\begin{definition}[Definition:Rank Axioms (Matroid)/Definition 2]
Let $S$ be a finite set.
Let $\rho: \powerset S \to \Z$ be a mapping from the power set of $S$ into the integers
$\rho$ is said to satisfy the '''rank axioms''' {{iff}}:
{{begin-axiom}}
{{axiom | n = \text R 1'
        | q = \forall X \in \powerset S
        | mr = 0 \le \map \rho X \le \size X
}}
{{axiom | n = \text R 2'
        | q = \forall X, Y \in \powerset S
        | mr = X \subseteq Y \implies \map \rho X \le \map \rho Y
}}
{{axiom | n = \text R 3'
        | q = \forall X, Y \in \powerset S
        | mr = \map \rho {X \cup Y} + \map \rho {X \cap Y} \le \map \rho X + \map \rho Y
}}
{{end-axiom}}
\end{definition}