Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms/Condition 4 Implies Condition 1

Theorem

Let $S$ be a finite set.

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

Let $\mathscr C$ be the set of circuits of a matroid $M = \struct{S, \mathscr I}$ on $S$

Then:

$\mathscr C$ satisfies the circuit axioms:

\((C1)\)	$:$						\(\displaystyle \O \notin \mathscr C \)
\((C2)\)	$:$		\(\displaystyle \forall C_1, C_2 \in \mathscr C:\)				\(\displaystyle C_1 \neq C_2 \implies C_1 \not \subseteq C_2 \)
\((C3)\)	$:$		\(\displaystyle \forall C_1, C_2 \in \mathscr C:\)				\(\displaystyle C_1 \neq C_2 \land z \in C_1 \cap C_2 \implies \exists C_3 \in \mathscr C : C_3 \subseteq \paren{C_1 \cup C_2} \setminus \set z \)

Proof

$\blacksquare$

Sources

• 1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 9.$ Circuits