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\section*{Delvise løsningsforslag til gamle eksamensspørsmål}
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{\large\bf EN}
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\noindent Vi setter opp et bevis (i ND1750) for 
$((P\rightarrow R)\land (Q\rightarrow R))\rightarrow ((P\lor Q)\rightarrow R).$


\begin{equation*}
\begin{fitch}
\fa (P\rightarrow R)\land (Q\rightarrow R) & P \\
\fa  P\rightarrow R & 1, Simp  \\
\fa  Q\rightarrow R & 1, Simp  \\
\fa \fa P\lor Q & P\\
\fa \fa R\lor R &  4,2,3, CD  \\
\fa \fa R & 5, Contr\\
\fa (P\lor Q)\rightarrow R & 4,6, CP\\
((P\rightarrow R)\land (Q\rightarrow R))\rightarrow ((P\lor Q)\rightarrow R) & 1,7, CP
\end{fitch}
\end{equation*}


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\noindent {\large\bf TO}

\noindent
\begin{tabbing}
1. $\ $ \= $\forall x (epost(x)\rightarrow$ \= $\exists y (sendt(y,x)\ \land \forall z (sendt(z,x)\rightarrow y=z)))\ \land$\\
\> \> $\exists y\ mottatt(y,x))$\\ \\
2. \>  $\exists x (epost(x)\land sendt(per,x))\ \land$ \\
\> $\exists x (epost(x)\land mottatt(per,x))\ \land$\\
\> $ \neg\exists x(epost(x)\land sendt(per,x)\land mottatt(per,x))$
\end{tabbing}

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\noindent {\large\bf TRE}
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\noindent Vi setter opp et bevis (i ND1750PRED) for 
$\forall x\exists y (P(x)lor P(y)) \rightarrow \exists x \ P(x)$.

\begin{equation*}
\begin{fitch}
\fa \forall x\exists y (P(x)\lor P(y)) & P \\
\fa \exists y (P(a)\lor P(y)) & 1, UI \\
\fa P(a)\lor P(b) & 2, EI \\
\fa \fa P(a) & P\\
\fa \fa \exists x\ P(x) & 4, EG\\
\fa P(a)\rightarrow \exists x\ P(x) & 4,5, CP \\
\fa \fa P(b) & P\\
\fa \fa \exists x\ P(x) & 7, EG\\
\fa P(b)\rightarrow \exists x\ P(x) & 7,8, CP \\
\fa \exists x\ P(x)\lor \exists x \ P(x) & 3,7,9, CD\\
\fa \exists x \ P(x) & 11, Contr\\
\forall x\exists y (P(x)\lor P(y)) \rightarrow \exists x \ P(x) & 1,12 CP\\
\end{fitch}
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