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?

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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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Essentials of DISCRETE MATHEMATICS:

Section 2.5- Partial Orderdings

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 x, y EX are two different products, we say
x 3y if the customer prefers y over x. (In order to satisfy the reflexive
property, we stipulate that x 3x for all a E X.) Suppose you know the
following things about your customer.
that
Customer prefers:
Over:
lettuce
broccoli
cabbage broccoli
cabbage
tomatoes
carrots
cabbage
carrots
lettuce
asparagus
lettuce
mushrooms
tomatoes
corn
tomatoes
corn
carrots
eggplant
carrots
eggplant
asparagus
onions
mushrooms 2
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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