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\newcommand{\lequiv}{\Leftrightarrow} %equivalence as logical connective
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\newcommand{\elc}{\models}  % semantic logical consequence
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{\bf Logic 1, WS 2017.
Homework 4, given Nov. 16, due Nov 23.}

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\noindent
1.
Prove by definition:
$$ \not \exi_x \varphi\ \equiv\ \all \not \varphi.$$

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\noindent
2. 
Find a counterexample for the (false) equivalence:
$$ \exi_x (\varphi\ \and \psi)\ \equiv\ (\exi_x \varphi) \and (\exi_x \psi).$$


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\noindent
3.
Transform the following formula into prenex normal form:
$$ \all_x \all_y ((\exi_z P[x, y, z]) \and ((\exi_u Q[x, u]) \impl (\exi_v Q[y, v]))).$$

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