Don R Walsh

Exercise B.1-1

Draw Venn diagrams that illustrate the first of the distributive laws (B.1).

\[A \cap (B \cup C) = ( A \cap B ) \cup (A \cap C)\]

B.1-1 Venn Diagram a

\[=\]

B.1-1 Venn Diagram b

\[=\]

B.1-1 Venn Diagram c

The following LaTeX code was used to generate Venn diagram \(A\):

\documentclass[tikz, margin=3mm]{standalone}

\begin{document}
    \begin{tikzpicture}[
dot/.style = {circle, inner sep=0pt, minimum size=1mm, fill,
              node contents={}}
                        ]
\def\firstcircle{(-1.2,0) coordinate (a) circle (2cm)}
\def\secondcircle{(1.2,0) coordinate (b)  circle (2cm)}
\def\thirdcircle{(0,-2.0) coordinate (c) circle (2cm)}
    \begin{scope}
\fill[gray] \firstcircle;
    \end{scope}
\draw \firstcircle  node[text=black,above] {$A$};
\draw \secondcircle node[text=black,above] {$B$}; 
\draw \thirdcircle node[text=black,below] {$C$};
\node [below=4cm, align=flush center,text width=8cm]
        {
            $A$
        };
    \end{tikzpicture}
  \end{document}

The following LaTeX code was used to generate Venn diagram \(B \cap C\):

\documentclass[tikz, margin=3mm]{standalone}

\begin{document}
    \begin{tikzpicture}[
dot/.style = {circle, inner sep=0pt, minimum size=1mm, fill,
              node contents={}}
                        ]
\def\firstcircle{(-1.2,0) coordinate (a) circle (2cm)}
\def\secondcircle{(1.2,0) coordinate (b)  circle (2cm)}
\def\thirdcircle{(0,-2.0) coordinate (c) circle (2cm)}
    \begin{scope}
\fill[gray] \secondcircle;
\fill[gray] \thirdcircle;
    \end{scope}
\draw \firstcircle  node[text=black,above] {$A$};
\draw \secondcircle node[text=black,above] {$B$}; 
\draw \thirdcircle node[text=black,below] {$C$};
\node [below=4cm, align=flush center,text width=8cm]
        {
            $B \cup C$
        };
    \end{tikzpicture}
  \end{document}

The following LaTeX code was used to generate Venn diagram \(A \cap (B \cup C)\):

\documentclass[tikz, margin=3mm]{standalone}

\begin{document}
    \begin{tikzpicture}[
dot/.style = {circle, inner sep=0pt, minimum size=1mm, fill,
              node contents={}}
                        ]
\def\firstcircle{(-1.2,0) coordinate (a) circle (2cm)}
\def\secondcircle{(1.2,0) coordinate (b)  circle (2cm)}
\def\thirdcircle{(0,-2.0) coordinate (c) circle (2cm)}
     \begin{scope}
    \clip \secondcircle;
    \fill[gray] \firstcircle;
      \end{scope}
      \begin{scope}
    \clip \firstcircle;
    \fill[gray] \thirdcircle;
      \end{scope}
\draw \firstcircle  node[text=black,above] {$A$};
\draw \secondcircle node[text=black,above] {$B$}; 
\draw \thirdcircle node[text=black,below] {$C$};
\node [below=4cm, align=flush center,text width=8cm]
        {
            $A \cap (B \cup C)$
        };
    \end{tikzpicture}
  \end{document}

The following LaTeX code was used to generate Venn diagram \(A \cap B\):

\documentclass[tikz, margin=3mm]{standalone}

\begin{document}
    \begin{tikzpicture}[
dot/.style = {circle, inner sep=0pt, minimum size=1mm, fill,
              node contents={}}
                        ]
\def\firstcircle{(-1.2,0) coordinate (a) circle (2cm)}
\def\secondcircle{(1.2,0) coordinate (b)  circle (2cm)}
\def\thirdcircle{(0,-2.0) coordinate (c) circle (2cm)}
     \begin{scope}
    \clip \secondcircle;
    \fill[gray] \firstcircle;
      \end{scope}
\draw \firstcircle  node[text=black,above] {$A$};
\draw \secondcircle node[text=black,above] {$B$}; 
\draw \thirdcircle node[text=black,below] {$C$};
\node [below=4cm, align=flush center,text width=8cm]
        {
            $A \cap B$
        };
    \end{tikzpicture}
  \end{document}

The following LaTeX code was used to generate Venn diagram \(A \cap C\):

\documentclass[tikz, margin=3mm]{standalone}

\begin{document}
    \begin{tikzpicture}[
dot/.style = {circle, inner sep=0pt, minimum size=1mm, fill,
              node contents={}}
                        ]
\def\firstcircle{(-1.2,0) coordinate (a) circle (2cm)}
\def\secondcircle{(1.2,0) coordinate (b)  circle (2cm)}
\def\thirdcircle{(0,-2.0) coordinate (c) circle (2cm)}
     \begin{scope}
    \clip \thirdcircle;
    \fill[gray] \firstcircle;
      \end{scope}
\draw \firstcircle  node[text=black,above] {$A$};
\draw \secondcircle node[text=black,above] {$B$}; 
\draw \thirdcircle node[text=black,below] {$C$};
\node [below=4cm, align=flush center,text width=8cm]
        {
            $A \cap C$
        };
    \end{tikzpicture}
  \end{document}