User:Leigh.Samphier/Sandbox/Bound for Cardinality of Matroid Circuit

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

Let $C \subseteq S$ be a circuit of $M$.

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

Then:
 * $\card C \le \map \rho S + 1$

Proof
By definition of a circuit:
 * $C$ is dependent

By matroid axiom $(\text I 1)$:
 * $C \ne \O$

Let $x \in C$.

From Set Difference is Subset and Set Difference with Disjoint Set:
 * $C \setminus \set x \subsetneq C$

From User:Leigh.Samphier/Sandbox/Proper Subset of Matroid Circuit is Independent and matroid axiom $(\text I 1)$:
 * $C \setminus \set x \in \mathscr I$

We have: