Leigh.Samphier/Sandbox/Matroid Satisfies Circuit Axioms

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $S$ be a finite set.

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


Then $\mathscr C$ is the set of circuits of a matroid on $S$ if and only if $\mathscr C$ satisfies the circuit axioms:

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


Proof

Necessary Condition

$\Box$

Sufficient Condition

$\blacksquare$


Sources