Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms/Condition 2 Implies Condition 3
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Theorem
Let $S$ be a finite set.
Let $\mathscr C$ be a non-empty set of subsets of $S$.
Let $\mathscr C$ satisfy the circuit axioms:
\((C1)\) | $:$ | \(\displaystyle \O \notin \mathscr C \) | ||||||
\((C2)\) | $:$ | \(\displaystyle \forall C_1, C_2 \in \mathscr C:\) | \(\displaystyle C_1 \neq C_2 \implies C_1 \not \subseteq C_2 \) | |||||
\((C3')\) | $:$ | \(\displaystyle \forall C_1, C_2 \in \mathscr C:\) | \(\displaystyle C_1 \neq C_2 \land z \in C_1 \cap C_2 \land w \in C_1 \setminus C_2 \implies \exists C_3 \in \mathscr C : y \in C_3 \subseteq \paren{C_1 \cup C_2} \setminus \set z \) |
Then:
- $\mathscr C$ satisfies the circuit axioms:
\((C1)\) | $:$ | \(\displaystyle \O \notin \mathscr C \) | ||||||
\((C2)\) | $:$ | \(\displaystyle \forall C_1, C_2 \in \mathscr C:\) | \(\displaystyle C_1 \neq C_2 \implies C_1 \not \subseteq C_2 \) | |||||
\((C3'')\) | $:$ | \(\displaystyle \forall X \subseteq S \land \forall x \in S:\) | \(\displaystyle \paren{\forall C \in \mathscr C : C \not \subseteq X} \implies \paren{\exists \text{ at most one } C \in \mathscr C : C \subseteq X \cup \set x} \) |
Proof
$\blacksquare$