Equivalence of Definitions of Matroid Circuit Axioms/Lemma 2

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Theorem

Let $S$ be a finite set.

Let $\mathscr C$ be a non-empty set of subsets of $S$ that satisfies the circuit axioms:

\((\text C 1)\)	$:$							\(\ds \O \notin \mathscr C \)
\((\text C 2)\)	$:$			\(\ds \forall C_1, C_2 \in \mathscr C:\)				\(\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 \)
\((\text C 3')\)	$:$			\(\ds \forall C_1, C_2 \in \mathscr C:\)				\(\ds C_1 \ne C_2 \land z \in C_1 \cap C_2 \land w \in C_1 \setminus C_2 \implies \exists C_3 \in \mathscr C : w \in C_3 \subseteq \paren {C_1 \cup C_2} \setminus \set z \)

For any ordered tuple $\tuple{x_1, \ldots, x_q}$ of elements of $S$, let $\map \theta {x_1, \ldots, x_q}$ be the ordered tuple defined by:

$\forall i \in \set{1, \ldots, q} : \map \theta {x_1, \ldots, x_q}_i = \begin{cases}

0 & : \exists C \in \mathscr C : x_i \in C \subseteq \set{x_1, \ldots, x_i}\\ 1 & : \text {otherwise} \end{cases}$

Let $t$ be the mapping from the set of ordered tuple of $S$ defined by:

$\map t {x_1, \ldots, x_q} = \ds \sum_{i = 1}^q \map \theta {x_1, \ldots, x_q}_i$

Let $\rho : \powerset S \to \Z$ be the mapping defined by:

$\forall A \subseteq S$:

$\map \rho A = \begin{cases}

0 & : \text{if } A = \O \\ \map t {x_1, \ldots, x_q } & : \text{if } A = \set{x_1, \ldots, x_q} \end{cases}$

Let $X \subseteq S$ and $y \in S \setminus X$.

Then:

$\map \rho {X \cup \set y} = \map \rho X$ if and only if $\exists C \in \mathscr C : y \in C \subseteq X \cup \set y$

Proof

Let $X = \set{x_1, \ldots, x_q}$

We have:

\(\ds \map \rho {X \cup \set y}\)

\(=\)

\(\ds \map t {x_1, \ldots, x_q, y}\)

Definition of $\rho$

\(\ds \)

\(=\)

\(\ds \map t {x_1, \ldots, x_q} + \map \theta {x_1, \ldots, x_q, y}_{q+1}\)

Definition of $t$

\(\ds \)

\(=\)

\(\ds \map \rho X + \map \theta {x_1, \ldots, x_q, y}_{q+1}\)

Definition of $t$

Hence:

\(\ds \map \rho {X \cup \set y} = \map \rho X\)

\(\leadstoandfrom\)

\(\ds \map \theta {x_1, \ldots, x_q, y}_{q+1} = 0\)

\(\ds \)

\(\leadstoandfrom\)

\(\ds \exists C \in \mathscr C : y \in C \subseteq X \cup \set y\)

Definition of $\theta$

$\blacksquare$