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\noindent
\parbox{2.25in}{\bf Math 2510-410}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Jan. $27^\mathrm{th}$, 2015}

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\begin{enumerate}
\item Consider the theorem ``$A, \neg B \vdash_L \neg (A \rightarrow B) $''. 
    \begin{enumerate}[(a)]
    \item Write down the truth table for ``$(A \wedge \neg B ) \rightarrow \neg (A \rightarrow B)$'' to show it is a tautology. 
    \item Suppose that in the middle of my ``proof'' of this proposition, I have a line which says 
              ``$ \neg B \rightarrow (A \rightarrow B) $''. How do I know there is a mistake in my proof?
    \end{enumerate}
\item My brother claims to have shown ``$A \vdash_L A \rightarrow B $'' in proof system $L$.
          Could he possibly have found a proof? Explain why or why not.
\item Consider Theorem L14 (on page 20) which says ``$\vdash_L A \rightarrow ((A \rightarrow B) \rightarrow B)$''
    \begin{enumerate}[(a)]
    \item Show this proposition is a tautology by writing down its truth table (an abbreviated truth table is ok).
    \item Prove this proposition in $L$ using the deduction theorem and modus ponens (no lemmas allowed).
    \item Prove this proposition in $L$ without the deduction theorem (you may use any lower numbered theorems).\\
              {\it Annoying hint:} Using previous lemmas, there is a 2 line proof.
    \end{enumerate}
\item Prove at least 3 of the 7 theorems L19 through L25 (pages 21 and 22) -- skip any one I do in class. 
         You may use the deduction theorem and theorems with lower numbers (including ones you've skipped) in your proofs.
\end{enumerate}

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