Axiom:Base Axiom (Matroid)/Formulation 3
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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:
\((\text B 3)\) | $:$ | \(\ds \forall B_1, B_2 \in \mathscr B:\) | \(\ds \exists \text{ a bijection } \pi : B_1 \to B_2 : \forall x \in B_1 : \paren {B_1 \setminus \set x } \cup \set {\map \pi x} \in \mathscr B \) |
See also
Sources
- 1969: Richard A. Brualdi: Comments on bases in dependence structures (Bulletin of the Australian Mathematical Society Vol. 1, no. 2: pp. 161 – 167)