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\renewcommand{\not}{\neg}
\newcommand{\aand}{\wedge}
\newcommand{\oor}{\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}}}


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\begin{document}

{\bf Logic 1, WS 2012.
Homework 2, given Oct 25, due Nov 8}

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1.
Prove the following equivalence by reducing both sides to CNF:
$$(A \aand B) \impl C\ \equiv\ (A \impl C) \oor (B \impl C).$$

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2.
Write equivalences for formulae with $\impl$ and $\lequiv$ containing truth constants.

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3.
Define the truth value of a conjunctive set ($\aand {\cal S}$) and find the truth value when the set is empty.

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4.
Prove by refutation, CNF, and resolution:
$$(A \aand B) \impl C\ \elc\ (A \impl C) \oor (B \impl C).$$


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