Distinct Matroid Elements are Parallel iff Each is in Closure of Other/Lemma

Theorem

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

Let $a, b \in S$.

Let $\set a$ and $\set b$ be independent.

Then $\set {a, b}$ is dependent if and only if:

$a \in \map \sigma {\set b}$

and

$b \in \map \sigma {\set a}$

where $\sigma: \powerset S \to \powerset S$ denotes the closure operator of $M$.

$\ds \set {a, b}$

$\notin$

$\ds \mathscr I$

$\ds \leadstoandfrom \ \ $

$\ds \set a \cup \set b$

$\notin$

$\ds \mathscr I$

$\ds \leadstoandfrom \ \ $

$\ds a$

$\in$

$\ds \map \sigma {\set b}$

$\, \ds \land \, $

$\ds b$

$\in$

$\ds \map \sigma {\set a}$

$\blacksquare$