Set Difference of Matroid Dependent Set with Independent Set is Non-empty/Corollary 3

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Theorem

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

Let $B$ be an base of $M$.

Let $C$ be a circuit of $M$.

Then:

$C \setminus B \ne \O$

Proof

By definition of matroid base:

$B$ is an independent subset of $M$

By definition of matroid circuit:

$C$ is a dependent subset of $M$

From Set Difference of Matroid Dependent Set with Independent Set is Non-empty:

$C \setminus B \ne \O$

$\blacksquare$