Rank of Empty Set is Zero

Theorem

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

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

Then:

$\map \rho \O = 0$

Proof

By matroid axiom $(\text I 1)$:

$\O$ is independent

From Rank of Independent Subset Equals Cardinality:

$\map \rho \O = \size \O$

From Cardinality of Empty Set:

$\size \O = 0$

The result follows.

$\blacksquare$

Sources

• 1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 6.$ Properties of the rank function