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%
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%
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\def\PA{{\mathrm{PA}}}

%
% Logical syntax and symbols
%
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\def\limplies{\rightarrow}
\def\tautEq{\vDash\Dashv}
\def\True{{\mathrm T}}
\def\False{{\mathrm F}}

\def\bigand{\bigwedge}
\def\bigor{\bigvee}

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\begin{document}


\begin{center}
{\bf Math 160A - Fall 2021 -
\bf Homework \#1 - Due Wednesday, October 6, 10:00pm} \\
(Hand in by uploading as three problems to Gradescope)
\end{center}

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\noindent
{\bf 1.}
(Distributivity or non-distributivity for $\land$ over $\liff$.) For each
of the following either prove it is true, or give a truth assignment~$\varphi$
which shows it is false.  (To give a proof that it is true, you may either argue
informally, or use truth tables or reduced truth tables.)
\begin{enumerate}
\item $p \land (q \liff r) \vDash (p\land q) \liff (p \land r)$.
\item $(p\land q) \liff (p \land r) \vDash p \land (q \liff r)$.
\end{enumerate}



\noindent{\bf 2.} (Semantics of $\liff$.)
\begin{enumerate}
\item
Show that $\liff$ is associative by proving that
$p \liff (q \liff r) \vDash \Dashv (p \liff q)\liff r$.
\item
Consider a formula
\[
p_1 \liff p_2 \liff p_3 \liff \cdots \liff p_k.
\]
Give a simple characterization of when
$\varphi(p_1 \liff p_2 \liff p_3 \liff \cdots \liff p_k) = \True$.
Your answer should be in terms of the number of~$i$'s such
that $\varphi(p_i)$ is equal to $\True$ or~$\False$.
\end{enumerate}

\noindent{\bf 3.} (Equivalence to a single formula.) Let $\Gamma$ be a set of formulas.
\begin{enumerate}
\item Suppose $\Gamma$ is finite. Prove that there exists a
formula~$A$ such that $\Gamma \vDash A$ and $A\vDash \Gamma$.
(The latter means that for all $B\in \Gamma$, $A\vDash B$.)
\item Give an example of an infinite, satisfiable~$\Gamma$ for which
there does not exist a satisfiable formula~$A$ such that
$A\vDash \Gamma$.
\end{enumerate}

\end{document}
