Independent Set can be Augmented by Larger Independent Set/Corollary
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Theorem
Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $X \subseteq S$ be an independent subset of $M$.
Let $B \subseteq S$ be a base of $M$.
Then:
- $\exists Z \subseteq B \setminus X : \card{X \cup Z} = \card B : X \cup Z$ is a base of $M$
Proof
From Cardinality of Independent Set of Matroid is Smaller or Equal to Base:
- $\card X \le \card B$
Case 1: $\card X < \card B$
From Independent Set can be Augmented by Larger Independent Set:
- $\exists Z \subseteq B \setminus X : X \cup Z \in \mathscr I : \card {X \cup Z} = \card B$
$\Box$
Case 2: $\card X = \card B$
Let $Z = \O$.
Then:
- $Z \subseteq B \setminus X : X \cup Z \in \mathscr I : \card {X \cup Z} = \card B$
$\Box$
From Independent Subset is Base if Cardinality Equals Cardinality of Base:
- $X \cup Z$ is a base of $M$
$\blacksquare$