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

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 $S$


 * $\mathscr C$ satisfies the circuit axioms

Proof
From Equivalence of Definitions of Matroid Circuit Axioms it is sufficient to show:
 * $(\text a)\quad$if $\mathscr C$ is the set of circuits of a matroid then $\mathscr C$ satisfies circuit axioms (formulation 1)

and
 * $(\text b)\quad$if $\mathscr C$ satisfies circuit axioms (formulation 2) then $\mathscr C$ is the set of circuits of a matroid

Circuits of Matroids implies Formulation 1
Let $\mathscr C$ be the set of circuits of a matroid $M = \struct{S, \mathscr I}$ on $S$

It is shown that $\mathscr C$ satisfies the circuit axioms (formulation 1):

Formulation 2 implies Circuits of Matroid
Let $\mathscr C$ be a non-empty set of subsets of $S$ that satisfies the circuit axioms (formulation 2):

Also see

 * Equivalence of Definitions of Matroid Circuit Axioms