# File:CartDistr.pdf

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## Summary

 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 21 September 2014 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

I, the copyright holder of this work, hereby publish it under the following licenses: 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.   This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported 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.

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current15:06, 21 September 2014 708 × 531 (13 KB)Jochen Burghardt{{Information |Description=Distributivity of Cartesian product and intersection (and union). |Source={{own}} |Date=2014-09-21 |Author=Jochen Burghardt |Permission= |other_versions= }}