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\begin{document}


\noindent
{\bf Solutions to HW 6.}

\bigskip

  
\begin{enumerate}
\item[1.] Write the following propositions in
  disjunctive normal form, assuming that each
  proposition is a function of $p$, $q$ and $r$. 
  
  \medskip
  
\noindent
\color{blue}
Note: In DNF, every variable should appear in every monomial,
and the number of monomials should equal the number
of $1$'s in the truth table of the proposition.
\color{black}

  \medskip

  \begin{enumerate}
\item[(i)] $p\to r$
\color{blue}
\[
((\neg p)\wedge (\neg q)\wedge (\neg r)) \vee
((\neg p)\wedge (\neg q)\wedge r) \vee
((\neg p)\wedge q\wedge (\neg r)) \vee
((\neg p)\wedge q\wedge r) \vee
%(p\wedge (\neg q)\wedge (\neg r)) \vee
(p\wedge (\neg q)\wedge r) \vee
%(p\wedge q\wedge (\neg r)) \vee
(p\wedge q\wedge r)
\]
\color{black}
  \medskip

\item[(ii)] $((p\to q)\to ((\neg p)\leftrightarrow r))$.
\color{blue}
\[
%%  ((\neg p)\wedge (\neg q)\wedge (\neg r)) \vee
((\neg p)\wedge (\neg q)\wedge r) \vee
%%((\neg p)\wedge q\wedge (\neg r)) \vee
((\neg p)\wedge q\wedge r) \vee
(p\wedge (\neg q)\wedge (\neg r)) \vee
(p\wedge (\neg q)\wedge r) \vee
(p\wedge q\wedge (\neg r))
%%(p\wedge q\wedge r)
\]
\color{black}
  \medskip

\item[(iii)] $q$
\color{blue}
\[
%((\neg p)\wedge (\neg q)\wedge (\neg r)) \vee
%((\neg p)\wedge (\neg q)\wedge r) \vee
((\neg p)\wedge q\wedge (\neg r)) \vee
((\neg p)\wedge q\wedge r) \vee
%(p\wedge (\neg q)\wedge (\neg r)) \vee
%(p\wedge (\neg q)\wedge r) \vee
(p\wedge q\wedge (\neg r)) \vee
  (p\wedge q\wedge r)
\]
\color{black}
  \medskip

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

  \medskip

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

\color{blue}
\[
(\forall A)(\forall B)((A=B)\leftrightarrow (\forall z)((z\in A)\leftrightarrow (z\in B)))
\]
\color{black}
  \medskip

\item[(ii)] Pairing.

\color{blue}
\[
(\forall A)(\forall B)(\exists P)(\forall z)((z\in P)\leftrightarrow ((z=A)\vee(z=B)))
\]
\color{black}
  \medskip

\item[(iii)] Power set.

\color{blue}
\[
(\forall A)(\exists P)(\forall z)((z\in P)\leftrightarrow \underline{(z\subseteq A)})
\]

or

\medskip

\[
(\forall A)(\exists P)(\forall z)((z\in P)\leftrightarrow \underline{((\forall w)((w\in z)\to (w\in A))}))
\]

\color{black}
  \medskip

\end{enumerate}
\end{enumerate}  

\vfill

\begin{enumerate}
\item[3.]
  In 1959, Pete Seeger took lines from the
  Book of Ecclesiastes to write a song,
  which was made famous by the Byrds in 1965. One line is:
  \medskip
  
  {\it To every thing there is a season,
    and a time to every purpose under heaven.}
  \medskip

  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 $p$''.

\end{enumerate}

\color{blue}
\[((\forall t)(\exists s) S(s,t))\wedge ((\forall p)(\exists T) P(T,p))\]
\color{black}

\vfill  


\end{document}
