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\pset{1}{February 4, 1996}{Start of Class, Tuesday, February 11}

\begin{problems}

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% 1.1 - 8
\problem

Let $p$, $q$, and $r$ be the following propositions:
\begin{center}
\begin{tabular}{cc}
$p:$ & You are a good water-polo player. \\
$q:$ & You are a good swimmer. \\
$r:$ & You are in the water-polo team. \\
\end{tabular}
\end{center}

Write the following propositions using $p$, $q$, and $r$ and logical
connectives (e.g.  $\vee$, $\wedge$, $\neg$, $\Rightarrow$).

\begin{problemparts}

\problempart 
To be in the water-polo team, it is sufficient for you to be a good water-polo player.

\problempart 
Being a good swimmer and a good water-polo player is necessary
for getting into the water-polo team.

\problempart 
You will get in the water polo team if and only if you 
are a good swimmer or you are a good water-polo player.

\problempart
You are neither a good swimmer nor a good water-polo player, yet you are in the water-polo team.

\end{problemparts}

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% 1.2 - 25 modified 
\problem Find a compound proposition involving the propositions $p$,
$q$, and $r$ that is false when exactly two of $p$, $q$, and $r$ are
false.  The compound proposition is true otherwise. 


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\problem 
% 1.3 # 8

Let $T(x,y)$ be the statement ``$x$ has proven $y$.''
Express each of the following sentences in terms of $T(x,y)$,
quantifiers and logical connectives, where the universe of the
discourse for $x$ is the set of all mathematicians and for
$y$ is the set of all theorems.

\begin{description}
\item[a)] There is a mathematician who has never proven a theorem. 
\item[b)] Every mathematician has proven a theorem.
\item[c)] Every theorem has been proven by a mathematician.
\item[d)] There is a mathematician who has proven the 
{\it Dominated Convergence Theorem} and the {\it Bounded Convergence Theorem}.
\item[e)] At least two mathematicians have proven the {\it Hahn-Banach
    Theorem} (Don't introduce numbers.  You will have to add one more
  standard symbol for this one).

\end{description}



\problem  (Due to
Lewis Carrol).  Define predicates and state the following arguments as
predicate logic formulas.
\begin{problemparts}
\problempart No frogs are poetical; some ducks are unpoetical.  Thus,
some ducks are not frogs.
\problempart Gold is heavy; nothing but gold will silence him.  Thus,
nothing light will silence him.
\problempart All lions are fierce; some lions do not drink coffee.
Thus, some creatures that drink coffee are not fierce.
\problempart Some pillows are soft, no pokers are soft.  Thus, some
pokers are not pillows.
\end{problemparts}


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\problem Prove the correct arguments from the previous problem and
show the incorrect ones are false.

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\problem 
Determine whether each of the following arguments is valid.  If an
argument is correct, what rule of inference (from Rosen, section 3.1)
is being used?  If it is not, what fallacy (from Rosen) occurs?
If an argument is not correct, provide the correct argument.
\begin{problemparts}
\problempart
{\em Theorem:} if $n^2>1$, then $n>1$.  {\em Proof:}
If $n$ is a real number such that $n>1$, then $n^2>1$.  Suppose that
$n^2>1$, then $n>1$.

\problempart
{\em Theorem:} The number $\log_2 3$ is irrational. {\em 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.

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

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


\end{problemparts}

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\problem 
Prove by contradiction: the square root of an irrational number 
is irrational.

\problem

Prove that $\sqrt[3]{5}$ is irrational.


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% 3.1 - 19
\problem 
Prove by cases that if $x$ and $y$ are real numbers, then 
\[ \max(x,y) = { x + y \over 2} + { |x - y| \over 2 } \]


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\problem

Using the axioms and inference rules for equality given in class,
prove that 
\[
(\forall xyz) ((x = y) \wedge (y=z))\Rightarrow (x=z)
\]

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\end{problems}
\end{document}
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