17. Suppose you want to write a program that will collect information on a customer's tastes and customize web content accordingly. By monitoring online shopping habits, you are able to collect a set of pairwise preferences on a set X of products. If a, y EX are two different products, we say that x y if the customer prefers y over a. (In order to satisfy the reflexive property, we stipulate that a 3 a for all a E X.) Suppose you know the following things about your customer. Customer prefers: Over: lettuce broccoli cabbage broccoli cabbage tomatoes carrots cabbage carrots lettuce asparagus lettuce mushrooms tomatoes corn tomatoes corn carrots eggplant eggplant carrots asparagus onions mushrooms 8 onions corn In order for (X,<) to be a poset, we must also assume that the customer's preferences are transitive. (a) Draw the Hasse diagram for (X,3). (b) What is/are the customer's favorite vegetable(s)? (I.e., what are the maximal element(s)?) What is/are the least favorite? (c) Use topological sorting to rank order these vegetables according to the customer's preferences. Is the ranking unique?

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.