Let S be a finite set and let S₁, S₂, . . . ,S_k be a partition of S into nonempty disjoint subsets. Define the structure (S, I) by the condition that I = {A :  |A ⋂ S_i | ≤ 1 for i = 1, 2, . . . ,k}. Show that (S, I) is a matroid. That is, the set of all sets A that contain at most one member of each subset in the partition determines the independent sets of a matroid.