User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Formulation 1 Iff Formulation 5

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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 $5$ of base axiom:

\((\text B 5)\)

$:$

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

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

Proof

Follows immediately from:

• User:Leigh.Samphier/Matroids/Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom
• User:Leigh.Samphier/Matroids/Matroid Bases Iff Satisfies Formulation 5 of Matroid Base Axiom

$\blacksquare$