Singleton is Dependent implies Rank is Zero/Corollary
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Theorem
Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $x \in S$.
Then:
- $x$ is a loop if and only if $\map \rho {\set x} = 0$
where $\rho$ denotes the rank function of $M$.
Proof
By definition of a loop:
- $x$ is a loop if and only if $\set x \notin \mathscr I$
From Singleton is Dependent implies Rank is Zero:
- if $\set x \notin \mathscr I$ then $\map \rho {\set x} = 0$
From Singleton is Independent implies Rank is One:
- if $\set x \in \mathscr I$ then $\map \rho {\set x} = 1$
It follows that:
- $\set x \notin \mathscr I$ if and only if $\map \rho {\set x} = 0$
$\blacksquare$
Sources
- 1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 4.$ Loops and parallel elements