# Independent Set can be Augmented by Larger Independent Set

## Theorem

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

Let $X, Y \in \mathscr I$ such that:

$\size X < \size Y$

Then there exists non-empty $Z \subseteq Y \setminus X$ such that:

$X \cup Z \in \mathscr I$
$\size {X \cup Z} = \size Y$

### Corollary

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

Let $\mathscr Z = \set {Z \subseteq Y \setminus X : X \cup Z \in \mathscr I}$

Note that $\O \in \mathscr Z$

So $\mathscr Z \ne \O$

Let $Z_0 \in \mathscr Z : \size {Z_0} = \max \set {\size Z : Z \in \mathscr Z}$

$\size {X \cup Z_0} < \size Y$
$\exists y \in Y \setminus \paren{X \cup Z_0} : X \cup Z_0 \cup \set y \in \mathscr I$

By choice of $Z_0$ and $y$:

$Z_0 \cup \set y \subseteq Y \setminus X$

So:

$Z_0 \cup \set y \in \mathscr Z$

Then:

 $\displaystyle \size {Z_0 \cup \set y}$ $=$ $\displaystyle \size {Z_0} + \size {\set y}$ Corollary to Cardinality of Set Union $\displaystyle$ $=$ $\displaystyle \size {Z_0} + 1$ Cardinality of Singleton $\displaystyle$ $>$ $\displaystyle \size {Z_0}$

This contradicts the choice of $Z_0$.

It follows that:

 $\displaystyle \size Y$ $\le$ $\displaystyle \size {X \cup Z_0}$ $\displaystyle$ $=$ $\displaystyle \size X + \size {Z_0}$ Corollary to Cardinality of Set Union $\displaystyle \leadsto \ \$ $\displaystyle \size {Z_0}$ $\ge$ $\displaystyle \size Y - \size X$
$\exists Z \subseteq Z_0 : \size Z = \size Y - \size X$

Then:

 $\displaystyle \size {X \cup Z}$ $=$ $\displaystyle \size X + \size Z$ Corollary to Cardinality of Set Union $\displaystyle$ $=$ $\displaystyle \size Y$

We have:

 $\displaystyle \card Z$ $=$ $\displaystyle \card Y - \card X$ $\displaystyle$ $>$ $\displaystyle 0$ As $\card X < \card Y$ $\displaystyle \leadsto \ \$ $\displaystyle Z$ $\neq$ $\displaystyle \O$ Cardinality of Empty Set

$\blacksquare$