Represent intersection, union, difference of two sets. - TeX

TopAnswers TeX

Top Answer

user 3.14159

You can adjust the `excluded` style in the second interval.

```
\documentclass[12pt]{standalone}
\usepackage{tikz}

\usepackage{amsmath}
\usetikzlibrary{patterns.meta}
\tikzset{pics/interval/.style={code={%
\def\pv##1{\pgfkeysvalueof{/tikz/interval/##1}}%
\tikzset{interval/.cd,#1}%
\iftikzintervalscaled
\pgfmathsetmacro{\myintervalscale}{\pv{width}/(\pv{xmax}-\pv{xmin})/1cm}%
\else
\pgfmathsetmacro{\myintervalscale}{1}%
\fi
\pgfmathsetmacro{\xmid}{\pv{xmax}/2+\pv{xmin}/2}%
\begin{scope}[xscale=\myintervalscale,shift={(-\xmid,0)}]
 \draw[interval/axis] (\pv{xmin},0) -- (\pv{xmax},0);
 \pgfmathtruncatemacro{\itest}{(\pv{xmin}<\pv{imin})}%
 \ifnum\itest=1
  \path[interval/excluded]
   (\pv{xmin},-\pv{h}/2) rectangle (\pv{imin},\pv{h}/2);
  \path  ([xshift=0.2ex/\myintervalscale]\pv{imin},0) node {$\pv{left}$}
  	node[interval/both labels,interval/left label]{$\pv{l}$};
 \fi
 \pgfmathtruncatemacro{\itest}{(\pv{xmax}>\pv{imax})}%
 \ifnum\itest=1
  \path[interval/excluded]
   (\pv{imax},-\pv{h}/2) rectangle ([xshift=-1em/\myintervalscale]\pv{xmax},\pv{h}/2);
  \path  ([xshift=-0.2ex/\myintervalscale]\pv{imax},0) node {$\pv{right}$}
  	node[interval/both labels,interval/right label]{$\pv{r}$};
 \fi
\end{scope}
}},
interval/.cd,axis domain/.code args={#1:#2}{%
\tikzset{interval/.cd,xmin=#1,xmax=#2}%
},interval domain/.code args={#1:#2}{%
\tikzset{interval/.cd,imin=#1,imax=#2}%
},xmin/.initial=-5,xmax/.initial=5,
imin/.initial=0,imax/.initial=1,
left/.initial={(},right/.initial={)},
scaled/.is if=tikzintervalscaled,scaled/.default=true,
width/.initial=8cm,h/.initial=9pt,
axis/.style={thick,-stealth},
excluded/.style={pattern={Lines[angle=45,distance={2pt}]}},
both labels/.style={below,text=blue,text height=1.2em},
left label/.style={},right label/.style={},
l/.initial={\pv{imin}},r/.initial={\pv{imax}}
}
\newif\iftikzintervalscaled
\tikzintervalscaledtrue
\begin{document}
\begin{tikzpicture}
\path (0,0) pic{interval={axis domain=-5:5,interval domain=-2:2}}
  pic{interval={excluded/.style={pattern={Lines[angle=-45,distance={2pt}]}},
  	axis domain=-5:5,interval domain=-3:1}};
\end{tikzpicture}
\end{document}
```

![Screen Shot 2020-09-21 at 9.47.50 PM.png](/image?hash=2d034f0cd3ce20f986e02aa6dba31aa974598410f4419a03c79da550c7cdca2c)

I think if you want to cover all cases, LaTeX may not be the right tool. Once you allow one intersection, you may want to intersect it with another interval, and so on. While it is *possible* to incorporate all of them somehow, the code will become a monster, which only a subset of the LaTeX users will understand, and probably much less users want to extend and maintain.

In this case, intersection is `{-3}`. ``` \begin{tikzpicture} \path (0,0) pic{interval={axis domain=-5:5,%interval domain=-2:2, intersection of={(-5, -3] and [-3,1]}}}; \end{tikzpicture} ```

