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