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%
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\begin{document}


\begin{center}
{\bf Math 160A - Fall 2021 -
\bf Homework \#3 - Due Wednesday, October 20, 10:00am (MORNING!)} \\
(Hand in by uploading to Gradescope)
\end{center}

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These are slightly modified versions of Exercises III.14, III.9, III.11, III.12, III.13 and III.21
of draft A.3.c of the PDF text. (The exercise numbers are changing with each draft...)

Your answers may use theorems from the PDF text up through Section III.4, but you may
{\bf not} use the Completeness or Soundness Theorems. For the last question, you may use
the Compactness Theorem.

\noindent
{\bf 1.} Prove that $\vdash ((p\limplies q) \limplies p)\limplies p$.
This tautology is known as Pierce's Law, and is sometimes used
as an axiom related to the Law of the Excluded Middle.
\bigskip

\noindent{\bf 2.} Prove that $\Gamma$ is inconsistent if and only if
$\Gamma \vdash \lnot (p_1 \limplies p_1)$.
\bigskip

\noindent
{\bf 3.}
Show that the following formulas have $\PL$-proofs.
\begin{enumerate}
\item $(A \land B) \limplies A$.
\item $(A \land B) \limplies B$.
\item $A \limplies B \limplies A\land B$.
\end{enumerate}

\noindent{\bf 4.}
Prove that $\Gamma\cup \{A \land B\}$ is inconsistent
if and only if $\Gamma\cup\{A,B\}$ is inconsistent.
[You may use the results from Exercise~3 in your answer.]
\bigskip

\noindent{\bf 5.}
Prove that $\Gamma\cup \{A \lor B\}$ is inconsistent
if and only if $\Gamma\cup\{A\}$ and $\Gamma\cup\{B\}$ are both inconsistent.
\bigskip

\noindent{\bf 6.} Two sets $\Gamma$ and $\Delta$ of formulas are
called ``complementary'' if every truth assignment~$\varphi$,
satisfies exactly one of $\Gamma$ and~$\Delta$.
Suppose $\Gamma$ and $\Delta$ are complementary.
Prove there are subsets $\Gamma'\subseteq \Gamma$
and $\Delta' \subseteq \Delta$ such that $\Gamma'$ and~$\Delta'$
are finite and such that $\Gamma' \vDash \Gamma$ and
$\Delta' \vDash \Delta$.
(The notation $\Gamma' \vDash \Gamma$ means $\Gamma' \vDash A$ for
all $A\in \Gamma$. You will need to use the Compactness Theorem
for this problem.)



\end{document}
