Definition:Matroid/Definition 3

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:

$(\text I 3)$

$:$

$\ds \forall U, V \in \mathscr I:$

$\ds \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren {V \cup Z \in \mathscr I} \land \paren {\size {V \cup Z} = \size U} $

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 $
$(\text I 5)$	$:$	$\ds \forall U, V \in \mathscr I:$	$\ds \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren{V \cup Z \in \mathscr I} \land \paren{\size{V \cup Z} = \size U} $

\((\text I 1)\)	$:$		\(\ds \O \in \mathscr I \)
\((\text I 2)\)	$:$	\(\ds \forall X \in \mathscr I: \forall Y \subseteq S:\)	\(\ds Y \subseteq X \implies Y \in \mathscr I \)
\((\text I 5)\)	$:$	\(\ds \forall U, V \in \mathscr I:\)	\(\ds \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren{V \cup Z \in \mathscr I} \land \paren{\size{V \cup Z} = \size U} \)