Math 181: A Mathematical World
Fall 2008
Instructor: Tom Cooney
Email: tcooney@math.uiuc.edu
Classroom: 143 Altgeld
Time: 9:00-9:50pm Monday, Wednesday, Friday
Office: 150 Altgeld
Office Hours: 2:00-2:50pm Wednesday, 2-2:50pm Friday, 3-3:50pm Thursday or by appointment.
Textbook: For All Practical Purposes: Mathematical Literacy in Today's World, Seventh Edition, W.H. Freeman and Company.
Course Description: Introduction to selected areas of mathematical sciences through
application to modeling and solution of problems involving networks, circuits, trees, linear
programming, random samples, regression, probability, inference, voting systems, game
theory, symmetry and tilings, geometric growth, comparison of algorithms, codes, and data
management.
Homework assignments, exam dates and other course details will be posted on
this website.
Here is the syllabus.
You can check your grades at the compass website.
This course adheres strictly to the University's Student Code Part 4 Policy on Academic Integrity. Cheating of any kind on the homeworks, exams, or the final will result in a letter being placed in your permanent school file and may result in an immediate F in the course.
Links This week's homework assignment
Solutions, review sheets to Hour Exams 1 and 2
Quiz 10: Friday, the 21st of November
Hour Exam 3: Friday, the 5th of December
Final Exam: Monday, 15th of December
Homework Assignments
Homework Assignment 1
Chapter 1: (p. 25) Problems 4, 8, 9, 21, 24, 26, 36, 37, 48, 53.
Due Date: Friday, the 5th of September.
Homework Assignment 2
Chapter 2: (p. 64) Problems 1, 11, 23, 27, 29a), 35, 36, 40a)b), 42.
Due Date: Friday, the 12th of September.
Homework Assignment 3
Chapter 2: (p. 72) Problems 49, 51a)b)c), 63 (and EXPLAIN your answers!), 68
Chapter 3: (p. 110) Problems 4, 5
Chapter 3: (p. 112) Problems 12, 19
Chapter 3: (Skill Check, p. 108) 2 (Explain) , 3 (Explain)
Due Date: Friday, the 19th of September.
Homework Assignment 4
Chapter 3: (p. 112) 15, 18a)b), 34, 36, 43, 46, 47
(and depending on how much we cover: 63)
Chapter 3: (Skills Check, p.108) 12
(and depending on how much we cover 18, 20)
Due Date: Friday, the 26th of September.
Comments on Homework 4 and Quiz 4
Homework Assignment 5
Chapter 3: (p.110) 45, 46, 63, 65.
Due Date: Friday, the 3rd of October.
Homework Assignment 6
Chapter 9: (p. 364)
Problems 2, 3, 5, 7, 9a)b), 10a)b), [but not 24a)]
Due Date: Friday, the 10th of October.
Homework Assignment 7
Chapter 9: (p. 364)
Problems 8, 10c)d), 13, 17, 20, 24, 27, 28, 33
Due Date: Friday, the 17th of October.
Homework Assignment 8
Chapter 10:
Problems 2, 4, 7, 8, 9, 12
Due Date: Friday, the 24th of October.
Homework Assignment 9
Chapter 11: See this pdf file for this week's homework assigment.
As announced in Wednesday's class, question 8 of the homework assignment will not be due until next week. The rest of the assignment is still due Friday but question 8 can wait until we have discussed that topic further.
My thanks to Melissa Dennison for problems on Homework Assignments 9 and 10.
Due Date: Friday, the 7th of November.
Homework Assignment 10
Chapter 11: See this pdf file for this week's homework assignment.
Problem 5 (Banzhaf power index with many voters) is postponed and will be due Friday, 21st instead of Friday, 14th. All other problems are still due Friday, 14th.
Due Date: Friday, the 14th of November.
Use combinations to compute the Banzhaf power index of the weighted voted
system
[7:3,2,2,2,2,2].
(This is the postponed problem from homework 10.)
Chapter 11: Problem 17, p.426 - a little more practice with combinations.
Chapter 13: Problems 2,3,7, p. 498
Due Date: Friday, 21st of November.
Some comments on this homework:
Adjusted winner procedure: Sometimes it will be necessary to transfer more than one item in order to make both people have the same number of points. Look at the point ratios (of the items belonging to the person with more points). We give (all or part of) the item with the lowest point ratio to the person with less points. If this person still has fewer points, we move on to the item with the next lowest point ratio and give (all or part of ) that item to the person with fewer points.
