Matroid Unique Circuit Property/Corollary

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Theorem

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

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

Let $x \in S \setminus B$.


Then there exists a unique circuit $C$ such that:

$x \in C \subseteq B \cup \set x$


That is, $C$ is the fundamental circuit of $x$ in $B$.


Proof

From Union of Matroid Base with Element of Complement is Dependent:

$B \cup \set x$ is dependent.

From Matroid Unique Circuit Property there exists a unique circuit $C$ such that:

$x \in C \subseteq B \cup \set x$

$\blacksquare$