It is easy to add this case but I am afraid that there are more, and of course there is the problem that the user could enter conflicting information. ``` \documentclass[12pt]{standalone} \usepackage{tikz} \usepackage{amsmath} \usetikzlibrary{patterns.meta} \makeatletter \tikzset{pics/interval/.style={code={% \def\pv##1{\pgfkeysvalueof{/tikz/interval/##1}}% \tikzset{interval/.cd,#1}% \iftikzintervalscaled \pgfmathsetmacro{\myintervalscale}{\pv{width}/(\pv{xmax}-\pv{xmin})/1cm}% \else \pgfmathsetmacro{\myintervalscale}{1}% \fi \pgfmathsetmacro{\xmid}{\pv{xmax}/2+\pv{xmin}/2}% \begin{scope}[xscale=\myintervalscale,shift={(-\xmid,0)}] \draw[interval/axis] (\pv{xmin},0) -- (\pv{xmax},0); \pgfmathtruncatemacro{\itest}{(\pv{xmin}<\pv{imin})}% \ifnum\itest=1 \path[interval/excluded] (\pv{xmin},-\pv{h}/2) rectangle (\pv{imin},\pv{h}/2); \path ([xshift=0.2ex/\myintervalscale]\pv{imin},0) node {$\pv{left}$} node[interval/both labels,interval/left label]{$\pgfmathparse{\pv{l}}\pgfmathprintnumber\pgfmathresult$}; \fi \pgfmathtruncatemacro{\itest}{(\pv{xmax}>\pv{imax})}% \ifnum\itest=1 \path[interval/excluded] (\pv{imax},-\pv{h}/2) rectangle ([xshift=-1em/\myintervalscale]\pv{xmax},\pv{h}/2); \path ([xshift=-0.2ex/\myintervalscale]\pv{imax},0) node {$\pv{right}$} node[interval/both labels,interval/right label]{$\pgfmathparse{\pv{r}}\pgfmathprintnumber\pgfmathresult$}; \fi \pgfmathtruncatemacro{\itest}{(\pv{xmax}<\pv{imax}&&\pv{xmin}>\pv{imin})}% \ifnum\itest=1 \path[interval/excluded] (\pv{xmin},-\pv{h}/2) rectangle (\pv{xmax},\pv{h}/2); \fi \end{scope} }}, interval/.cd,axis domain/.code args={#1:#2}{% \tikzset{interval/.cd,xmin=#1,xmax=#2}% },interval domain/.code args={#1:#2}{% \tikzset{interval/.cd,imin=#1,imax=#2}% },xmin/.initial=-5,xmax/.initial=5, imin/.initial=0,imax/.initial=1, left/.initial={(},right/.initial={)}, scaled/.is if=tikzintervalscaled,scaled/.default=true, width/.initial=8cm,h/.initial=9pt, axis/.style={thick,-stealth}, excluded/.style={pattern={Lines[angle=45,distance={2pt}]}}, both labels/.style={below,text=blue,text height=1.2em}, left label/.style={},right label/.style={}, l/.initial={\pv{imin}},r/.initial={\pv{imax}}, intersection of/.code args={#1 and #2}{% \expanded{\noexpand\pgfutil@in@{[}{#1}}% test if [ is contained in #1 \ifpgfutil@in@% \edef\pgf@util@tmpc{1}% \else \edef\pgf@util@tmpc{0}% \fi \expanded{\noexpand\pgfutil@in@{]}{#1}}% test if [ is contained in #1 \ifpgfutil@in@% \edef\pgf@util@tmpc{\the\numexpr\pgf@util@tmpc+2}% \fi \ifcase\pgf@util@tmpc \def\pgf@util@tmpp(##1,##2);{\pgfmathsetmacro\pgf@util@tmpa{##1}\pgfmathsetmacro\pgf@util@tmpb{##2}}% \expandafter\pgf@util@tmpp#1; \or \def\pgf@util@tmpp[##1,##2);{\pgfmathsetmacro\pgf@util@tmpa{##1}\pgfmathsetmacro\pgf@util@tmpb{##2}}% \expandafter\pgf@util@tmpp#1; \or \def\pgf@util@tmpp(##1,##2];{\pgfmathsetmacro\pgf@util@tmpa{##1}\pgfmathsetmacro\pgf@util@tmpb{##2}}% \expandafter\pgf@util@tmpp#1; \or \def\pgf@util@tmpp[##1,##2];{\pgfmathsetmacro\pgf@util@tmpa{##1}\pgfmathsetmacro\pgf@util@tmpb{##2}}% \expandafter\pgf@util@tmpp#1; \fi % \expanded{\noexpand\pgfutil@in@{[}{#2}}% test if [ is contained in #1 \ifpgfutil@in@% \edef\pgf@util@tmpc{1}% \else \edef\pgf@util@tmpc{0}% \fi \expanded{\noexpand\pgfutil@in@{]}{#2}}% test if [ is contained in #1 \ifpgfutil@in@% \edef\pgf@util@tmpc{\the\numexpr\pgf@util@tmpc+2}% \fi \ifcase\pgf@util@tmpc \def\pgf@util@tmpp(##1,##2);{\pgfmathsetmacro\pgf@util@tmpx{##1}\pgfmathsetmacro\pgf@util@tmpy{##2}}% \expandafter\pgf@util@tmpp#2; \or \def\pgf@util@tmpp[##1,##2);{\pgfmathsetmacro\pgf@util@tmpx{##1}\pgfmathsetmacro\pgf@util@tmpy{##2}}% \expandafter\pgf@util@tmpp#2; \or \def\pgf@util@tmpp(##1,##2];{\pgfmathsetmacro\pgf@util@tmpx{##1}\pgfmathsetmacro\pgf@util@tmpy{##2}}% \expandafter\pgf@util@tmpp#2; \or \def\pgf@util@tmpp[##1,##2];{\pgfmathsetmacro\pgf@util@tmpx{##1}\pgfmathsetmacro\pgf@util@tmpy{##2}}% \expandafter\pgf@util@tmpp#2; \fi \pgfmathtruncatemacro\pgf@util@tmpi{(min(\pgf@util@tmpb,\pgf@util@tmpy)>max(\pgf@util@tmpa,\pgf@util@tmpx)?1:0)}% \ifnum\pgf@util@tmpi=0\relax \typeout{The intersection between #1 and #2 is empty.}% \tikzset{interval/.cd,imin/.evaluated=\pgfkeysvalueof{/tikz/interval/xmin}-0.1, imax/.evaluated=\pgfkeysvalueof{/tikz/interval/xmax}+0.1}% \else \pgfmathsetmacro{\pgf@util@tmpd}{max(\pgf@util@tmpa,\pgf@util@tmpx)}% \pgfmathsetmacro{\pgf@util@tmpe}{min(\pgf@util@tmpb,\pgf@util@tmpy)}% \tikzset{interval/.cd,imin=\pgf@util@tmpd,imax=\pgf@util@tmpe}% \fi}} \newif\iftikzintervalscaled \tikzintervalscaledtrue \makeatother \begin{document} \begin{tikzpicture} \path (0,0) pic{interval={axis domain=-5:5,%interval domain=-2:2, intersection of={(-4, -3) and [-3,1]}}}; \end{tikzpicture} \end{document} ```

