It seems to be lots of problems like this in the archive but none seems to match mine.
I have a TikZ figure that depends on \draw. I know that keys off an assumed Cartesian plane and maybe that is the problem with centering -- if so I might be screwed.
Anyway, here is the code:
\documentclass{amsbook}
\usepackage{tikz}
\begin{document}
\def\SetA{(0,0) circle (1.5cm)}
\def\SetB{(1,2) circle (1.5cm)}
\def\SetC{(2,0) circle (1.5cm)}
\begin{center}
\begin{figure}
\begin{tikzpicture}
\draw (-3,-2) rectangle (5,4);
\draw (4.5,3.5) node {$\mathcal{U}$};
\draw (-2,2) node {$A \cap B$};
\draw (4,2) node {$A \cap C$};
\draw \SetA node[below] {$A$};
\draw \SetB node [above] {$B$};
\draw \SetC node [below] {$C$};
\begin{scope}
\clip \SetA;
\fill[red] \SetB;
\end{scope}
\begin{scope}
\clip \SetA;
\clip \SetB;
\fill[green] \SetC;
\end{scope}
\end{tikzpicture}
\caption{Examples of Intersection}
\end{figure}
\end{center}
\end{document}
And here is the output:

