Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Necessary Condition

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Theorem

Let $M = \struct {S, \mathscr I}$ be a matroid.

Let $\mathscr B$ be the set of bases of the matroid $M$.

Then $\mathscr B$ satisfies formulation $1$ of base axiom:

\((\text B 1)\)

$:$

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

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

Proof

Let $B_1, B_2 \in \mathscr B$.

Let $x \in B_1 \setminus B_2$.

We have:

\(\ds \size {B_1 \setminus \set x}\)

\(=\)

\(\ds \size {B_1} - \size {\set x}\)

Cardinality of Set Difference with Subset

\(\ds \)

\(=\)

\(\ds \size {B_2} - \size {\set x}\)

All Bases of Matroid have same Cardinality

\(\ds \)

\(=\)

\(\ds \size {B_2} - 1\)

Cardinality of Singleton

\(\ds \)

\(<\)

\(\ds \size {B_2}\)

By matroid axiom $(\text I 3)$:

$\exists y \in B_2 \setminus \paren{B_1 \setminus \set x} : \paren{ B_1 \setminus \set x} \cup \set y \in \mathscr I$

We have:

\(\ds B_2 \setminus \paren{B_1 \setminus \set x}\)

\(=\)

\(\ds \paren{B_2 \setminus B_1} \cup \paren{B_2 \cap \set x}\)

Set Difference with Set Difference is Union of Set Difference with Intersection

\(\ds \)

\(=\)

\(\ds \paren{B_2 \setminus B_1} \cup \O\)

Intersection With Singleton is Disjoint if Not Element

\(\ds \)

\(=\)

\(\ds B_2 \setminus B_1\)

Union with Empty Set

Then:

$\exists y \in B_2 \setminus B_1 : \paren{ B_1 \setminus \set x} \cup \set y \in \mathscr I$

We have:

\(\ds \size{\paren { B_1 \setminus \set x} \cup \set y}\)

\(=\)

\(\ds \size{B_1 \setminus \set x} + \size{\set y}\)

Corollary to Cardinality of Set Union

\(\ds \)

\(=\)

\(\ds \size{B_1} - \size{\set x} + \size{\set y}\)

Cardinality of Set Difference with Subset

\(\ds \)

\(=\)

\(\ds \size{B_1} - 1 + 1\)

Cardinality of Singleton

\(\ds \)

\(=\)

\(\ds \size{B_1}\)

From Independent Subset is Base if Cardinality Equals Cardinality of Base:

$\paren { B_1 \setminus \set x} \cup \set y \in \mathscr B$

Since $x$, $B_1$ and $B_2$ were arbitrary then the result follows.

$\blacksquare$