\problemdata     
{proofs-4}           
{quantifiers, logic}              
{Rosen 1.3 no.8}              
{S97, S01 PS-1} 		
{S01 Karger}        	

\begin{problem}
Let $T(x,y)$ be the statement ``$x$ has proven $y$.''
Express each of the following sentences in terms of $T(x,y)$,
quantifiers and logical connectives, where the universe of the
discourse for $x$ is the set of all mathematicians and for
$y$ is the set of all theorems.

\begin{description}
\item[a)] There is a mathematician who has never proven a theorem. 
\item[b)] Every mathematician has proven a theorem.
\item[c)] Every theorem has been proven by a mathematician.
\item[d)] There is a mathematician who has proven the 
{\it Dominated Convergence Theorem} and the {\it Bounded Convergence Theorem}.
\item[e)] At least two mathematicians have proven the {\it Hahn-Banach
    Theorem} (Don't introduce numbers.  You will have to add one more
  standard symbol for this one).

\end{description}
\solution{
\begin{description}
\item[a)] $ \exists x \forall y (\neg T(x,y)) $
\item[b)] $ \forall x \exists y T(x,y) $

\item[c)] $ \forall y \exists x T(x,y)$

\item[d)]$ \exists x (T(x, \textrm{Dominated\_Conv.\_Thm}) \wedge
 T(x, \textrm{Bounded\_Conv.\_Thm})) $

\item[e)]  $ \exists x_1, x_2 ((x_1 \neq x_2) \wedge
 T(x_1, \textrm{Hahn-Banach\_Thm}) \wedge
 T(x_2, \textrm{Hahn-Banach\_Thm})) $

\end{description}
}
        
\end{problem}