Definition:Matroid/Definition 4

Definition

Let $M = \struct {S, \mathscr I}$ be an independence system.

$M$ is called a matroid on $S$ if and only if $M$ also satisfies:

$(\text I 3)$

$:$

$\ds \forall A \subseteq S:$

$\ds \text {all maximal subsets $Y \subseteq A$ with $Y \in \mathscr I$ have the same cardinality} $

The properties of a matroid are as follows.

For a given matroid $M = \struct {S, \mathscr I}$ these statements hold true:

$(\text I 1)$	$:$		$\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 $
$(\text I 3)$	$:$	$\ds \forall A \subseteq S:$	$\ds \text{ all maximal subsets } Y \subseteq A \text{ with } Y \in \mathscr I \text{ have the same cardinality} $

\((\text I 1)\)	$:$		\(\ds \O \in \mathscr I \)
\((\text I 2)\)	$:$	\(\ds \forall X \in \mathscr I: \forall Y \subseteq S:\)	\(\ds Y \subseteq X \implies Y \in \mathscr I \)
\((\text I 3)\)	$:$	\(\ds \forall A \subseteq S:\)	\(\ds \text{ all maximal subsets } Y \subseteq A \text{ with } Y \in \mathscr I \text{ have the same cardinality} \)