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\renewcommand{\not}{\neg}
\newcommand{\aand}{\wedge}
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\newcommand{\lequiv}{\Leftrightarrow} %equivalence as logical connective
\newcommand{\all}{\forall}
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\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 2015.
Homework 2, given Oct 20, due Nov 29}

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

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2.
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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3.
Prove $P,\ \not Q \impl \not P\ \ \elc\ \ Q$ by
constructing the truth table of the associated implication.

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4. Write the equivalences of implication over one constant and one variable.

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5. Define the syntax and the semantics of disjunction over sets. Study the special cases when the set is empty and when the set contains only one element.

(See the style used in the lecture for studying the conjunction.)

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6. Express the distributivity of disjunction over conjunction when they are
applied to sets. Study the special cases when sets are empty or contaion only
one element.

(See the style used in the lecture for studying the distributivity of conjunction over disjunction.)

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