User:Leigh.Samphier/Matroids/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 $S$.

Formulation 1
$\mathscr C$ satisfies the circuit axioms:

Formulation 2
$\mathscr C$ satisfies the circuit axioms:

Formulation 3
$\mathscr C$ satisfies the circuit axioms:

Also see

 * User:Leigh.Samphier/Matroids/Circuits of Matroid iff Matroid Circuit Axioms