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{\bf Logic 1, WS 2004.
Homework 1, given Oct 07, due Oct 14}

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\noindent
1. Read carefully the first chapter of the script and write a question or a
negative comment related to this chapter.

\bigskip

\noindent
2. In the definition given during the lecture for the language of propositional
logic as a set, express formally the last statement
(``${\cal F}$ is the smallest set which has the above properties'').

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\noindent
3. Write the grammar (in the sense of formal language theory) which generates
the language of propositional logic over the propositional variables $A, B, C.$



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\noindent
4. Prove:

{\bf For any} propositional formulae
$\varphi_1, \varphi_2, \ldots, \varphi_n, \psi,$

{\bf if} 
$ \varphi_1 \and \varphi_2 \and \ldots \and \varphi_n \elc \psi$
{\bf then} 
$(\varphi_1 \and \varphi_2 \and \ldots \and \varphi_n) \impl \psi$
is valid



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\noindent
5. Prove:

{\bf For any} propositional formulae
$\varphi_1, \varphi_2, \ldots, \varphi_n, \psi,$

$ \varphi_1 \and \varphi_2 \and \ldots \and \varphi_n \elc \psi$
{\bf if and only if}
$\varphi_1 \and \varphi_2 \and \ldots \and \varphi_n \and \not \psi$
is inconsistent.

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