Element of Matroid Base and Circuit has a Substitute/Lemma 3
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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 $(\text I 3')$:
- $\exists y \in \paren{\paren{C \setminus \set x} \cup X} \setminus \paren{B \setminus \set x} : \paren{B \setminus \set x} \cup \set y \in \mathscr I : \card{\paren{B \setminus \set x} \cup \set y} = \card {\paren{ C \setminus \set x} \cup X}$
$\blacksquare$