Yes, this is what I meant by "loose ends". There are special cases one could tackle one by one. In this case you should get a message saying that the intervals do not intersect, don't you? If you tell me what the output should be in this case I will be happy to change things accordingly. So far the code just falls back to the initial values.

Thank you very much. Please see this case ``` \begin{tikzpicture} \path (0,0) pic{interval={axis domain=-5:5,%interval domain=-2:2, intersection of={(-4, -3) and [-3,1]}}}; \end{tikzpicture} ``` The result is empty set. I got ![ScreenHunter 1071.png](/image?hash=3e49d5448736ceb842eafdcef3c2320f1f4eec80954b896fbf294dbd0c96d698)

There are loose ends and all that, but this may be a proof of principle.: ``` \documentclass[12pt]{standalone} \usepackage{tikz} \usepackage{amsmath} \usetikzlibrary{patterns.meta} \makeatletter \tikzset{pics/interval/.style={code={% \def\pv##1{\pgfkeysvalueof{/tikz/interval/##1}}% \tikzset{interval/.cd,#1}% \iftikzintervalscaled \pgfmathsetmacro{\myintervalscale}{\pv{width}/(\pv{xmax}-\pv{xmin})/1cm}% \else \pgfmathsetmacro{\myintervalscale}{1}% \fi \pgfmathsetmacro{\xmid}{\pv{xmax}/2+\pv{xmin}/2}% \begin{scope}[xscale=\myintervalscale,shift={(-\xmid,0)}] \draw[interval/axis] (\pv{xmin},0) -- (\pv{xmax},0); \pgfmathtruncatemacro{\itest}{(\pv{xmin}<\pv{imin})}% \ifnum\itest=1 \path[interval/excluded] (\pv{xmin},-\pv{h}/2) rectangle (\pv{imin},\pv{h}/2); \path ([xshift=0.2ex/\myintervalscale]\pv{imin},0) node {$\pv{left}$} node[interval/both labels,interval/left label]{$\pgfmathparse{\pv{l}}\pgfmathprintnumber\pgfmathresult$}; \fi \pgfmathtruncatemacro{\itest}{(\pv{xmax}>\pv{imax})}% \ifnum\itest=1 \path[interval/excluded] (\pv{imax},-\pv{h}/2) rectangle ([xshift=-1em/\myintervalscale]\pv{xmax},\pv{h}/2); \path ([xshift=-0.2ex/\myintervalscale]\pv{imax},0) node {$\pv{right}$} node[interval/both labels,interval/right label]{$\pgfmathparse{\pv{r}}\pgfmathprintnumber\pgfmathresult$}; \fi \end{scope} }}, interval/.cd,axis domain/.code args={#1:#2}{% \tikzset{interval/.cd,xmin=#1,xmax=#2}% },interval domain/.code args={#1:#2}{% \tikzset{interval/.cd,imin=#1,imax=#2}% },xmin/.initial=-5,xmax/.initial=5, imin/.initial=0,imax/.initial=1, left/.initial={(},right/.initial={)}, scaled/.is if=tikzintervalscaled,scaled/.default=true, width/.initial=8cm,h/.initial=9pt, axis/.style={thick,-stealth}, excluded/.style={pattern={Lines[angle=45,distance={2pt}]}}, both labels/.style={below,text=blue,text height=1.2em}, left