Category:Axioms/Matroid Axioms

This category contains axioms related to Matroid Axioms.

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

Pages in category "Axioms/Matroid Axioms"

The following 5 pages are in this category, out of 5 total.

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

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

\((\text I 1)\)	$:$		\(\ds \O \in \mathscr I \)
\((\text I 2)\)	$:$	\(\ds \forall X \in \mathscr I: \forall Y \subseteq S:\)	\(\ds Y \subseteq X \implies Y \in \mathscr I \)
\((\text I 3'')\)	$:$	\(\ds \forall U, V \in \mathscr I:\)	\(\ds \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren{V \cup Z \in \mathscr I} \land \paren{\size{V \cup Z} = \size U} \)

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