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\begin{document}


\begin{center}
{\bf Math 160A - Fall 2021 -
\bf Homework 4 - Due Wednesday, October 27, 10:00pm} \\
(Hand in by uploading to Gradescope)
\end{center}

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\noindent
{\bf 1.} Suppose that
$\Gamma \vDash p_i$ or $\Gamma \vDash \lnot p_i$, for every~$i$.
Prove that,
for every formula~$A$, $\Gamma \vDash A$ or $\Gamma \vDash \lnot A$.
(This property of~$\Gamma$ is similar to being complete; however, instead of having
one of $A$ or~$\lnot A$ a member of~$\Gamma$, we have one of
$A$ or~$\lnot A$ tautologically implied by~$\Gamma$.)

\bigskip

\noindent{\bf 2.} Use the Compactness Theorem for propositional logic to
prove that a graph is 3-colorable if and only if every finite
subgraph is 3-colorable. (``3-colorable'' means there is an assignment of three
colors to the vertices of the graph so that no edge connects vertices assigned the
same color.) For this, fix a graph~$G$.
Use propositional variables
$r_i, g_i, b_i$ whose intended meanings are that
``Vertex~$i$ is red'', ``Vertex~$i$ is green'', and ``Vertex~$i$ is blue'', respectively.
Let $\Gamma$ be a set of formulas using these variables that expresses the
conditions that (a)~each vertex has a color assigned to it, and
(b)~if two vectices $i$ and~$j$ are joined by an edge in~$G$, then they
are not assigned the same color. The set~$\Gamma$ should
be satisfiable if and only if $G$ is 3-colorable.
Then apply the Compactness Theorem.

This is mostly a conceptual problem.
Feel free to discuss this on piazza and discord.
What to hand-in to be graded:
Describe what formulas are in the set~$\Gamma$ in terms of the
vertices and edges of~$G$.


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