TU Wien:Discrete Mathematics VU (Gittenberger)/Exam 2025-12-18

Comment regarding the scope: we were told the content of the last lectures covering Applications was not going to be part of this exam, but it might be included in future exams.

Exercise 1

Is $\mathbb {Z} _{7}[x]/(x^{2}+5)$ a field? Is $x+2$ a unit in it? If so, find its multiplicative inverse.

Exercise 2

Let $E$ be a set, $1\leq k\leq |E|$ an integer, and let $S$ denote the set of all subsets $X\subseteq E$ with cardinality at most 𝑘. Examine whether 𝑀=(𝐸,𝑆) is a matroid.

Exercise 3

1. How many multisets of $\{a,b,c\}$ are there of size 12?
2. How many multisets which are subsets of $M=\{a,a,a,a,b,b,b,b,b,b,c,c,c,c,c\}$ are there of size 12? Answer this question by
   1. Determining the number of multisets which are not subsets of $M$ because they contain too many a's;
   2. Using the principle of inclusion-exclusion.

Exercise 4

Unlabeled combinatorial structures, definition, why and how, we had 2 prominent examples - explain the steps and justify.