User:Leigh.Samphier/Matroids/Independent Subset Contains No Dependent Subset

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Theorem

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

Let $X \subseteq S$ be any independent subset of $M$.

Then:

No dependent subset $D$ of $M$ is a subset of $X$.

Corollary 1

Let $B \subseteq S$ be any base of $M$.

Then:

No dependent subset $D$ of $M$ is a subset of $B$.

Corollary 2

Let $X \subseteq S$ be any independent subset of $M$.

Then:

No circuit $C$ of $M$ is a subset of $X$.

Corollary 3

Let $B \subseteq S$ be any base of $M$.

Then:

No circuit $C$ of $M$ is a subset of $B$.

Proof

By definition of independent subset:

$X \in \mathscr I$

By definition of matroid, specifically matroid axiom $( \text I 2)$:

$\forall Y \subseteq X : Y \in \mathscr I$

By definition of dependent subset:

$\forall Y \subseteq X : Y$ is not a dependent subset

$\blacksquare$