Distinct Elements are Parallel iff Pair forms Circuit

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Theorem

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

Let $x, y \in S : x \ne y$.


Then:

$x$ and $y$ are parallel if and only if $\set {x, y}$ is a circuit

Proof

Necessary Condition

Let $x$ and $y$ be parallel.

By definition of parallel:

$\set x$ is independent
$\set y$ is independent
$\set {x, y}$ is dependent


Let $A \subseteq \set {x, y}$ be dependent.

Thus:

$A \ne \set x, \set y$


By matroid axiom $(I1)$:

$\O$ is independent

Thus:

$A \ne \O$


From Power Set of Doubleton:

$\powerset {\set {x, y}} = \big \{ \O, \set x, \set y, \set {x,y} \big \}$

Thus:

$A = \set{x,y}$


By definition of a minimal set:

$\set {x, y}$ is a minimal dependent subset

It follows that $\set {x, y}$ is a circuit by definition.

$\Box$

Sufficient Condition

Let $\set {x, y}$ by a circuit.

By definition of a circuit:

$\set {x, y}$ is a minimal dependent subset

By definition of a subset:

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

By definition of a minimal dependent subset:

$\set x, \set y$ are independent subsets

It follows that $x$ and $y$ are parallel by definition.

$\blacksquare$

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