\begin{definition}[Definition:Independence System Axioms]
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:
{{begin-axiom}}
{{axiom | n = \text I 1
        | mr= \O \in \mathscr F 
}}
{{axiom | n = \text I 2
        | q = \forall X \in \mathscr F: \forall Y \subseteq S
        | mr= Y \subseteq X \implies Y \in \mathscr F
}}
{{end-axiom}}
Category:Definitions/Matroid Theory
\end{definition}