TU Wien:Diskrete Mathematik für Informatik VO (Gittenberger)/Schriftliche Prüfung 2014-02-04

Task 1

$a_n=n+n\cdot 5^{n}n\ge 0$

$b_n=\sum _{k=0}^{n-1}{a}_{k}n\ge 1,b_0=0$

Find explicit expressions for $A(x)=\sum _{n\ge 0}{a}_n{x}^n$ and $B(x)=\sum _{n\ge 0}{b}_n{x}^n$.

Task 2

Let $R={\mathbb{Z}}_5[x]/(x^2+3x+1)$.

List all elements of $R$

Prove or disprove that $R$ is a field.

Examine whether $x+3$ is a unit, and if so, calculate its inverse element.

Task 3

Prove or disprove that the following functions are well-defined for all $m\ge 2$.

$f:{\mathbb{Z}}_m\to {\mathbb{Z}}_m,\bar{x}\mapsto \overline{x^2}$

$g:{\mathbb{Z}}_m\to {\mathbb{Z}}_m,\bar{x}\mapsto \overline{2^x}$

$h:{\mathbb{Z}}_m\to {\mathbb{Z}}_d,\bar{x}\mapsto \bar{x}d|m(\text{i.e. }xmodm\mapsto xmodd)$

Task 4

Calculate $\mu (0,1)$ where $\mu$ is the Möbius function for the poset defined by this Hasse diagram:

   1
 / | \
|  |  c
a  b  |
|  |  d
 \ | /
   0