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

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

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

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

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

List all elements of ${\displaystyle R}$

Prove or disprove that ${\displaystyle R}$ is a field.

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

Prove or disprove that the following functions are well-defined for all ${\displaystyle m\geq 2}$.

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

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

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

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