Rank of Empty Set is Zero
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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