Rank Function is Increasing

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Let $M = \struct {S, \mathscr I}$ be a matroid.

Let $\rho: \powerset S \to \Z$ be the rank function of $M$.

Let $A, B \subseteq S$ be subsets of $S$ such that $A \subseteq B$.


$\map \rho A \le \map \rho B$



\(\ds \map \rho A\) \(=\) \(\ds \max \set {\size X : X \subseteq A \land X \in \mathscr I}\) Definition of Rank Function
\(\ds \) \(=\) \(\ds \max \set {\size X : X \in \powerset A \land X \in \mathscr I}\) Definition of Power Set of $\O$
\(\ds \) \(=\) \(\ds \max \set {\size X : X \in \powerset A \cap \mathscr I}\) Definition of Set Intersection


\(\ds \map \rho B\) \(=\) \(\ds \max \set {\size X : X \in \powerset B \cap \mathscr I}\)

From Power Set of Subset:

$\powerset A \subseteq \powerset B$

From Set Intersection Preserves Subsets:

$\powerset A \cap \mathscr I \subseteq \powerset B \cap \mathscr I$

It follows that:

$\set {\size X : X \in \powerset A \cap \mathscr I} \subseteq \set {\size X : X \in \powerset B \cap \mathscr I}$

From Max of Subfamily of Operands Less or Equal to Max:

$\max \set {\size X : X \in \powerset B \cap \mathscr I} \le \max \set {\size X : X \in \powerset B \cap \mathscr I}$