Axiom:Circuit Axioms (Matroid)/Formulation 3
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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:
\((\text C 1)\) | $:$ | \(\ds \O \notin \mathscr C \) | |||||||
\((\text C 2)\) | $:$ | \(\ds \forall C_1, C_2 \in \mathscr C:\) | \(\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 \) | ||||||
\((\text C 3'')\) | $:$ | \(\ds \forall X \subseteq S \land \forall x \in S:\) | \(\ds \paren {\forall C \in \mathscr C : C \nsubseteq X} \implies \paren {\exists \text{ at most one } C \in \mathscr C : C \subseteq X \cup \set x} \) |