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\begin{document}
{\sf KTH Matematik}
\begin{center} 
 {\bf 5B1118 Diskret Matematik \\
Kontrollskrivning 4 \\
M{\aa}ndagen den 3 May, 2004 \\}
\end{center}

\vspace{.3in}

\begin{itemize}
\item {\it Skrivtid:}{\bf 10:15-11:15.}
\item {\it Till{\aa}tna hj\"alpmedel: Minir\"aknare med sifferdisplay.}
\item {\it Motivering kr\"avs!}
\item {\it F\"or godk\"ant resultat kr\"avs minst 5 po\"ang.}
\end{itemize}

\vspace{.1in}

\noindent\framebox[15cm]{Personnummer:\hspace{12cm}}\\
\framebox[15cm]{Namn:\hspace{12cm}}\\

\vspace{.1in}

\begin{enumerate}
\item (3p) Betrakta f\"oljande funktion:
$$f:\nat\to\nat\;\;\; f(n)=n+(-1)^n$$
\begin{enumerate}
\item \"Ar $f$ injektiv?
\item \"Ar $f$ surjektiv?
\item \"Ar $f$ bijektiv?
\end{enumerate}

\vspace{.2in}

\item(3p) Betrakta f\"oljande m\"angd:
$$F(\real)=\{f:\real\to \real\}$$
och f\"oljande bin\"ara operation:
$$+:F(\real)\times F(\real)\to F(\real)$$
d\"ar $f+g$ defineras som funktionen $(f+g)(x)=f(x)+g(x)$ f\"or varje \\
$(f,g)\in F(\real)\times F(\real)$ och f\"or varje $x\in\real$.

 Visa att $(F(\real), +)$ \"ar en grupp.

\vspace{.2in}

\item(3p) Betrakta f\"oljande permutationer $\sigma,\tau\in{\mathbb S}_8$:
$$\sigma=[35472816],\;\;\; \tau=[12436587]$$
\begin{enumerate}
\item Skriv $\sigma,\tau$ p{\aa} cykelform.
\item Best\"am $\sigma^{-1}$.
\item Best\"am $\tau\circ\sigma^{-1}$
\end{enumerate} 
\end{enumerate}
\vspace{.3in}


betyg:
\begin{tabular}{|c|c|c|c|}\hline
Upp.(1)& Upp. (2)&Upp. (3)&Tot.\\ \hline
&&&\\ \hline
\end{tabular}
\newpage 
KTH, Matematik, 5B1118, KS4.\hspace{1in}Namn: 
\newpage 
KTH, Matematik, 5B1118, KS4.\hspace{1in}Namn: 
\newpage 
KTH, Matematik, 5B1118, KS4.\hspace{1in}Namn:
\end{document}