\begin{definition}[Definition:Rank Function (Matroid)]
Let $M = \struct {S, \mathscr I}$ be a matroid.
The '''rank function''' of $M$ is the mapping $\rho : \powerset S \to \Z$ from the power set of $S$ into the integers defined by:
:$\forall A \subseteq S : \map \rho A = \max \set {\size X : X \subseteq A \land X \in \mathscr I}$
where $\size A$ denotes the cardinality of $A$.
\end{definition}