User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Formulation 1 Iff Formulation 7

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Theorem

Let $S$ be a finite set.

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

Then:

$\mathscr B$ satisfies formulation $1$ of base axiom:

\((\text B 1)\)

$:$

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

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

if and only if

$\mathscr B$ satisfies formulation $7$ of base axiom:

\((\text B 7)\)

$:$

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

\(\ds \exists \text{ a bijection } \pi : B_1 \to B_2 : \forall x \in B_1 : \paren {B_2 \setminus \set {\map \pi x} } \cup \set x \in \mathscr B \)

Proof

Necessary Condition

Follows immediately from:

• Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom

• Matroid Bases Satisfy Formulation 7 of Matroid Base Axiom

$\Box$

Sufficient Condition

From Formulation 3 Iff Formulation 7:

$\mathscr B$ satisfies formulation $7$ of base axiom:

\((\text B 7)\)

$:$

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

\(\ds \exists \text{ a bijection } \pi : B_1 \to B_2 : \forall x \in B_1 : \paren {B_2 \setminus \set {\map \pi x} } \cup \set x \in \mathscr B \)

if and only if

$\mathscr B$ satisfies formulation $3$ of base axiom:

\((\text B 3)\)

$:$

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

\(\ds \exists \text{ a bijection } \pi : B_1 \to B_2 : \forall x \in B_1 : \paren {B_1 \setminus \set x } \cup \set {\map \pi x} \in \mathscr B \)

By choosing $y = \map \pi x$ in formulation $3$, formulation $1$ follows immediately.

$\blacksquare$