# Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms

## Theorem

Let $S$ be a finite set.

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

The following definitions of the concept of Matroid Circuit Axioms are equivalent:

### Definition 1

$\mathscr C$ satisfies the circuit axioms:

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

### Definition 2

$\mathscr C$ satisfies the circuit axioms:

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

## Proof

### Definition 1 implies Definition 2

$\Box$

### Definition 2 implies Definition 1

$\blacksquare$