File:CartDistr.pdf

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CartDistr.pdf (708 × 531 pixels, file size: 13 KB, MIME type: application/pdf)

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Summary

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Description
English: Demonstrates the distributivity of the Cartesian product over set intersection, viz. A×(B∩C)=(A×B)∩(A×C), using example sets A = { y∈ℝ: 1≤y≤4 }, B = { x∈ℝ: 2≤x≤5 }, and C = { x∈ℝ: 4≤x≤7 }. Moreover, the distributivity of the Cartesian product over set union and set difference, viz. A×(B∪C)=(A×B)∪(A×C) and A×(B\C)=(A×B)\(A×C), respectively, can be seen from the same example.
Date
Source Own work
Author Jochen Burghardt
Other versions File:CartDistr.pdf — File:CartDistr svg.svg
LaTeX source code
\documentclass[12pt]{article}
\usepackage[pdftex]{color}
\usepackage[paperwidth=120mm,paperheight=90mm]{geometry}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{120mm}
\setlength{\textheight}{90mm}
\setlength{\evensidemargin}{-2.7cm}
\setlength{\oddsidemargin}{-2.5cm}
\setlength{\parindent}{0cm}
\setlength{\unitlength}{1mm}
\sloppy

\definecolor{cABf}      {rgb}{0.70,0.00,0.00}   % rectangle AxB foreground
\definecolor{cACf}      {rgb}{0.00,0.70,0.00}   % rectangle AxC foreground
\definecolor{cABb}      {rgb}{0.99,0.80,0.80}   % rectangle AxB background
\definecolor{cACb}      {rgb}{0.80,0.99,0.80}   % rectangle AxC background
\definecolor{cINb}      {rgb}{0.99,0.99,0.80}   % intersection background
\definecolor{cAf}       {rgb}{0.00,0.00,0.70}   % interval A
\definecolor{cAxe}      {rgb}{0.80,0.80,0.99}   % coordinate axes
\definecolor{cEmb}      {rgb}{0.90,0.90,0.90}   % embedding dots

