Definition:Matroid/Definition 1

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 x \in U \setminus V : V \cup \set x \in \mathscr I $

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 $
$(\text I 3)$	$:$	$\ds \forall U, V \in \mathscr I:$	$\ds \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I $

2018: Bernhard H. Korte and Jens Vygen: Combinatorial Optimization: Theory and Algorithms (6th ed.) Chapter $13$ Matroids $\S 13.1$ Independence Systems and Matroids, Definition $13.3$

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