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