Axiom:Base Axiom (Matroid)

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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:

\((\text B 1)\)   $:$     \(\ds \forall B_1, B_2 \in \mathscr B:\) \(\ds x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \cup \set y} \setminus \set x \in \mathscr B \)