        \problemdata       % Takes 5 *mandatory* arguments
        {subset-zeros}             % latex-friendly label for the prob.
        {proofs}  % The topic(s) of the problem content.
        {}               % Source (if known)
        {F01 PS1-3, S00} % Usage (list ones you are aware of).
        {}        % Last revision info (author, date).

\begin{problem}
\begin{claim*}
If every element of a set, $A$, is nonzero, and $A \subseteq B$, then
every element of $B$ is nonzero.
\end{claim*}

\begin{problemparts}
\problempart Give a counterexample to the Claim.

\solution{Let $A = \mathbb{N^+}$ and $B = \ints$.}

\problempart What is wrong with the following proof of the Claim?  (It is not
sufficient merely to observe that a proof is wrong because its conclusion
is wrong---you have to say where.  A proof could be wrong regardless of
whether its conclusion is correct; some famous historical examples of
wrong ``proofs'' whose conclusions eventually turned out to be right are
mentioned in the Lecture Notes.)

\begin{falseproof}
Let $x$ be an arbitrary element of $A$.  Since $\forall x \in A \; (x \ne
0)$ we can conclude $x \ne 0$.  Also, since $A \subseteq B$ we have $x \in
B$.  Since $x \in B$, $x \ne 0$, and $x$ was arbitrary, we can conclude
that $\forall x \in B \; (x \ne 0)$.
\end{falseproof}

\solution{We selected $x$ arbitrarily from set $A$, but are generalizing
to all $x$ over a different, larger set $B$.}

\end{problemparts}

\end{problem}