Axiom:Circuit Axioms (Matroid)/Formulation 2
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Definition
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:
\((\text C 1)\) | $:$ | \(\ds \O \notin \mathscr C \) | |||||||
\((\text C 2)\) | $:$ | \(\ds \forall C_1, C_2 \in \mathscr C:\) | \(\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 \) | ||||||
\((\text C 4)\) | $:$ | \(\ds \forall C_1, C_2 \in \mathscr C:\) | \(\ds C_1 \ne C_2 \land z \in C_1 \cap C_2 \land w \in C_1 \setminus C_2 \implies \exists C_3 \in \mathscr C : w \in C_3 \subseteq \paren {C_1 \cup C_2} \setminus \set z \) |
Also see
Sources
- 1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 9.$ Circuits