# 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 $M$.

## Proof

 $\ds \set {a, b}$ $\notin$ $\ds \mathscr I$ $\ds \leadstoandfrom \ \$ $\ds \set a \cup \set b$ $\notin$ $\ds \mathscr I$ Union of Disjoint Singletons is Doubleton $\ds \leadstoandfrom \ \$ $\ds a$ $\in$ $\ds \map \sigma {\set b}$ Element Depends on Independent Set iff Union with Singleton is Dependent $\, \ds \land \,$ $\ds b$ $\in$ $\ds \map \sigma {\set a}$ Element Depends on Independent Set iff Union with Singleton is Dependent

$\blacksquare$