TU Wien:Discrete Mathematics VO (Gittenberger)/Written Exam 2022-03-11

1. Let $A,B$ be bases of a matroid. Prove that $|A|=|B|$.

2. Is ${\sqrt {2}}+{\sqrt {3}}$ algebraic over $\mathbb {Q}$, determine its minimal polynomial.

3. Determine $\mu (0,1)$ of the following partial order:

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

4. Let $A,B$ be two finite sets with $|A|=n$ and $|B|=k$.

a) How many injective mappings $f:A\mapsto B$ are there?

b) Show that the number of surjective mappings $f:A\mapsto B$ equals $k!S_{n,k}$