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\lhead{Math 2001 \\ Assignment 6} 
\rhead{Solutions\\ October 23, 2023}

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\begin{enumerate}
\item[1.] Write the following propositions in
  disjunctive normal form, assuming that each
  proposition is a function of $p$, $q$ and $r$. 
  
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  \begin{enumerate}
\item[(i)] $p\to r$

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\item[(ii)] $((p\to q)\to ((\neg p)\leftrightarrow r))$.

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\item[(iii)] $q$

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\end{enumerate}
\end{enumerate}
 
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\begin{enumerate}
\item[2.]
  Write the following axioms of set theory as formal sentences.

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  \begin{enumerate}
\item[(i)] Extensionality.

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\item[(ii)] Pairing.

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\item[(iii)] Power set.

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\end{enumerate}
\end{enumerate}  

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\begin{enumerate}
\item[3.]
 In 1959, Pete Seeger
  wrote a song, Turn! Turn! Turn!, which was made 
famous by the Byrds in 1965. One line is:
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  {\it To every thing there is a season,
    and a time for every purpose under heaven.}
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Write this as a formal sentence using predicates 
$S(s,t)$ = "$s$ is the season for thing $t$" and 
$P(T,p)$ = "$T$ is the time for purpose $p$". 
(Note: this example shows that
the English language is more flexible than
mathematical language 
in the possible ways to order $\forall$ and $\exists$.)
\end{enumerate}


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