A pair (V, I) is called a matroid if V is a finite ground set and I is a nonempty collection of subsets of V that are independent.
i must satisfy (i) if X ⊂ Y and Y ∈ i then X ∈ i, (ii) if X, Y ∈ i and |X| < |Y | then X ∪{e} ∈ i for some e ∈ Y \X.