Element of Matroid Base and Circuit has Substitute/Lemma 2
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Lemma for Element of Matroid Base and Circuit has Substitute
Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $C \subseteq S$ be a dependent subset of $M$.
Let $x \in C$.
Let $X \subseteq S$ such that:
- $\paren {C \setminus \set x} \cup X \in \mathscr I$
Then:
- $x \notin \paren {C \setminus \set x} \cup X$
Proof
Aiming for a contradiction, suppose:
- $x \in \paren {C \setminus \set x} \cup X$
Then:
- $\set x, C \setminus \set x \subseteq \paren {C \setminus \set x} \cup X$
From Union of Subsets is Subset:
- $C \subseteq \paren {C \setminus \set x} \cup X$
From matroid axiom $(\text I 2)$:
- $C \in \mathscr I$
This contradicts:
- $C \notin \mathscr I$
It follows that:
- $x \notin \paren {C \setminus \set x} \cup X$
$\blacksquare$