View source for Matroid

==Rank==
We define the '''rank''' ρ(''A'') of a subset ''A'' of ''E'' to be the maximum [[cardinality]] of an independent subset of ''A''.  The rank satisfies the following

:<math>0 \le \rho(A) \le |A| ;\,</math>
:<math>A \subseteq B \Rightarrow \rho(A) \le \rho(B) ;\,</math>
:<math>\rho(A) + \rho(B) \ge \rho(A\cap B) + \rho(A \cup B) .\,</math>

The last of these is the ''submodular inequality''.  

A ''flat'' is a subset ''A'' of ''E'' such that the rank of ''A'' is strictly less than the rank of any proper [[superset]] of ''A''.