# Category:Axioms/Matroid Axioms

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This category contains axioms related to Matroid Axioms.

Let $S$ be a finite set.

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:

### Axioms 1

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

### Axioms 2

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

### Axioms 3

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

### Axioms 4

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

## Pages in category "Axioms/Matroid Axioms"

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