\begin{definition}[Definition:Circuit Axioms (Matroid)]
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''' {{iff}}:
{{begin-axiom}}
{{axiom | n = \text C 1
        | q = \forall C_1, C_2 \in \mathscr C
        | mr = C_1 \ne C_2 \implies C_1 \nsubseteq C_2
}}
{{axiom | n = \text C 2
        | q = \forall C_1, C_2 \in \mathscr C
        | mr = C_1 \ne 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
}}
{{end-axiom}}
\end{definition}