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 $S$.

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

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

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

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

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

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

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

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

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