In: Advanced Math
(a) Show that there are, up to isomorphism, exactly 8 matroids whose underlying set has three elements. Calling the elements a, b, c, exhibit, for each of these matroids, its bases, cycles and independent sets.
(b) Consider the matroid M on the set E = {a, b, c, d}, where the bases are the subsets of E having precisely two elements. Detrmine all the cycles of M, and show that there is no graph G such that M is the cycle matroid M(G).