\documentclass[12pt]{article}
\usepackage{amsfonts}

\voffset=-3.0cm
\hoffset=-2.6cm
\textwidth=17.5cm
\textheight=24cm

\renewcommand{\not}{\neg}
\renewcommand{\and}{\wedge}
\renewcommand{\or}{\vee}
\newcommand{\impl}{\Rightarrow}
\newcommand{\lequiv}{\Leftrightarrow} %equivalence as logical connective
\newcommand{\all}{\forall}
\newcommand{\exi}{\exists}
\newcommand{\elc}{\models}  % semantic logical consequence
\newcommand{\ylc}{\vdash}  % syntactic logical consequence
%\equiv is already defined for === semantic equivalence
\newcommand{\union}{\cup}
\newcommand{\intersect}{\cap}
\newcommand{\true}{{\mathbb{T}}}
\newcommand{\false}{{\mathbb{F}}}


\pagestyle{empty}

\begin{document}

{\bf Logic 1, WS 2007.
Homework 1, given Oct 11, due Oct 18}

\bigskip

\noindent
1.
Prove by definition that for any propositional formulae
$\phi_1, \ldots, \phi_n, \psi$:\\
if
the formula
$\phi_1 \and \ldots \and \phi_n \impl \psi$ is valid,\\
then
$\phi_1, \ldots, \phi_n\ \elc\ \psi$.

\bigskip

\noindent
2. 
Compute the truth value of the formula
$(A \or B) \impl C$
and compare the semantic function with the semantic function of the formula
$(A \impl C) \and (B \impl C)$ (which was computed during the lecture).

\bigskip

\noindent
3. 
Give equivalent formulae for all logical connectives when one of the
operands is $\false$.

\bigskip

\noindent
4.
(optional)
Give a formal proof of the correctness for the resolution inference rule.

\bigskip

\noindent
5.
Find the truth value of $\or \emptyset$ (the empty disjunction).




\end{document}
