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Getting back to my usually questions, I am looking to have fun with this question and pursue ideas how to write a better program for the tikz diagram(Venn Diagram) below. Marmot told me not to nest tikzpictures so this is one solution to do so. I am interested in other ideas. Modify as you wish!

documentclass{article}

%usepackage{geometry}

usepackage{tikz}



topmargin=-0.45in

evensidemargin=0in

oddsidemargin=0in

textwidth=6.5in

textheight=9.0in

headsep=0.25in



linespread{1.1}



newcommand{A}{mathbf{A}}

newcommand{B}{mathbf{B}}

newcommand{C}{mathbf{C}}

newcommand{x}{mathbf{x}}



begin{document}



defthirdcircle{(0,0) ellipse (1.5cm and .5cm)}     %%%%%% A

defsecondcircle{(0,0) ellipse (2.25cm and .75cm)}  %%%%%% B

deffirstcircle{(0,0) ellipse (3cm and 1cm)}        %%%%%% C

defExtracircle{(2,0) ellipse (1.1cm and 2cm)}      %%%%%% Extra Set 

                                                        C in part(c)



colorlet{circle edge}{blue!50}

colorlet{circle area}{blue!20}



tikzset{filled/.style={fill=circle area, draw=circle edge, thick},

 outline/.style={draw=circle edge, thick}}





begin{center}

begin{minipage}{.45textwidth}

begin{tikzpicture}

  begin{scope} 

        clip Extracircle;

            fill[filled] thirdcircle;

        end{scope} 

 draw[outline] firstcircle node[right=2.25cm] {$A$};

     draw[outline] thirdcircle node  {$B$};

   draw[fill] (0:1) circle (1pt) node[right=1pt] {scriptsize $x$};   

 draw[outline] Extracircle node[above=1cm] {$C$};

    node at (0,-2) [below] {$mathbf{B cap C}$};

    node at (4,-2) [below] {$mathbf{subset}$};

 end{tikzpicture}

end{minipage}

begin{minipage}{.45textwidth}

begin{tikzpicture}

begin{scope} 

        clip Extracircle;

            fill[filled] firstcircle;

        end{scope} 

 draw[outline] firstcircle node[right=2.25cm] {$A$};

     draw[outline] thirdcircle node  {$B$};

   draw[fill] (0:1) circle (1pt) node[right=1pt] {scriptsize $x$};   

 draw[outline] Extracircle node[above=1cm] {$C$};

node at (0,-2) [below] {$mathbf{A cap C}$};

end{tikzpicture}

end{minipage}

end{center}



end{document}

This outputs:

Let me start by saying that if you are fine with the outcome and you do not have any problems with it, you may just leave it as is and not try to "improve" it. Since you ask here, here come a few suggestions (focusing on the TikZ part and ignoring the page geometry issues, which can be addressed more elegantly by really using the geometry package.)

This is some code in which some of the things got changed.

1. Instead of minipages, here the two parts are put in scopes (with local bounding boxes). This can help when it comes to adjusting the distance, and in particular now the figure is really centered. In your example, the minipages but not necessarily the figure was centered. (I personally would probably use centering here.)


2. Instead of storing the various ellipses in macros, they get stored in TikZ styles. (This is not necessarily "better" but may be considered TikZier. ;-)


3. I also added some application of the even odd rule, which got mentioned by Dave in the mean time.

documentclass{article}

%usepackage{geometry}

usepackage{tikz}

usetikzlibrary{patterns}

topmargin=-0.45in

evensidemargin=0in

oddsidemargin=0in

textwidth=6.5in

textheight=9.0in

headsep=0.25in



linespread{1.1}



newcommand{A}{mathbf{A}}

newcommand{B}{mathbf{B}}

newcommand{C}{mathbf{C}}

newcommand{x}{mathbf{x}}



begin{document}



C in part(c)



colorlet{circle edge}{blue!50}

colorlet{circle area}{blue!20}



tikzset{filled/.style={fill=circle area, draw=circle edge, thick},

 outline/.style={draw=circle edge, thick}}





begin{center}

begin{tikzpicture}[standard ellipse/.style={insert path={(0,0) ellipse (#1*1.5cm and #1*.5cm)}},

standard ellipse/.default=1,

extra circle/.style={insert path={(2,0) ellipse (1.1cm and 2cm)}}]

 begin{scope}[local bounding box=left]

  coordinate (O) at (0,0);

  begin{scope} 

   clip[extra circle];

   fill[filled,standard ellipse];

  end{scope} 

  draw[outline,standard ellipse=2] node[right=2.25cm] {$A$};

  draw[outline,standard ellipse] node  {$B$};

  draw[fill] (0:1) circle (1pt) node[right=1pt,font=scriptsize] {$x$};   

  draw[outline,extra circle] node[above=1cm] {$C$};

  node at (0,-2) [below] {$mathbf{B cap C}$};

  node at (4,-2) [below] {$mathbf{subset}$};

 end{scope}

 begin{scope}[local bounding box=right,shift={([xshift=3cm]left.east|-O)}]

  begin{scope} 

   clip[extra circle];

   fill[filled,standard ellipse=2];

   path[pattern=north east lines,even odd rule,standard ellipse=2,standard

   ellipse=1];

  end{scope} 

  draw[outline,standard ellipse=2] node[right=2.25cm] {$A$};

  draw[outline,standard ellipse] node  {$B$};

  draw[fill] (0:1) circle (1pt) node[right=1pt,font=scriptsize] {$x$};   

  draw[outline,extra circle] node[above=1cm] {$C$};

  node at (0,-2) [below] {$mathbf{A cap C}$};

 end{scope} 

end{tikzpicture}

end{center}

end{document}