Let $\mathcal{M}$ be a matroid of rank $k$ on the set $X$. Recall that a set $A \subset X$ is independent if it is a subset of some basis of $\mathcal{M}$.
(a) Let $S \subset X$ be a nonempty subset. Show that the maximal (under inclusion) independent subsets of $S$ all have the same cardinality.
(b) Show that these maximal independent subsets form a matroid $\left.\mathcal{M}\right|_S$ on $S$.

A poset $P$ is graded (in the sense of the previous pset) if and only if we can assign an integer $\rho(x)$, called the rank, to each $x \in P$ so that if $x \lessdot y$ then $\rho(y)=\rho(x)+1$. (You may assume this.)

Let $P$ be a finite graded poset with a $\hat{0}$ and $\hat{1}$. We say that $P$ is Eulerian if each interval $[s, t]$ where $s<t$ has the same number of elements with odd rank as elements with even rank.
(a) What do intervals of length 2 (that is, $[s, t]$ where $\rho(t)=\rho(s)+2$ ) in Eulerian posets look like?
(b) Verify that the Boolean algebra $B_n$ is Eulerian.
(c) Prove that a poset is Eulerian if and only if the Mobius function is given by $\mu(s, t)=(-1)^{\rho(t)-\rho(s)}$.

Let $\mathcal{A}$ be the hyperplane arrangement consisting of the $n$ hyperplanes $x_i=0$ in $\mathbb{R}^n$, for $i=1,2, \ldots, n$.
(a) Show that the intersection poset $L(\mathcal{A})$ is isomorphic to the Boolean algebra.
(b) Compute the Mobius function of $L(\mathcal{A})$.
(c) Compute the characteristic polynomial of $\mathcal{A}$.

Let $G$ be a simple graph on $[n]$ and let $\mathcal{A}_G$ denote the corresponding graphical arrangement in $\mathbb{R}^n$. Prove that when $G$ has no cycles, the poset $L\left(\mathcal{A}_G\right)$ is isomorphic to a Boolean algebra. Deduce a formula for the number of regions and bounded regions in $\mathcal{A}_G$ in this case.

Let $\mathcal{A}$ be the hyperplane arrangement in $\mathbb{R}^n$ consisting of all hyperplanes $x_i=x_j$ for $i \neq j$ and the hyperplanes $x_i=0$ for $i=1,2, \ldots, n$. Prove that
$$
\chi_{\mathcal{A}}(t)=(t-1)(t-2)(t-3) \cdots(t-n) .
$$