File:CartUnion.pdf

Summary

Description
English: Demonstrates the non-compatibility of the Cartesian product and set union, viz. (A∪B)×(C∪D)≠(A×C)∪(B×D), using example sets A = { x∈ℝ: 2≤x≤5 }, B = { x∈ℝ: 3≤x≤7 }, C = { y∈ℝ: 1≤y≤3 }, and D = { y∈ℝ: 2≤y≤4 }.
Date
Source Own work
Author Jochen Burghardt
Other versions File:CartUnion.pdf — File:CartUnion 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{cACf}      {rgb}{0.70,0.00,0.00}   % rectangle AxC foreground
\definecolor{cBDf}      {rgb}{0.00,0.70,0.00}   % rectangle BxD foreground
\definecolor{cACb}      {rgb}{0.99,0.80,0.80}   % rectangle AxC background
\definecolor{cBDb}      {rgb}{0.80,0.99,0.80}   % rectangle BxD background
\definecolor{cINb}      {rgb}{0.99,0.99,0.80}   % intersection background
\definecolor{cUNf}      {rgb}{0.00,0.00,0.70}   % union foreground
\definecolor{cUNb}      {rgb}{0.89,0.89,0.99}   % union background
\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(45,7)(0,1){13}{\makebox(0,0){$\cdot$}}}% A
\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(55,12)(0,1){8}{\makebox(0,0){$\cdot$}}}% B
\textcolor{cEmb}{\multiput(55,26)(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(7,35)(1,0){13}{\makebox(0,0){$\cdot$}}}% C
\textcolor{cEmb}{\multiput(26,35)(1,0){19}{\makebox(0,0){$\cdot$}}}%
\textcolor{cEmb}{\multiput(11,55)(1,0){9}{\makebox(0,0){$\cdot$}}}%
\textcolor{cEmb}{\multiput(26,55)(1,0){19}{\makebox(0,0){$\cdot$}}}%
\textcolor{cEmb}{\multiput(12,45)(1,0){8}{\makebox(0,0){$\cdot$}}}% D
\textcolor{cEmb}{\multiput(26,45)(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){29}{\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{cACf}{\put(45,4){\line(0,1){2}}}% A
\textcolor{cACf}{\put(75,4){\line(0,1){2}}}%
\textcolor{cBDf}{\put(55,9){\line(0,1){2}}}% B
\textcolor{cBDf}{\put(95,9){\line(0,1){2}}}%
\textcolor{cUNf}{\put(45,6.5){\line(0,1){2}}}% A u B
\textcolor{cUNf}{\put(95,6.5){\line(0,1){2}}}%
% y-interval end markers
\textcolor{cACf}{\put(4,35){\line(1,0){2}}}% C
\textcolor{cACf}{\put(4,55){\line(1,0){2}}}%
\textcolor{cBDf}{\put(9,45){\line(1,0){2}}}% D
\textcolor{cBDf}{\put(9,65){\line(1,0){2}}}%
\textcolor{cUNf}{\put(6.5,35){\line(1,0){2}}}% C u D
\textcolor{cUNf}{\put(6.5,65){\line(1,0){2}}}%
\thicklines
% x-interval lines and captions
\textcolor{cACf}{\put(45,5){\line(1,0){30}}}% A
\textcolor{cACf}{\put(46,10){\makebox(0,0)[bl]{$A$}}}%
\textcolor{cBDf}{\put(55,10){\line(1,0){40}}}% B
\textcolor{cBDf}{\put(94,5){\makebox(0,0)[tr]{$B$}}}%
\textcolor{cUNf}{\put(45,7.5){\line(1,0){50}}}% A u B
% y-interval lines and captions
\textcolor{cACf}{\put(5,35){\line(0,1){20}}}% C
\textcolor{cACf}{\put(10,36){\makebox(0,0)[bl]{$C$}}}%
\textcolor{cBDf}{\put(10,45){\line(0,1){20}}}% D
\textcolor{cBDf}{\put(5,64){\makebox(0,0)[tr]{$D$}}}%
\textcolor{cUNf}{\put(7.5,35){\line(0,1){30}}}% C u D
% rectangle backgrounds
\textcolor{cUNb}{\put(45,35){\makebox(0,0)[bl]{\rule{50mm}{30mm}}}}% union
\textcolor{cACb}{\put(45,35){\makebox(0,0)[bl]{\rule{30mm}{20mm}}}}% A x C
\textcolor{cBDb}{\put(55,45){\makebox(0,0)[bl]{\rule{40mm}{20mm}}}}% B x D
\textcolor{cINb}{\put(55,45){\makebox(0,0)[bl]{\rule{20mm}{10mm}}}}% intersection
% rectangle borders and captions
\thinlines
\textcolor{cUNf}{\put(45,35){\framebox(50,30){}}}% union
\textcolor{cACf}{\put(45,35){\framebox(30,20){}}}% A x C
\textcolor{cACf}{\put(46,36){\makebox(0,0)[bl]{$A \times C$}}}%
\textcolor{cBDf}{\put(55,45){\framebox(40,20){}}}% B x D
\textcolor{cBDf}{\put(94,64){\makebox(0,0)[tr]{$B \times D$}}}%
\end{picture}
\end{document}

Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:
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Category:CC-BY-SA-4.0,3.0,2.5,2.0,1.0 Category:Files by User:Jochen Burghardt Category:GFDL Category:Images with LaTeX source code Category:License migration redundant Category:Other images that should use vector graphics Category:PDF that should be converted to SVG Category:Self-published work Category:Set product Category:Union (set theory)