# Flats of the dual matroid and union of circuits of the matroid

I'm struggling to formally demonstrate the hypothesis that every flat of the dual matroid ($$F \in \mathcal{F}(M^*)$$) is the complement of some circuit or union of circuits of the original matroid $$M$$.

I tried to translate the following statements into a simpler language and I think I figured out the idea but not how to write it in a generic way:

1. Hyperplanes are maximal proper flats
2. Every flat is the intersection of some collection of hyperplanes of the matroid
3. Circuits are minimal dependent sets
4. Hyperplanes of $$M^*$$ are the complements of the circuits of $$M$$.

If all the flats of the matroid can be constructed by forming all possible intersections of various collections of hyperplanes and $$E(M) - H^* = C$$, then all the unions of various collections of circuits can be constructed by $$E(M) - F^*$$.

Summing up:

$$F^* = H^*_1 \cap H^*_2 = {({H^*_1}^C \cup {H^*_2}^C)}^C = {(C_1 \cup C_2)}^C$$

Could you help me see the best way to write this?

My main reference is the book Matroids: A Geometric Introduction, by Gary Gordon and Jennifer McNulty.

$$F^*$$ is a flat of $$M^*$$ if and only if there are hyperplanes $$H^*_1, \dots, H^*_n$$ of $$M^*$$ ($$n\ge 1$$) such that $$F^* = \bigcap_{i=1}^nH^*_i$$. But $$H^*$$ is a hyperplane of the dual if and only if there is a circuit $$C$$ of $$M$$ such that $$E \setminus C= H^*$$. So $$F^*$$ is a flat of $$M^*$$ if and only if $$F^* = \bigcap_{i=1}^n(E \setminus C_i)$$, where $$C_i$$ is the complement of $$H_i$$. It follows that $$F^* = E \setminus \bigcup C_i$$ since the intersection of complements of sets in $$E$$ is the complement of the union of the sets.