Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 4 Iff Definition 5
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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 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:
\((\text B 5)\) | $:$ | \(\displaystyle \forall B_1, B_2 \in \mathscr B:\) | \(\displaystyle x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_2 \setminus \set y} \cup \set x \in \mathscr B \) |
$\blacksquare$