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\klausur{Mathematik für Informatiker 1}{Prof. M. Mendler}{Wintersemester 11/12}{90}{Wörterbuch, Taschenrechner ohne vollständige alphanumerische Tastatur oder Grafikdisplay}

\section{} 
Formalise the following statements in FOL.
\begin{enumerate}   
\item[(a)] In every country in which the Gross National Product decreases there is an economic 		recession (Jedes Land, in dem das Bruttosozialprodukt schrumpft, hat eine Rezession)
\item[(b)] There exists a country in which the GNP neither in- nor decreases (Es gibt ein Land, in dem das BSP weder wächst noch schrumpft)
\end{enumerate}
Give the signature $\Sigma = \{S, OP, REL\}$ and a $\Sigma$-sorted family of variables X;
translate (a) and (b) (Either English or German version, not both!) into FOL-Formulas from Form$_{\Sigma}$(X). Explain intended meaning of the chosen S, OP, REL \\
Note: You do not need to prove anything!
\section{}
Assume that 'has\_seen(p,m)' and 'Knows(p,c)' are predicates stating that a person p has seen a movie m and knows a concept c, respectively. Further, let you be a particular person, 'the\_matrix' a movie and 'deja\_vu' a concept. Prove the following argument using the Fitch proof calculus. The Fitch rules are given in Appendix I.\\
% fitch formula here
\[
\begin{nd}
\hypo{1}{\forall p(Has\_seen(p, the\_matrix) \rightarrow Knows(p,deja\_vu))}
\hypo{2}{\neg Knows(you, deja\_vu)}
\have[?]{3}{\forall m(Has\_seen(you,m) \rightarrow \neg (m = the\_matrix))}
\end{nd}
\]\\
Informally:
\begin{enumerate}
\item[1.] "'Everybody who has seen 'the\_matrix' knows what a 'deja\_vu' is"'
\item[2.] "'You do not know what a 'deja\_vu' is"'
\item[?.] "'Whatever movie you may have seen, it was not 'the\_matrix'"'
\end{enumerate}
Hint: negation $\neg$(d=the\_matrix) for arbitrary d can be derived from a contradiction between $\neg$Knows(you,deja\_vu) and Knows(you,deja\_vu). To obtain the latter it will be useful to derive Has\_seen(you,the\_matrix). This can be obtained from Has\_seen(you,d) applying the Identity Elimination Rule =Elim from the equation d = the\_matrix.
\section{}
Consider the propositional signature $ \Sigma = \{S,OP,REL\}$ with S = OP = $\emptyset$ and REL = $\{P,Q,R: Prop\}$.

\begin{enumerate}
\item[(a)]  Using a systematic truth table evaluation, or otherwise, show that both propositions

$o_1 = (P \rightarrow R) \vee (Q \rightarrow R)$ and $o_2 = (P \wedge Q) \rightarrow R $

evaluate to the same under all possible truth valuations, i.e., for all $\Sigma$-structures $\mathbf{A}$, [[$o_1$]]$\mathbf{A}$ = [[$o_2$]]$\mathbf{A}$. The semantic (truth) evaluation tables are given in Appendix II.

\item[(b)] Transform the propositional formula

$\neg(\neg((\neg P \vee Q) \wedge R)) \vee R$
	
into full disjunctive normal form (DNF) over $\Sigma$ and indicate which laws (see Appendix II) of Boolean Algebra you are using in each step.

\end{enumerate}
\section{}
Let $P \subset Person \times Person$ be the parent relation on the set of all persons, i.e., we have $(x,y) \in P$
if $x$ is a parent of $y$. What is the relation \\
\\
$R =_{df} P \circ ((P \circ P^T) \setminus \Delta_{Person})$ ? \\
\\
Recall that the operations are difference $R \setminus S$, converse $R^T$ and composition $R \circ S$ of relational algebra.
$\Delta_{Person} = \{(x,x) \mid x \in Person\}$ is the diagonal relation over the base set Person.
What family relationship does $R$ express? Justify your answer!
\section{}
Let the sequence $x_{0},x_{1},x_{2},\ldots$ of natural numbers be defined recursively by the following equations:
\\
\\
$x_{0} = 4$ \\
$x_{1} = 9$ \\
$x_{n} = 3 \cdot x_{n-1} - 2 \cdot x_{n-2}$ for $n \geq 2$\\
\\
Use strong induction (Werteverlaufsinduktion) to prove that for all natural numbers $n \in \mathbf{N}$ we have $X_{n} = 5 \cdot 2^{n} - 1$
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