How to draw a Venn diagram for X - (Y intersect Z)?

Question

I need to draw a Venn diagram for$ X - (Y \cap Z), but I can only figure out how to get $X - (Y \cup) Z)? Can anyone help?

\documentclass{article}

\usepackage{tikz}
\usetikzlibrary{shapes,backgrounds}
\begin{document}

\def\firstcircle{(0,0) circle (1.5cm)}
\def\secondcircle{(60:2cm) circle (1.5cm)}
\def\thirdcircle{(0:2cm) circle (1.5cm)}
\begin{tikzpicture}
\begin{scope}[shift={(6cm,0cm)}]
        \begin{scope}[even odd rule]% first circle without the second
            \clip \thirdcircle (-3, -3) rectangle (3,3);
            \clip \secondcircle (-3, -3) rectangle (3,3);
        \fill[yellow] \firstcircle;


        \end{scope}
        \draw \firstcircle node {$X$};
        \draw \secondcircle node {$Y$};
        \draw \thirdcircle node {$Z$};
    \end{scope}
  \end{tikzpicture}

Answer 1

Here is a different approach. We fill the second and third circles with white, then fill the first circle with yellow. Finally, we color the intersection of the three circles white.

\documentclass{article}

\usepackage{tikz}

\usetikzlibrary{shapes,backgrounds}

\begin{document}
\def\firstcircle{(0,0) circle (1.5 cm)}
\def\secondcircle{(0: 2 cm) circle (1.5 cm)}
\def\thirdcircle{(60:2 cm) circle (1.5 cm)}

\begin{tikzpicture}   

%Define the circles for the Venn diagram for three sets.

\draw[fill=white] \secondcircle;
\draw[fill=white] \thirdcircle;
\draw[fill=yellow] \firstcircle;

% Label the sets.

\node[left] at (0, 0) {\(X\)};
\node[above] at (60: 2 cm) {\(Y\)};
\node[right] at (0: 2 cm) {\(Z\)};    

%Color the intersection of all three circles white.

\begin{scope}
  \clip \firstcircle;
  \clip \secondcircle;
  \draw[fill=white] \thirdcircle;
\end{scope}

%Draw the circles.

\draw \firstcircle;
\draw \secondcircle;
\draw \thirdcircle;

\end{tikzpicture}

which produces the diagram

Answer 2

Just for fun!

\documentclass[border=10pt,pstricks]{standalone}
\begin{document}
\begin{pspicture}(-3,-4)(6,6)
\psclip[linestyle=none]{%
        \pscircle[fillstyle=solid,fillcolor=pink](0,0){3}}
    \psclip{%
        \pscircle[fillstyle=solid,fillcolor=pink](3;60){3}}
        \pscircle[fillstyle=solid,fillcolor=white](3,0){3}
    \endpsclip
\endpsclip
\pscircle(0,0){3} \rput(-1.5,-0.5){$X \setminus (Y \cap Z)$}
\pscircle(3,0){3} \rput(1.5,3.5){$Y$}
\pscircle(3;60){3} \rput(3.5,-0.5){$Z$}
\end{pspicture}
\end{document}

Answer 3

There is a package that specializes on Venn diagrams: venndiagram. One needs to "hack" it a bit since the intersection you are after seems not be in the predefined set, but this is what \setpostvennhook is for. This hack defines the interiors and the contours of the circles. They have the names A, B and C, but their labels can be set to X, Y and Z, say.

\documentclass[border=10pt]{standalone}
\usepackage{venndiagram}
\makeatletter% https://tex.stackexchange.com/a/499947
\tikzset{interior/.style={insert path={
 (\csname @venn@#1x\endcsname,\csname @venn@#1y\endcsname) circle[radius=\@venn@radius-\pgflinewidth/2]}},
 contour/.style={insert path={
 (\csname @venn@#1x\endcsname,\csname @venn@#1y\endcsname) circle[radius=\@venn@radius]}}}
\makeatother
\begin{document}
\begin{venndiagram3sets}[labelA=$X$,labelB=$Y$,labelC=$Z$]
\fillA
\setpostvennhook{
\clip[interior=B];
\fill[white,interior=C];
\draw[contour=A];}
\end{venndiagram3sets}
\end{document}

ADDENDUM: Since this is a repeating theme, I am trying to collect some standard tricks here, and also add some stuff to the venndiagram repertoire.

1. The reverse clip trick allows one to clip on everything outside a given path.
2. The even odd clip trick allows one to apply the even odd rule to clips.
3. I added the styles interior of, which produce a path that contains the interior of either A, B, or C, contour of which allows one to redraw the contour of either A, B, or C, and common of, which allows one to retrieve the intersection of two of the sets. The latter requires the calc library.

Using these tricks, one can avoid overpainting anything in white. We first clip on everything outside the intersection of B and C,

\clip[common of=B and C,reverse clip];

and then fill the interior of A,

\fill[\VennShade,interior of=A];

Complete MWE. (Notice that some care is taken not to add any hard coded distances, the idea being that you produce several diagrams and have common settings.)

