\begin{definition}[Definition:Matroid Axioms/Axioms 2]
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: 
{{begin-axiom}}
{{axiom | n = \text I 1
        | mr= \O \in \mathscr I 
}}
{{axiom | n = \text I 2
        | q = \forall X \in \mathscr I: \forall Y \subseteq S
        | mr= Y \subseteq X \implies Y \in \mathscr I
}}
{{axiom | n = \text I 3'
        | q = \forall U, V \in \mathscr I
        | mr= \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I
}}
{{end-axiom}}
Category:Definitions/Matroid Theory
\end{definition}