%
%
\put(30,78){\makebox(0,0)[r]{\textcolor{fSi}{not surjective,}}}%
\put(60,78){\makebox(0,0)[r]{\textcolor{fSi}{injective:}}}%
\put(65,78){\makebox(0,0)[l]{\textcolor{fSi}{$n \mapsto n$}}}%
%
\put(30,73){\makebox(0,0)[r]{\textcolor{fsI}{surjective,}}}%
\put(60,73){\makebox(0,0)[r]{\textcolor{fsI}{not injective:}}}%
\put(65,73){\makebox(0,0)[l]{\textcolor{fsI}{$n \mapsto n \tdiv 4$}}}%
#2 % highlighted mapping arros
\end{picture}
}
% now compose picture parts to generate the actual pages
% foregrounds
\definecolor{fsi} {rgb}{0.99,0.00,0.00} % +surj +inj *****
\definecolor{fSi} {rgb}{0.80,0.99,0.80} % -surj +inj
\definecolor{fsI} {rgb}{0.80,0.80,0.99} % +surj -inj
\doPicture{\SiMapping\sIMapping}{\siMapping}%
\newpage
% foregrounds
\definecolor{fsi} {rgb}{0.99,0.80,0.80} % +surj +inj
\definecolor{fSi} {rgb}{0.00,0.99,0.00} % -surj +inj *****
\definecolor{fsI} {rgb}{0.80,0.80,0.99} % +surj -inj
\doPicture{\siMapping\sIMapping}{\SiMapping}%
\newpage
% foregrounds
\definecolor{fSi} {rgb}{0.80,0.99,0.80} % -surj +inj
\definecolor{fsi} {rgb}{0.99,0.80,0.80} % +surj +inj
\definecolor{fsI} {rgb}{0.00,0.00,0.99} % +surj -inj *****
\doPicture{\SiMapping\siMapping}{\sIMapping}%
\end{document}
|
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.