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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?

documentclassarticle

usepackagetikz
usetikzlibraryshapes,backgrounds
begindocument

deffirstcircle(0,0) circle (1.5cm)
defsecondcircle(60:2cm) circle (1.5cm)
defthirdcircle(0:2cm) circle (1.5cm)
begintikzpicture
beginscope[shift=(6cm,0cm)]
 beginscope[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;


 endscope
 draw firstcircle node $X$;
 draw secondcircle node $Y$;
 draw thirdcircle node $Z$;
 endscope
 endtikzpicture

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.

documentclassarticle

usepackagetikz

usetikzlibraryshapes,backgrounds

begindocument
deffirstcircle(0,0) circle (1.5 cm)
defsecondcircle(0: 2 cm) circle (1.5 cm)
defthirdcircle(60:2 cm) circle (1.5 cm)

begintikzpicture 

%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.

beginscope
 clip firstcircle;
 clip secondcircle;
 draw[fill=white] thirdcircle;
endscope

%Draw the circles.

draw firstcircle;
draw secondcircle;
draw thirdcircle;

endtikzpicture

which produces the diagram

Just for fun!

documentclass[border=10pt,pstricks]standalone
begindocument
beginpspicture(-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)$Z$
endpspicture
enddocument

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
usepackagevenndiagram
makeatletter% https://tex.stackexchange.com/a/499947
tikzsetinterior/.style=insert path=
 (csname @venn@#1xendcsname,csname @venn@#1yendcsname) circle[radius=@venn@radius-pgflinewidth/2],
 contour/.style=insert path=
 (csname @venn@#1xendcsname,csname @venn@#1yendcsname) circle[radius=@venn@radius]
makeatother
begindocument
beginvenndiagram3sets[labelA=$X$,labelB=$Y$,labelC=$Z$]
fillA
setpostvennhook
clip[interior=B];
fill[white,interior=C];
draw[contour=A];
endvenndiagram3sets
enddocument

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
usepackagevenndiagram
usetikzlibrarycalc,backgrounds
makeatletter% https://tex.stackexchange.com/a/499947
tikzsetinterior of/.style=insert path=
 (csname @venn@#1xendcsname,csname @venn@#1yendcsname) circle[radius=@venn@radius-pgflinewidth/2],
 contour of/.style=insert path=
 (csname @venn@#1xendcsname,csname @venn@#1yendcsname)
 circle[radius=@venn@radius],
 midpoint of/.style=insert path=
 (csname @venn@#1xendcsname,csname @venn@#1yendcsname)
 ,
 common of/.style args=#1 and #2insert path=
 let p1=($(csname @venn@#2xendcsname,csname @venn@#2yendcsname)-(csname @venn@#1xendcsname,csname @venn@#1yendcsname)$),
 n1=veclen(y1,x1)/2,n2=sqrt(@venn@radius*@venn@radius-n1*n1),
 n3=atan2(y1,x1),n4=atan2(n2,n1) in 
 ($(csname @venn@#1xendcsname,csname @venn@#1yendcsname)+(n3+n4:@venn@radius)$)
 arc(n3+n4:n3-n4:@venn@radius) arc(n3+180+n4:n3+180-n4:@venn@radius) 
 -- cycle
 %pgfextratypeoutn2,n3,n4,@venn@radius
 ,
 reverse clip/.style=insert path=
 (@venn@w,@venn@h) -,
 even odd clip/.code=% https://tex.stackexchange.com/a/76216
 pgfseteorule
defVennShade@venn@shade
makeatother
begindocument
beginvenndiagram3sets[labelA=$X$,labelB=$Y$,labelC=$Z$]
setpostvennhook
beginscope[on background layer]
 clip[common of=B and C,reverse clip];
 fill[VennShade,interior of=A];
endscope
endvenndiagram3sets
enddocument

Another solution with venndiagram:

documentclass[svgnames, tikz, border=6pt]standalone
usepackage[utf8]inputenc
usepackage[T1]fontenc
usepackagelmodern, amssymb, nccmath

usepackagevenndiagram

begindocument

 beginvenndiagram3sets[radius=1.5cm, overlap=1.2cm]%
 setkeysvennshade=Lavender, labelA=, labelB=
 fillOnlyAfillACapBNotCfillACapCNotB
 setkeysvennshade=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]$medmathAsmallsetminus(Bcap C)$;
 %
 endvenndiagram3sets

enddocument 

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
begindocument
begintikzpicture
defr1.5 defd1
deffirstC(90:d) circle(r)
defsecondC(-30:d) circle(r)
defthirdC(210:d) circle(r)

fill[magenta!30] firstC; 
beginscope
clip firstC;
clip secondC;
fill[white] thirdC;
endscope
draw firstC secondC thirdC;

path
(90:1.2*d) node[magenta,scale=.6]$Asetminus (Bcap C)$ 
(90:2*d) node$A$
(210:2*d) node$B$
(-30:2*d) node$C$;
endtikzpicture
enddocument

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))*"$Asetminus (Bcap C)$",1.2A,magenta);
add(bbox(5mm),Fill(white));