Singleton is Independent implies Rank is One

Theorem

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

Let $x \in S$.

Let $\set x$ be independent.

Then:

$\map \rho {\set x} = 1$

where $\rho$ denotes the rank function of $M$.

Corollary

$\set x$ is an independent subset if and only if $\map \rho {\set x} = 1$

Proof

From Rank of Independent Subset Equals Cardinality:

$\map \rho {\set x} = \size {\set x}$

From Cardinality of Singleton:

$\size {\set x} = 1$

The result follows.

$\blacksquare$