User:Leigh.Samphier/Matroids/Axiom:Base Axiom (Matroid)/Formulation 3

Definition

Let $S$ be a finite set.

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

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

\((\text B 3)\)

$:$

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

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

See also

• User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms