Axiom:Independence System Axioms

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Definition

Let $S$ be a finite set.

Let $\mathscr F$ be a set of subsets of $S$.

The independence system axioms are the conditions on $\mathscr F$ which are satisfied for all elements of $\mathscr F$ in order for the ordered pair $\struct {S, \mathscr F}$ to be an independence system:

\((\text I 1)\)   $:$   \(\ds \O \in \mathscr F \)      
\((\text I 2)\)   $:$     \(\ds \forall X \in \mathscr F: \forall Y \subseteq S:\) \(\ds Y \subseteq X \implies Y \in \mathscr F \)