Definition:Rank Axioms (Matroid)/Definition 2

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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:

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