# Definition:Rank Axioms (Matroid)/Definition 2

## Definition

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 if and only if:

 $(\text R 1')$ $:$ $\ds \forall X \in \powerset S:$ $\ds 0 \le \map \rho X \le \size X$ $(\text R 2')$ $:$ $\ds \forall X, Y \in \powerset S:$ $\ds X \subseteq Y \implies \map \rho X \le \map \rho Y$ $(\text R 3')$ $:$ $\ds \forall X, Y \in \powerset S:$ $\ds \map \rho {X \cup Y} + \map \rho {X \cap Y} \le \map \rho X + \map \rho Y$