Parallel Elements Depend on Same Subsets

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Theorem

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

Let $A \subseteq S$.

Let $x, y \in S$.

Let $x$ be parallel to $y$.


Then:

$x$ depends on $A$ if and only if $y$ depends on $A$


Proof

This follows directly from Closure of Subset Contains Parallel Elements.

$\blacksquare$


Sources