Set Difference of Matroid Dependent Set with Independent Set is Non-empty

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Theorem

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

Let $I$ be an independent subset of $M$.

Let $D$ be a dependent subset of $M$.

Then:

$D \setminus I \ne \O$

Corollary 1

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

Let $D$ be a dependent subset of $M$.

Let $B$ be a base of $M$.

Then:

$D \setminus B \ne \O$

Corollary 2

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

Let $I$ be an independent subset of $M$.

Let $C$ be a circuit of $M$.

Then:

$C \setminus I \ne \O$

Corollary 3

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

Let $B$ be an base of $M$.

Let $C$ be a circuit of $M$.

Then:

$C \setminus B \ne \O$

Proof

From Independent Subset Contains No Dependent Subset:

$D \nsubseteq I$

By definition of subset:

$\exists x \in D : x \notin I$

By definition of set difference:

$\exists x \in D \setminus I$

The result follows.

$\blacksquare$