User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Axiom B4 Iff Axiom B5

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Theorem

Let $S$ be a finite set.

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


The following definitions for the Matroid Base Axioms are equivalent:

Axiom $(\text B 4)$

\((\text B 4)\)   $:$     \(\ds \forall B_1, B_2 \in \mathscr B:\) \(\ds x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \setminus \set x} \cup \set y, \paren {B_2 \setminus \set y} \cup \set x \in \mathscr B \)      

Axiom $(\text B 5)$

\((\text B 5)\)   $:$     \(\ds \forall B_1, B_2 \in \mathscr B:\) \(\ds x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_2 \setminus \set y} \cup \set x \in \mathscr B \)      

Proof

Necessary Condition

Follows immediately from Axiom $(\text B 4)$ and Axiom $(\text B 5)$.

$\Box$


Sufficient Condition

Let $\mathscr B$ satisfy the base axiom:

\((\text B 4)\)   $:$     \(\ds \forall B_1, B_2 \in \mathscr B:\) \(\ds x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \setminus \set x} \cup \set y, \paren {B_2 \setminus \set y} \cup \set x \in \mathscr B \)      


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It follows that $\mathscr B$ satisfies the base axiom:

\((\text B 5)\)   $:$     \(\ds \forall B_1, B_2 \in \mathscr B:\) \(\ds x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_2 \setminus \set y} \cup \set x \in \mathscr B \)      

$\blacksquare$

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