A local food critic has recently published her official ranking for delicious-ness of mushrooms. She gives her ranks relatively as follows: - Cremini is better than Button - Cremini is better than Portobello - Chanterelle is better than Cremini - Morel is better than Chanterelle - Shitake is better than Enoki - Enoki is better than Cremini - Morel is better than Porcini - Porcini is better than Cremini Assume these rankings are in fact covers of an ordering on the set of mushrooms M={ Porcini, Chanterelle, Enoki, Shitake, Morel, Cremini, Portobello, Button }. Answer the following: (a) This ordering is clearly a partial order. List all the pairs of incomparable mushrooms. You do not need to list symmetric pairs; if (A,B) is an incomparable pair, then obviously so is (B,A). (b) Which mushrooms are highest ranked? Which mushrooms are lowest ranked? (Find the minimal and maximal elements). (c) With the addition of exactly two relative rankings (e.g. "A is better than B"), this ranking can be transformed into one with a unique maximum and minimum. Give two such rankings. Explain why these two choices work by stating the now unique maximal and minimal elements and explaining why each is the maximal or minimal. (Hint: use the definition of maximal and minimal). (d) Unfortunately, there are still incomparable pairs of mushrooms from the new ordering produced in part (c). Which are those incomparable pairs? (e) Draw a Hasse diagram of your ordering from part (c)