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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
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{\bf Logic 1, WS 2016.
Homework 2, given Oct 20, due Oct 27}

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1. Write the boolean functions corresponding to negation, disjunction, implication,
and equivalence.

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2. Show by truth table that 
%   $(P \and Q) \impl R\ \equiv\ (P \impl R) \or (Q \impl R).$ 
   $(P \aand Q) \impl R\ \equiv\ (P \impl R) \oor (Q \impl R).$ 

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\noindent
3.
Prove that
for any propositional formulae
$\varphi_1, \varphi_2, \ldots, \varphi_n, \psi,$
if
$(\varphi_1 \aand \varphi_2 \aand \ldots \aand \varphi_n) \impl \psi$ is valid,
then
$\varphi_1, \varphi_2, \ldots, \varphi_n\ \elc \psi.$

(See the style used in the lecture for proving the opposite implication.)

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