Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 2

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Let $S$ be a finite set.

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:

\((C1)\)   $:$   \(\ds \O \notin \mathscr C \)             
\((C2)\)   $:$     \(\ds \forall C_1, C_2 \in \mathscr C:\) \(\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 \)             
\((C3')\)   $:$     \(\ds \forall C_1, C_2 \in \mathscr C:\) \(\ds C_1 \ne C_2 \land z \in C_1 \cap C_2 \land w \in C_1 \setminus C_2 \implies \exists C_3 \in \mathscr C : y \in C_3 \subseteq \paren {C_1 \cup C_2} \setminus \set z \)