Axiom:Circuit Axioms (Matroid)/Formulation 2

Definition

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

$\mathscr C$ is said to satisfy the circuit axioms if and only if:

$(\text C 1)$	$:$		$\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 $
$(\text C 4)$	$:$	$\ds \forall C_1, C_2 \in \mathscr C:$	$\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 $

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

\((\text C 1)\)	$:$		\(\ds \O \notin \mathscr C \)
\((\text C 2)\)	$:$	\(\ds \forall C_1, C_2 \in \mathscr C:\)	\(\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 \)
\((\text C 4)\)	$:$	\(\ds \forall C_1, C_2 \in \mathscr C:\)	\(\ds C_1 \ne C_2 \land z \in C_1 \cap C_2 \land w \in C_1 \setminus C_2 \implies \exists C_3 \in \mathscr C : w \in C_3 \subseteq \paren {C_1 \cup C_2} \setminus \set z \)