For a set A, let C = {Pi | Pl is a partition of A}. Define relation

Question:

For a set A, let C = {Pi | Pl is a partition of A}. Define relation R on C by Pi R Pj if Pi ≤ Pj - that is, Pi is a refinement of Pj.
(a) Verify that R is a partial order on C.
(b) For A = {1, 2, 3, 4, 5}, let Pi, 1 ≤ i ≤ 4, be the following partitions: P1:{1, 2}, {3, 4, 5}; P2: {1, 2}, {3, 4}, {5}; P3: {1}, {2}, {3, 4, 5}; P4: {1, 2}, {3}, {4}, {5}. Draw the Hasse diagram for C = {Pt | 1 ≤ i ≤ 4}, where C is partially ordered by refinement.