Axiom:Matroid Axioms

Definition
Let $S$ be a finite set.

Let $\mathscr I$ be a set of subsets of $S$.

The matroid axioms are the conditions on $\mathscr I$ which are satisfied for all elements of $\mathscr I$ in order for the ordered pair $\struct{S, \mathscr I}$ to be a matroid:
 * $(I1) \quad \O \in \mathscr I$
 * $(I2) \quad X \in \mathscr I \land Y \subseteq X \implies Y \in \mathscr I$
 * $(I3) \quad U, V \in \mathscr I \land \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I$