Axiom:Matroid Axioms/Axioms 1

Definition

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

The matroid axioms are the conditions on $S$ and $\mathscr I$ in order for the ordered pair $\struct {S, \mathscr I}$ to be a matroid:

$(\text I 1)$	$:$		$\ds \O \in \mathscr I $
$(\text I 2)$	$:$	$\ds \forall X \in \mathscr I: \forall Y \subseteq S:$	$\ds Y \subseteq X \implies Y \in \mathscr I $
$(\text I 3)$	$:$	$\ds \forall U, V \in \mathscr I:$	$\ds \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I $

\((\text I 1)\)	$:$		\(\ds \O \in \mathscr I \)
\((\text I 2)\)	$:$	\(\ds \forall X \in \mathscr I: \forall Y \subseteq S:\)	\(\ds Y \subseteq X \implies Y \in \mathscr I \)
\((\text I 3)\)	$:$	\(\ds \forall U, V \in \mathscr I:\)	\(\ds \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I \)