Equivalence of Definitions of Matroid Circuit Axioms/Lemma 4

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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 satisfying the matroid rank axioms 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 $M = \struct{S, \mathscr I}$ be the matroid corresponding to the rank function $\rho$.

Let $\mathscr C_M$ be the set of circuits of $M$

Then:

$\forall C \in \mathscr C_M : \exists C' \in \mathscr C : C' \subseteq C$

Proof

Let $C \in \mathscr C_M$.

By definition of circuit:

$C$ is dependent

By matroid axiom $(\text I 1)$:

$C \neq \O$

Let $x \in C$.

From Proper Subset of Matroid Circuit is Independent:

$C \setminus \set x$ is independent

We have:

\(\ds \map \rho C\)

\(=\)

\(\ds \card C - 1\)

Rank of Matroid Circuit is One Less Than Cardinality

\(\ds \)

\(=\)

\(\ds \card C - \card{\set x}\)

Cardinality of Singleton

\(\ds \)

\(=\)

\(\ds \card {C \setminus \set x}\)

Cardinality of Set Difference

\(\ds \)

\(=\)

\(\ds \map \rho {C \setminus \set x}\)

Rank of Independent Subset Equals Cardinality

Lemma 2

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

Then:

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

$\Box$

From Lemma 2:

$\exists C' \in \mathscr C : x \in C' \subseteq C$

$\blacksquare$