Distinct Elements are Parallel iff Pair forms Circuit

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 $(\text I 1)$:

$\O$ is independent

Thus:

$A \ne \O$

From Power Set of Doubleton:

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

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.

$\blacksquare$

Sources

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