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