\problemdata       % Takes 5 *mandatory* arguments
  {logic-newquantifier}    % latex-friendly label for the prob.
  {logic, quantifiers}     % The topic(s) of the problem content.
  {}                       % Source (if known)
  {S02 cp2M-4;}            % Usage (list ones you are aware of).
  {Nick Matsakis, S02}     % Last revision info (author, date).
\begin{problem}
A self-proclaimed ``great logician'' has invented a new quantifier, on par
with $\exists$ (``there exists'') and $\forall$ (``for all'').  The new
quantifier is symbolized by $U$ and read ``there exists a unique''.
The proposition $Ux\, P(x)$ is true iff there is \emph{exactly one}
$x$ in the domain of discourse for which $P(x)$ is true.  The logician has
noted, ``There used to be two quantifiers, but now there are three!  I
have extended the whole field of mathematics by 50\%!''

\begin{problemparts}

\problempart Rewrite the proposition $U x\, P(x)$ using only the
$\exists$ quantifier.

\solution{
\[
\exists x\ (P(x) \wedge \neg (\exists y\ (x \neq y \wedge P(y)))
\]
}

\problempart Rewrite the proposition $U x\, P(x)$ using only the
$\forall$ quantifier.

\solution{
\[
\neg \forall x\ (\neg P(x) \vee \neg \forall y\ (x = y \vee \neg P(y)))
\]
}
\end{problemparts}
\end{problem} 
