User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms

Theorem
Let $S$ be a finite set.

Let $\mathscr B$ be a non-empty set of subsets of $S$.

Axiom 1
$\mathscr B$ is said to satisfy the base axiom :

Axiom 2
$\mathscr B$ is said to satisfy the base axiom :

Axiom 3
$\mathscr B$ is said to satisfy the base axiom :

Axiom 4
$\mathscr B$ is said to satisfy the base axiom :

Axiom 5
$\mathscr B$ is said to satisfy the base axiom :

Axiom 6
$\mathscr B$ is said to satisfy the base axiom :

Axiom 7
$\mathscr B$ is said to satisfy the base axiom :

Definition 1 iff Definition 2
Axiom 1 holds Axiom 2 holds follows immediately from the lemma.

Definition 5 iff Definition 6
Axiom 5 holds Axiom 6 holds follows immediately from the lemma.