If there is an item that both people value equally, give it to the person with fewer points. If they now have different number of points, look at the point ratios to decide which items they should share ownership of.
Let's look at Question 7, allocation 1 (and you can do the other allocations by yourselves).
Proportional (for three people) means that each person thinks he or she is receiving one-third or more of the total. It is proportional if every person thinks they are getting one-third or more of 100 = 33.333... points. In this allocation, Bob has 10 points and does not think he is getting his fair share of 33 or more points. So allocation 1 is not proportional. (If there were 4 people, proportional would mean each person believes they are receiving one-fourth or more of the total, and so on.)
Envy-free: What does envy mean? It means you look at what someone else has and want it for yourself; you think that they are getting a better share than you are; you prefer what they have to what you have. In Allocation 1, Bob has 10 points. Carol has received Y. In Bob's opinion, Y is worth 50 points. Bob feels envy; he thinks he has received 10 points and that Carol has received 50 points. He thinks Carol is getting more than he is. He prefers Carol's share to his share. Allocation 1 is not envy-free.
Equitable: This means everyone receives the same number of points. If everyone thought they were receiving 50 points worth of goods, that would be an equitable distribution. If everyone thought they were receiving 20 points, that would be an equitable distribution. In allocation 1, Bob thinks he is getting 10 points and Carol thinks she is getting 40 points and Ted thinks he is getting 30 points. This is not equitable - they are not all walking away with the same number of points.
Pareto Optimal: Here's an example of a Pareto-optimal allocation. You split up $20 as follows, $10 to A, $10 to B. You can improve this (from A's point of view) by giving $15 to A and $5 to B, but B will not be happy about this. There is no way of improving upon this 10-10 split that will make one person happier without making someone else unhappy. Any way of changing things to help one person will harm the other.
Here's an example of an allocation that is not Pareto-optimal: allocation 1.
Bob gets Z for 10 points.
Carol gets Y for 40 points.
Ted gets X for 30 points.
Question asks you to change this allocation in a way that makes someone happier without hurting anyone else.
How about this?
Bob gets X for 40 points. 40>10 so Bob is happier.
Carol gets Y for 40 points. Still has 40 points, so Carol is as happy as before and has not been hurt by this change.
Ted gets Z for 40 points. 40>30 so Ted is happier.
This allocation is an improvement upon allocation 1: the changes made some people happier and the changes did not make anyone unhappy.
It would not work to give Y to Bob and Z to Carol. Yes, Bob would be happier but Carol would be unhappy as she went down from 40 points to 30 points.
Hour Exams 1 and 2
Hour Exam 2: Friday, the 31st of October
List of Topics for Exam 2
Review sheet from Wednesday's class
As noted, in the email I sent to the class, these two pdf files should also include Approval voting, which is definitely examinable material for Friday's exam. Good luck! Happy Halloween!
Solutions to Exam 2
Hour Exam 1: Friday, the 3rd of October
List of Topics for Exam 1
Review Sheet from Wednesday's class
Solutions to Exam 1
Probable Exam Dates
You will be given at least a week's notice of the actual exam dates.
The hour exams will be held (in class) near the following dates:
Hour Exam 1: Friday, 3rd of October.
Hour Exam 2: Friday, 31st of October.
Hour Exam 3: Friday, 5th of December.
The final will be held (in 143 Altgeld):
1:30-4:30 p.m. on Monday, 15th of December 2008. The final will cover the entire course.
Links
Tom Cooney's homepage
Solutions to and Comments on Quizzes and Homework Assignments
Check your grades at the compass website.
Wikipedia
A good source for interesting and often also true mathematical tidbits. Try
searching for
Euler, Konigsberg Bridge Problem, William Rowan Hamilton, Voting systems, Borda, Condorcet
The publisher of the textbook has computer applets online here, which you can use to get some more practice with the ideas in the course.
The very first graph theory problem ever, the bridges of Konigsberg
Lots of mathematical info from
Wolfram Research ,
one of the world leaders in mathematical computer software,
based right here in Urbana-Champaign.
The MacTutor
History of Mathematics archive : Mathematicians' biographies and more!
This is the webpage from the last time I taught this course. Approach with caution! Things will be different this semester.
