# Definition:Matroid Axioms

## Definition

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

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

### Axioms 2

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

### Axioms 3

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

### Axioms 4

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