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

Task 1

$\displaystyle {a_{n}=n+n\cdot 5^{n}\quad n\geq 0}$

$\displaystyle {b_{n}=\sum _{k=0}^{n-1}a_{k}\quad n\geq 1,b_{0}=0}$

Find explicit expressions for $\displaystyle {A(x)=\sum _{n\geq 0}a_{n}x^{n}}$ and $\displaystyle {B(x)=\sum _{n\geq 0}b_{n}x^{n}}$.

Task 2

Let $\displaystyle {R=\mathbb {Z} _{5}[x]/(x^{2}+3x+1)}$.

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 $m\geq 2$.

$f:\mathbb {Z} _{m}\rightarrow \mathbb {Z} _{m},{\overline {x}}\mapsto {\overline {x^{2}}}$

$g:\mathbb {Z} _{m}\rightarrow \mathbb {Z} _{m},{\overline {x}}\mapsto {\overline {2^{x}}}$

$h:\mathbb {Z} _{m}\rightarrow \mathbb {Z} _{d},{\overline {x}}\mapsto {\overline {x}}\quad d|m\quad ({\text{i.e. }}x{\bmod {m}}\mapsto x{\bmod {d}})$

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