Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 4 Iff Definition 5

Theorem

Let $S$ be a finite set.

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

The following definitions of the concept of Matroid Base Axiom are equivalent:

Definition 4

$\mathscr B$ is said to satisfy the base axiom if and only if:

\((\text B 4)\)

$:$

\(\displaystyle \forall B_1, B_2 \in \mathscr B:\)

\(\displaystyle x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \setminus \set x} \cup \set y, \paren {B_2 \setminus \set y} \cup \set x \in \mathscr B \)

Definition 5

$\mathscr B$ is said to satisfy the base axiom if and only if:

\((\text B 5)\)

$:$

\(\displaystyle \forall B_1, B_2 \in \mathscr B:\)

\(\displaystyle x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_2 \setminus \set y} \cup \set x \in \mathscr B \)

Proof

Necessary Condition

Follows immediately from Definition 4 and Definition 5.

$\Box$

Sufficient Condition

Let $\mathscr B$ satisfy the base axiom:

\((\text B 5)\)

$:$

\(\displaystyle \forall B_1, B_2 \in \mathscr B:\)

\(\displaystyle x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_2 \setminus \set y} \cup \set x \in \mathscr B \)

$\blacksquare$