\problemdata
{S00 ps1}
{true or false}
{intro}

\begin{problem} Are these statements true or false?  The universe of
discourse is the natural numbers.  
(The set of natural numbers is $\{0,1,2, \ldots\}$.)

%\begin{minipage}[t]{3in}
\begin{problemparts}
\problempart $\forall x \exists y (2x-y=0)$

\solution{
True. (Given a natural number $x$, consider $y = 2x$; $y$ is a natural number.)
}
\problempart $\exists y \forall x (2x-y=0)$
\solution{
False. (Fix any $y$; there is always an $x > y/2$.)
}
\problempart $\forall x (x < 10 \implies \forall y (y < x \implies y < 9))$
\solution{
True.  (Remember the universe of discourse is the natural numbers.)
}
\problempart $\forall x \exists y (y > x \land \exists z (y+z=100))$
\solution{ False. (If $x > 100$ then $y > 100$ so no $z \in \mathbb{N}$
works.)
}
\end{problemparts}
\end{problem}
