\problemdata       % Takes 5 *mandatory* arguments
  {truth-v-domains}        % latex-friendly label for the prob.
  {logic}                  % The topic(s) of the problem content.
  {}                       % Source (if known)
  {S02 cp2M-1}             % Usage (list ones you are aware of).
  {Nick Matsakis, S02}     % Last revision info (author, date).
\begin{problem}
For each of the logical formulas, indicate whether or not it is true when
the domain of discourse is $\naturals$ (the natural numbers 0, 1, 2,
\dots), $\integers$ (the integers), $\rationals$ (the rationals), $\reals$
(the real numbers), and $\complexes$ (the complex numbers).

\[\begin{array}{c}
\exists x\ (x^2=2)\\
\forall x \exists y\ (x^2 = y)\\
\forall y \exists x\ (x^2 = y)\\
\forall x\neq 0\exists y\ (xy=1)\\
\exists x \exists y\ (x + 2y = 2) \wedge (2 x + 4 y = 5)
\end{array}\]

\solution{\[
\begin{array}{llllll}
Statement & \naturals & \integers & \rationals & \reals & \complexes\\
\exists x\ (x^2=2) & f & f & f & t\ (x=\sqrt{2}) & t\\
\forall x \exists y\ (x^2 = y) & t & t & t & t\ (y=x^2)& t\\
\forall y \exists x\ (x=y^2) & f & f & f & f\ (\text{take }y<0) & t\\
\forall x\neq 0\exists y\ (xy=1) & f & f & t & t\ (y=1/x) & t\\
\exists x \exists y\ (x + 2y = 2) \wedge (2 x + 4 y = 5) & f & f & f & f & f
\end{array}
\]
}
\end{problem}
