Axiom:Matroid Axioms/Axioms 4

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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:

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