Rank Axioms for Matroid Rank Functions is a result in [matroid theory](/page/Matroid%20Theory). It formalizes the assertion that let be a finite matroid, and let be its rank function, defined by \begin{align } r(A) = \max\{|I| : I \subset A \text{ and } I \in \mathcal I\} \end{align } for each subset . Then, for all subsets and every element , the following hold: 1..