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\noindent
\parbox{2in}{\bf Math 2510} 
\hfill {\Large \bf Test \#1} \hfill
\parbox{2in}{\bf \hfill February $27^{\mathrm{th}}$, 2015}

\vspace{0.3in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein}\\
Don't merely state answers, prove your statements. {\bf Be sure to show your work!}

\begin{description}
   \item[\large 1. (10 points)] Consider theorem L15: $\vdash \neg A \to (A \to B)$
\begin{enumerate}[(a)]

   \item Show L15 is a tautology using by filling out an abbreviated truth table.

\vspace{4in}

   \item Prove L15 in system L. [You may use the deduction theorem and theorems L1 -- L14.]\\ {\it Hint:} I have a quick proof using the deduction theorem and lemmas L5 and L10 in mind.

\end{enumerate}

\newpage
   \item[\large 2. (10 points)] Still in System L\dots

\begin{enumerate}[(a)]
\item Is ``$A \vee B \vdash A$'' provable in $L$? What about ``$A \vdash A \vee B$''? Justify your answer(s).

\vfill

\item Here is my proof of theorem L7:  $A \to (B \to C) \vdash B \to (A \to C)$\dots

\vspace{0.2in}

\begin{enumerate}[1:]
   \item $A \to (B \to C)$ \hfill \underline{1: \hspace*{2.5in}}

\vspace{0.2in}

   \item $B \to (A \to (B \to C))$ \hfill \underline{2: \hspace*{2.5in}}

\vspace{0.2in}

   \item $(A \to (B \to C)) \to ((A \to B) \to (A \to C))$ \hfill \underline{3: \hspace*{2.5in}}

\vspace{0.2in}

   \item $(A \to B) \to (A \to C)$ \hfill \underline{4: \hspace*{2.5in}}

\vspace{0.2in}

   \item $B \to ((A \to B) \to (A \to C))$ \hfill \underline{5: \hspace*{2.5in}}

\vspace{0.2in}

   \item $B \to (A \to B)$ \hfill \underline{6: \hspace*{2.5in}}

\vspace{0.2in}

   \item $B \to (A \to C)$ \hfill \underline{7:  L3 with $A:=$ \hspace*{0.5in},  $B:=$ \hspace{0.5in}, and $C:=$ \hspace{0.5in}}

\vspace{0.2in}

\end{enumerate}

Fill in justifications for each line. I used the usual rules of system L and restricted myself to using lemmas among L1 -- L6.

\end{enumerate}

\newpage
   \item[\large 3. (10 points)] Construct models whose objects are $\mathbb{Z}$ (integers) to show that 
    $$(\forall x \, P(x)) \to (\exists y \, Q(\underline{c},f(y)))$$ is satisfiable but not logically valid. [Note: $P(x)$ and $Q(x,y)$ are predicates, $f(x)$ is a function, and $\underline{c}$ is a constant.]

\newpage
   \item[\large 4. (10 points)] Proofs in K. You may use the deduction theorem and lower numbered theorems.
   
   \begin{enumerate}[(a)]
      \item Prove theorem K7: $\vdash \forall x (A(x) \to B(x)) \to (\forall x A(x) \to \forall x B(x))$
      
      \vspace{4.5in}
      
      \item Prove theorem K21: $\vdash \exists x A(x) \to \exists x (A(x) \vee B(x))$
   \end{enumerate}   

\newpage
   \item[\large 5. (10 points)] How about...more proofs?
   
   \begin{enumerate}[(a)]
      \item Use induction to show that $1+2+\cdots+n = \frac{n(n+1)}{2}$ for all positive integers $n$. 
      
      \vspace{4.5in}
      
      \item Use proof by contradiction to show: For all $x,y \in \mathbb{Z}$, if $xy$ is even, then either $x$ or $y$ is even.  
   \end{enumerate}   

\end{description}

\newpage
    
\noindent
\parbox{1.5in}{\bf Math 2510} 
\hfill {\Large \bf Test \#1 -- TAKE HOME} \hfill
\parbox{1.5in}{\bf \hfill Due: Mar. $2^{\mathrm{nd}}$, 2015}

\vspace{0.3in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein}\\
You may use notes and your textbook, but no help from other people [except myself and Noah] especially your classmates.
 
\vspace{0.2in} 
   
\begin{description}
   \item[\large 6. (25 points)]  Redo the in-class portion of the test.
   
\vspace{0.5in}   

   \item[\large 7. (15 points)]  A few more proofs in K.  Prove (at least) 5 of the theorems K27 -- K37.

\vspace{0.5in}

   \item[\large 8. (15 points)]  More proofs !?!
   \begin{enumerate}[(a)]
      \item Prove that $n! > 2^n$ for all integers $n \geq 4$.

\vspace{0.2in}

      \item Prove that $\sqrt[3]{2}$ is irrational.

\vspace{0.2in}

      \item For any integer $n$, prove that $n^2+2$ is not divisible by 4.
   \end{enumerate}


\end{description}
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