Axiom:Matroid Axioms/Axioms 3

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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:

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