label/.style={},right label/.style={}, l/.initial={\pv{imin}},r/.initial={\pv{imax}}, intersection of/.code args={#1 and #2}{% \expanded{\noexpand\pgfutil@in@{[}{#1}}% test if [ is contained in #1 \ifpgfutil@in@% \edef\pgf@util@tmpc{1}% \else \edef\pgf@util@tmpc{0}% \fi \expanded{\noexpand\pgfutil@in@{]}{#1}}% test if [ is contained in #1 \ifpgfutil@in@% \edef\pgf@util@tmpc{\the\numexpr\pgf@util@tmpc+2}% \fi \ifcase\pgf@util@tmpc \def\pgf@util@tmpp(##1,##2);{\pgfmathsetmacro\pgf@util@tmpa{##1}\pgfmathsetmacro\pgf@util@tmpb{##2}}% \expandafter\pgf@util@tmpp#1; \or \def\pgf@util@tmpp[##1,##2);{\pgfmathsetmacro\pgf@util@tmpa{##1}\pgfmathsetmacro\pgf@util@tmpb{##2}}% \expandafter\pgf@util@tmpp#1; \or \def\pgf@util@tmpp(##1,##2];{\pgfmathsetmacro\pgf@util@tmpa{##1}\pgfmathsetmacro\pgf@util@tmpb{##2}}% \expandafter\pgf@util@tmpp#1; \or \def\pgf@util@tmpp[##1,##2];{\pgfmathsetmacro\pgf@util@tmpa{##1}\pgfmathsetmacro\pgf@util@tmpb{##2}}% \expandafter\pgf@util@tmpp#1; \fi % \expanded{\noexpand\pgfutil@in@{[}{#2}}% test if [ is contained in #1 \ifpgfutil@in@% \edef\pgf@util@tmpc{1}% \else \edef\pgf@util@tmpc{0}% \fi \expanded{\noexpand\pgfutil@in@{]}{#2}}% test if [ is contained in #1 \ifpgfutil@in@% \edef\pgf@util@tmpc{\the\numexpr\pgf@util@tmpc+2}% \fi \ifcase\pgf@util@tmpc \def\pgf@util@tmpp(##1,##2);{\pgfmathsetmacro\pgf@util@tmpx{##1}\pgfmathsetmacro\pgf@util@tmpy{##2}}% \expandafter\pgf@util@tmpp#2; \or \def\pgf@util@tmpp[##1,##2);{\pgfmathsetmacro\pgf@util@tmpx{##1}\pgfmathsetmacro\pgf@util@tmpy{##2}}% \expandafter\pgf@util@tmpp#2; \or \def\pgf@util@tmpp(##1,##2];{\pgfmathsetmacro\pgf@util@tmpx{##1}\pgfmathsetmacro\pgf@util@tmpy{##2}}% \expandafter\pgf@util@tmpp#2; \or \def\pgf@util@tmpp[##1,##2];{\pgfmathsetmacro\pgf@util@tmpx{##1}\pgfmathsetmacro\pgf@util@tmpy{##2}}% \expandafter\pgf@util@tmpp#2; \fi \pgfmathtruncatemacro\pgf@util@tmpi{(min(\pgf@util@tmpb,\pgf@util@tmpy)>max(\pgf@util@tmpa,\pgf@util@tmpx)?1:0)}% \ifnum\pgf@util@tmpi=0\relax \typeout{The intersection between #1 and #2 is empty.}% \else \pgfmathsetmacro{\pgf@util@tmpd}{max(\pgf@util@tmpa,\pgf@util@tmpx)}% \pgfmathsetmacro{\pgf@util@tmpe}{min(\pgf@util@tmpb,\pgf@util@tmpy)}% \tikzset{interval/.cd,imin=\pgf@util@tmpd,imax=\pgf@util@tmpe}% \fi}} \newif\iftikzintervalscaled \tikzintervalscaledtrue \makeatother \begin{document} \begin{tikzpicture} \path (0,0) pic{interval={axis domain=-5:5,%interval domain=-2:2, intersection of={(-2, 2) and [-3,1]}}}; \end{tikzpicture} \end{document} ```

1 Answer