\begin{document}
\begin{picture}(115,85)
% embedding from intervals to rectangles
\textcolor{cEmb}{\multiput(45,10)(0,1){10}{\makebox(0,0){$\cdot$}}}% B
\textcolor{cEmb}{\multiput(45,26)(0,1){9}{\makebox(0,0){$\cdot$}}}%
\textcolor{cEmb}{\multiput(75,11)(0,1){9}{\makebox(0,0){$\cdot$}}}%
\textcolor{cEmb}{\multiput(75,26)(0,1){9}{\makebox(0,0){$\cdot$}}}%
\textcolor{cEmb}{\multiput(65,12)(0,1){8}{\makebox(0,0){$\cdot$}}}% C
\textcolor{cEmb}{\multiput(65,26)(0,1){9}{\makebox(0,0){$\cdot$}}}%
\textcolor{cEmb}{\multiput(95,12)(0,1){8}{\makebox(0,0){$\cdot$}}}%
\textcolor{cEmb}{\multiput(95,26)(0,1){19}{\makebox(0,0){$\cdot$}}}%
\textcolor{cEmb}{\multiput(12,35)(1,0){8}{\makebox(0,0){$\cdot$}}}% A
\textcolor{cEmb}{\multiput(26,35)(1,0){19}{\makebox(0,0){$\cdot$}}}%
\textcolor{cEmb}{\multiput(12,65)(1,0){8}{\makebox(0,0){$\cdot$}}}%
\textcolor{cEmb}{\multiput(26,65)(1,0){19}{\makebox(0,0){$\cdot$}}}%
% y axis
\textcolor{cAxe}{\put(25.000,20.000){\vector(0,1){60.000}}}%
\textcolor{cAxe}{\multiput(25.000,35.000)(0.000,10.000){5}{\line(-1,0){1.000}}}%
\textcolor{cAxe}{\put(27,80){\makebox(0,0)[l]{$y$}}}%
\textcolor{cAxe}{\put(20,35){\makebox(0,0)[l]{$1$}}}%
\textcolor{cAxe}{\put(20,45){\makebox(0,0)[l]{$2$}}}%
\textcolor{cAxe}{\put(20,55){\makebox(0,0)[l]{$3$}}}%
\textcolor{cAxe}{\put(20,65){\makebox(0,0)[l]{$4$}}}%
\textcolor{cAxe}{\put(20,75){\makebox(0,0)[l]{$5$}}}%
% x axis
\textcolor{cAxe}{\put(20.000,25.000){\vector(1,0){90.000}}}%
\textcolor{cAxe}{\multiput(35.000,25.000)(10.000,0.000){8}{\line(0,-1){1.000}}}%
\textcolor{cAxe}{\put(110,27){\makebox(0,0)[b]{$x$}}}%
\textcolor{cAxe}{\put(35,20){\makebox(0,0)[b]{$1$}}}%
\textcolor{cAxe}{\put(45,20){\makebox(0,0)[b]{$2$}}}%
\textcolor{cAxe}{\put(55,20){\makebox(0,0)[b]{$3$}}}%
\textcolor{cAxe}{\put(65,20){\makebox(0,0)[b]{$4$}}}%
\textcolor{cAxe}{\put(75,20){\makebox(0,0)[b]{$5$}}}%
\textcolor{cAxe}{\put(85,20){\makebox(0,0)[b]{$6$}}}%
\textcolor{cAxe}{\put(95,20){\makebox(0,0)[b]{$7$}}}%
\textcolor{cAxe}{\put(105,20){\makebox(0,0)[b]{$8$}}}%
% x-interval end markers
\textcolor{cABf}{\put(45,7){\line(0,1){2}}}% B
\textcolor{cABf}{\put(75,7){\line(0,1){2}}}%
\textcolor{cACf}{\put(65,9){\line(0,1){2}}}% C
\textcolor{cACf}{\put(95,9){\line(0,1){2}}}%
% y-interval end markers
\textcolor{cAf}{\put(9,35){\line(1,0){2}}}% A
\textcolor{cAf}{\put(9,65){\line(1,0){2}}}%
\thicklines
% x-interval lines and captions
\textcolor{cABf}{\put(45,8){\line(1,0){30}}}% B
\textcolor{cABf}{\put(46,10){\makebox(0,0)[bl]{$B$}}}%
\textcolor{cACf}{\put(65,10){\line(1,0){30}}}% C
\textcolor{cACf}{\put(94,8){\makebox(0,0)[tr]{$C$}}}%
% y-interval lines and captions
\textcolor{cAf}{\put(12,36){\makebox(0,0)[bl]{$A$}}}%
\textcolor{cAf}{\put(10,35){\line(0,1){30}}}% A
% rectangle backgrounds
\textcolor{cABb}{\put(45,35){\makebox(0,0)[bl]{\rule{30mm}{30mm}}}}% A x B
\textcolor{cACb}{\put(65,35.2){\makebox(0,0)[bl]{\rule{30mm}{30mm}}}}% A x C
\textcolor{cINb}{\put(65,35){\makebox(0,0)[bl]{\rule{10mm}{30mm}}}}% intersection
% rectangle borders and captions
\thinlines
\textcolor{cABf}{\put(45,35){\framebox(30,30){}}}% A x B
\textcolor{cABf}{\put(46,36){\makebox(0,0)[bl]{$A \times B$}}}%
\textcolor{cACf}{\put(65,35.2){\framebox(30,30){}}}% A x C
\textcolor{cACf}{\put(94,36){\makebox(0,0)[br]{$A \times C$}}}%
\end{picture}
\end{document}

Licensing

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I, the copyright holder of this work, hereby publish it under the following licenses:
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
w:en:Creative Commons
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Date/TimeThumbnailDimensionsUserComment
current15:06, 21 September 2014Thumbnail for version as of 15:06, 21 September 2014708 × 531 (13 KB)Jochen Burghardt (talk | contribs){{Information |Description=Distributivity of Cartesian product and intersection (and union). |Source={{own}} |Date=2014-09-21 |Author=Jochen Burghardt |Permission= |other_versions= }}

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