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\newcommand{\lequiv}{\Leftrightarrow} %equivalence as logical connective
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%\equiv is already defined for === semantic equivalence
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{\bf Logic 1, WS 2006.
Additional Homework 1, given Oct 12, due Oct 19}

\bigskip

\noindent
1. 
Using the DeMorgan rules for quantified formulae,
prove $\not \all_{C[x]} \phi[x] \equiv \exi_{C[x]} \not \phi[x]$ and its dual,
where $C[x]$ is a condition on $x$ (like e. g. $x$ is in a certain set), and
$\phi[x]$ is a formula containing $x$.

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\noindent
2.
Prove $\not \and \Phi \equiv \or {\overline{\Phi}}$,
where $\Phi$ is a set of formulae and $\overline{\Phi}$ is the set of the
respective negated formulae.



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