\problemdata     
{proofs-2}           
{proofs, logic}              
{}              
{F95 PS-1, F99, S01 PS-1} 		
{S01 Karger}        	


\begin{problem}

Determine whether each of the following arguments is valid.  If an
argument is correct, what rule of inference (from Rosen, \S~3.1) is
being used? If it is not, what fallacy (again from Rosen \S~3.1)
occurs?

\begin{problemparts}
\problempart
{\em Theorem:} if $n^2>1$, then $n>1$. \\
{\em Suggested proof:} If $n$ is a real number such that $n>1$, then $n^2>1$.
Suppose that $n^2>1$, then $n>1$.

\solution{Invalid. Fallacy of affirming the conclusion.}

\problempart
{\em Theorem:} The number $\log_2 3$ is irrational. \\
{\em Suggested proof:} A number is irrational if it is not the ratio of
two integers. Therefore, since $\log_2 3$ cannot be written in the form
$a/b$ where $a$ and $b$ are integers, it is irrational.

\solution{Invalid. Fallacy of affirming the conclusion.}

\problempart
{\em Theorem:} If $n^2 \leq 9$ then, $n \leq 3$. \\
{\em Suggested proof:} Because if $n$ is a real number with $n>3$,
then $n^2 >9$.

\solution{Valid. Modus tollens.}

\problempart
{\em Theorem} If $n\leq 2$, then $n^2\leq 4$.\\
{\em Suggested proof:} Because if $n$ is a real number with $n>2$,
then $n^2>4$.

\solution{Invalid. Fallacy of denying the hypothesis.}

\end{problemparts}      
\end{problem}



