Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms

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Theorem

Let $S$ be a finite set.

Let $\mathscr C$ be a non-empty set of subsets of $S$.


The following definitions of the concept of Matroid Circuit Axioms are equivalent:

Definition 1

$\mathscr C$ satisfies the circuit axioms:

\((C1)\)   $:$     \(\displaystyle \forall C_1, C_2 \in \mathscr C:\) \(\displaystyle C_1 \neq C_2 \implies C_1 \not \subseteq C_2 \)             
\((C2)\)   $:$     \(\displaystyle \forall C_1, C_2 \in \mathscr C:\) \(\displaystyle C_1 \neq C_2 \land z \in C_1 \cap C_2 \implies \exists C_3 \in \mathscr C : C_3 \subseteq \paren{C_1 \cup C_2} \setminus \set z \)             


Definition 2

$\mathscr C$ satisfies the circuit axioms:

\((C1)\)   $:$     \(\displaystyle \forall C_1, C_2 \in \mathscr C:\) \(\displaystyle C_1 \neq C_2 \implies C_1 \not \subseteq C_2 \)             
\((C2')\)   $:$     \(\displaystyle \forall C_1, C_2 \in \mathscr C:\) \(\displaystyle C_1 \neq C_2 \land z \in C_1 \cap C_2 \land w \in C_1 \setminus C_2 \implies \exists C_3 \in \mathscr C : y \in C_3 \subseteq \paren{C_1 \cup C_2} \setminus \set z \)             


Proof

Definition 1 implies Definition 2

$\Box$


Definition 2 implies Definition 1

$\blacksquare$

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