Independent Subset is Contained in Maximal Independent Subset
Jump to navigation
Jump to search
Theorem
Let $M = \struct{S, \mathscr I}$ be a matroid.
Let $A \subseteq S$.
Let $X \in \mathscr I$ such that $X \subseteq A$.
Then:
- $\exists Y \in \mathscr I : X \subseteq Y \subseteq A : \size Y = \map \rho A$
where $\rho$ is the rank function on $M$.
Proof
By definition of the rank function on $M$:
- $\size X \le \map \rho A$
Case 1 : $\size X = \map \rho A$
Let $\size X = \map \rho A$.
Let $Y = X$ and the result follows.
$\Box$
Case 2 : $\size X < \map \rho A$
Let $\size X < \map \rho A$.
By definition of the rank function on $M$:
- $\map \rho A = \max \set{\size I : I \subseteq A \land I \in \mathscr I}$
From Max Operation Equals an Operand:
- $\exists Z \in \mathscr I : Z \subseteq A$ and $\size Z = \map \rho A$
From Independent Set can be Augmented by Larger Independent Set:
- $\exists Y' \subseteq Z \setminus X: \size {X \cup Y'} = \size Z$ and $X \cup Y' \in \mathscr I$
We have:
\(\ds Y'\) | \(\subseteq\) | \(\ds Z \setminus X\) | ||||||||||||
\(\ds \) | \(\subseteq\) | \(\ds Z\) | Set Difference is Subset | |||||||||||
\(\ds \) | \(\subseteq\) | \(\ds A\) | by choice of $Z$ |
From Union of Subsets is Subset:
- $X \cup Y' \subseteq A$
Let $Y = X \cup Y'$ and the result follows.
$\Box$
In either case, the result follows.
$\blacksquare$