Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 3

From ProofWiki
Jump to navigation Jump to search

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 axiom if and only if:

\((\text B 3)\)   $:$     \(\displaystyle \forall B_1, B_2 \in \mathscr B:\) \(\displaystyle \exists \text{ a bijection } \pi : B_1 \setminus B_2 \to B_2 \setminus B_1 : \forall x \in B_1 \setminus B_2 : \paren {B_1 \setminus \set x } \cup \set {\map \pi x} \in \mathscr B \)             


Sources