\documentclass[border=10pt]{standalone}
\usepackage{venndiagram}
\usetikzlibrary{calc,backgrounds}
\makeatletter% https://tex.stackexchange.com/a/499947
\tikzset{interior of/.style={insert path={
 (\csname @venn@#1x\endcsname,\csname @venn@#1y\endcsname) circle[radius=\@venn@radius-\pgflinewidth/2]}},
 contour of/.style={insert path={
 (\csname @venn@#1x\endcsname,\csname @venn@#1y\endcsname)
 circle[radius=\@venn@radius]}},
 midpoint of/.style={insert path={
 (\csname @venn@#1x\endcsname,\csname @venn@#1y\endcsname)
 }},
 common of/.style args={#1 and #2}{insert path={
 let \p1=($(\csname @venn@#2x\endcsname,\csname @venn@#2y\endcsname)-(\csname @venn@#1x\endcsname,\csname @venn@#1y\endcsname)$),
  \n1={veclen(\y1,\x1)/2},\n2={sqrt(\@venn@radius*\@venn@radius-\n1*\n1)},
  \n3={atan2(\y1,\x1)},\n4={atan2(\n2,\n1)} in 
 ($(\csname @venn@#1x\endcsname,\csname @venn@#1y\endcsname)+(\n3+\n4:\@venn@radius)$)
 arc(\n3+\n4:\n3-\n4:\@venn@radius) arc(\n3+180+\n4:\n3+180-\n4:\@venn@radius) 
 -- cycle
 %\pgfextra{\typeout{\n2,\n3,\n4,\@venn@radius}}
 }},
 reverse clip/.style={insert path={{% https://tex.stackexchange.com/a/127045
  [overlay] (0,0)  -|
  (\@venn@w,\@venn@h) -| cycle
    }}},
  even odd clip/.code={% https://tex.stackexchange.com/a/76216
  \pgfseteorule}}
\def\VennShade{\@venn@shade}
\makeatother
\begin{document}
\begin{venndiagram3sets}[labelA=$X$,labelB=$Y$,labelC=$Z$]
\setpostvennhook{
\begin{scope}[on background layer]
 \clip[common of=B and C,reverse clip];
 \fill[\VennShade,interior of=A];
\end{scope}}
\end{venndiagram3sets}
\end{document}

Answer 4

Another solution with venndiagram:

\documentclass[svgnames, tikz, border=6pt]{standalone}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern, amssymb, nccmath}

\usepackage{venndiagram}

\begin{document}

   \begin{venndiagram3sets}[radius=1.5cm, overlap=1.2cm]%
        \setkeys{venn}{shade=Lavender, labelA=, labelB=}
        \fillOnlyA\fillACapBNotC\fillACapCNotB
        \setkeys{venn}{shade=WhiteSmoke!80!Thistle, labelA=, labelB=}
        \fillBCapC
        \setpostvennhook
        {%
        \draw (labelA) ++(180:1.1cm) node{$A$};
         \draw (labelB) ++(0:1.1cm) node{$B$};
        \draw (labelABC) ++(-20:0.6cm) node[text=MediumVioletRed]{$\medmath{ B ∩ C}$};
        \draw (labelA) ++(-100:1.2cm) node[text=NavyBlue]{$\medmath{A\smallsetminus(B\cap C)}$};
        }%
        \end{venndiagram3sets}

\end{document} 

Answer 5

This is usual clip of TikZ as Taussig did above. Maybe I am a bit difficult when considering plainess and flexibility of the TikZ code.

\documentclass[tikz,border=5mm]{standalone}
\begin{document}
\begin{tikzpicture}
\def\r{1.5} \def\d{1}
\def\firstC{(90:\d) circle(\r)}
\def\secondC{(-30:\d) circle(\r)}
\def\thirdC{(210:\d) circle(\r)}

\fill[magenta!30] \firstC;   
\begin{scope}
\clip \firstC;
\clip \secondC;
\fill[white] \thirdC;
\end{scope}
\draw \firstC \secondC \thirdC;

\path
(90:1.2*\d) node[magenta,scale=.6]{$A\setminus (B\cap C)$} 
(90:2*\d)  node{$A$}
(210:2*\d) node{$B$}
(-30:2*\d) node{$C$};
\end{tikzpicture}
\end{document}

Here are translation to Asymptote code (exactly the same picture).

unitsize(1cm);
real r=1.5,d=1;
pair A=d*dir(90),B=d*dir(210),C=d*dir(-30);
path firstC=circle(A,r);
path secondC=circle(B,r);
path thirdC=circle(C,r);

fill(firstC,.3magenta+.7white);   
picture BcapC; 
fill(BcapC,thirdC,white);   
clip(BcapC,firstC);
clip(BcapC,secondC);
add(BcapC);

draw(firstC^^secondC^^thirdC);

label("$A$",2A);
label("$B$",2B);
label("$C$",2C);
label((scale(.6))*"$A\setminus (B\cap C)$",1.2A,magenta);
add(bbox(5mm),Fill(white));