Set with Two Parallel Elements is Dependent

Theorem

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

Let $A \subseteq S$.

Let $x, y \in S$.

Let $x, y$ be parallel elements.

If $x, y \in A$ then $A$ is dependent.

Proof

Let $x, y \in A$.

From Doubleton of Elements is Subset:

$\set{x, y} \subseteq A$

By the definition of parallel elements:

$\set {x, y}$ is dependent

From Superset of Dependent Set is Dependent:

$A$ is dependent

$\blacksquare$

Sources

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