Superset of Dependent Set is Dependent/Corollary

Theorem

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

Let $A \subseteq S$.

Let $x \in A$.

If $x$ is a loop then $A$ is dependent.

Proof

Let $x$ be a loop.

By definition of a loop:

$\set x \notin \mathscr I$

By definition of a dependent subset:

$\set x$ is a dependent subset

From Singleton of Element is Subset:

$\set x \subseteq A$

From Superset of Dependent Set is Dependent:

$A$ is a dependent subset

$\blacksquare$

Sources

• 1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 4.$ Loops and parallel elements