Element of Matroid Base and Circuit has a Substitute/Lemma 3

Theorem

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

Let $B \subseteq S$ be a base of $M$.

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

Let $x \in B \cap C$.

Let $X \subseteq S$ such that:

$\paren{ C \setminus \set x} \cup X$ is a base of $M$.

Then:

$\exists y \in \paren{\paren{C \setminus \set x} \cup X} \setminus \paren{B \setminus \set x} : \paren{B \setminus \set x} \cup \set y \in \mathscr I : \card{\paren{B \setminus \set x} \cup \set y} = \card {\paren{ C \setminus \set x} \cup X}$

Proof

From Set Difference is Subset:

$B \setminus \set x \subseteq B$

From matroid axiom $(\text I 2)$:

$B \setminus \set x \in \mathscr I$

We have

\(\ds \card {B \setminus \set x} + 1\)

\(=\)

\(\ds \paren{\card B - \card{\set x} } + 1\)

Cardinality of Set Difference with Subset

\(\ds \)

\(=\)

\(\ds \paren{\card B - 1 } + 1\)

Cardinality of Singleton

\(\ds \)

\(=\)

\(\ds \card B\)

\(\ds \)

\(=\)

\(\ds \card {\paren{ C \setminus \set x} \cup X}\)

All Bases of Matroid have same Cardinality

From matroid axiom $(\text I 3')$:

$\exists y \in \paren{\paren{C \setminus \set x} \cup X} \setminus \paren{B \setminus \set x} : \paren{B \setminus \set x} \cup \set y \in \mathscr I : \card{\paren{B \setminus \set x} \cup \set y} = \card {\paren{ C \setminus \set x} \cup X}$

$\blacksquare$