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

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## 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 |

$\blacksquare$