Matroid (Independence Axioms)

A matroid is a set called the ground set, together with a collection \(\mathscr{I}\) of subsets called independent sets such that the following hold:

(1). \( \emptyset \in \mathscr{I} \)

(2). If \( I \in \mathscr{I} \) and \( J \subseteq I \), then \( J \in \mathscr{I} \).

(3). If \(I_1, I_2 \in \mathscr{I} \), and \( |I_1|<|I_2| \), then there exists an \( e \in {I_2 \backslash I_1} \) for which \(I_1 \cup \{e\} \in \mathscr{I} \).

We call axiom (3) the augmentation axiom.