Math 302, Transition to Higher Mathematics
(5:35-6:50 MW in C-8)


Syllabus: Here is the syllabus.

Online resources:
 Homework:

Assigned
 Due date Read
 Exercises
8/30
 9/1
 To the Student, 1.1-2
 Turn in on 9/1: 1.2.3
9/1
 9/8
 1.3
 1.2.1, 4-6, 14, 15
Turn in on 9/8: 1.2.13
9/8
 9/15
 1.4
 Prove Theorem 1.3.1.(i)-(iii), (xi)-(xiii) and Theorem 1.3.2.(iv), (vii)-(ix), (xii)-(xiv)
1.3.5, 9, 12
Turn in on 9/13: 1.3.4
No new problem to turn in on 9/15.
9/15
 9/27
 1.5
 1.4.1-4
1.5.2, 3, 5, 6
Turn in on 9/20: 1.5.4
Turn in on 9/27: Translate the following sentences into statements that Block World understands:
(a) Every triangle is smaller than every pentagon.
(b) Every pentagon is the same size as some square.
(c) There is a triangle that is between a square and a pentagon.
(d) There is only one large pentagon.
(e) Every medium pentagon is in the same column as a large triangle.
9/27
 10/4
 1.5
2.1
 1.5.7-12
Turn in on 9/29: Choose sets S and T of objects that you can describe in Block World. Choose an open sentence P(x,y). Encode into Block World syntax each of the eight doubly quantified statements in Figure 1.5.1 on p. 48 in your textbook.
There is no new exercise to turn in on 10/4, but be sure to finish all your homework assigned up to now because it is all fair game for the exam on 10/6.
10/4
 10/11
 2.1, 2
 Homework holiday. Prepare for your upcoming exam. New HW will be assigned next week.
10/11
 10/18
 2.2-4
 2.1.1
2.2.3-6
2.3.2
Turn in on 10/13: 2.2.7
Turn in on 10/18: 2.3.2
10/18
 10/25
 2.4-6
 2.3.5-8
2.4.7,8
Turn in on 10/20: 2.3.4
10/25
 11/3
 3.2-3
 2.4.9, 10
2.5.4-6, 9, 10
3.2.4, 6, 8
Turn in on 10/20: 3.2.7
Turn in on 11/3: 3.2.11
11/3
 11/8
 3.3
 3.2.13, 15
3.3.6
No exercise to turn in this time. This is a short assignment so as to leave you time to prepare for the upcoming exam on 11/10.
Also, there is no new problem of the week. I haven't gotten any solutions to the last, so I have extended its deadline to 11/8.
11/8
 11/15
 3.3
 Turn in on 11/15: Prove Theorem 3.3.5.(v).
11/15
 11/22
 3.4, 4.1
 3.3.7, 8, 11, 12, 17, 22
Turn in on 11/17: 3.3.13
Turn in on 11/22: Prove Theorem 3.4.3.(iv).
11/22
 11/29
 4.2-4
 4.1.2, 5, 7
4.2.3, 6, 8, 13
Turn in on 11/29: Prove Theorem 4.2.3.(viii).
11/29
 12/6
 4.4, 5.1, 2
 3.4.3-5
4.4.12, 13
Turn in on 12/1: 4.4.1
Turn in on 12/6: 4.4.13. (1) and (2)
12/6
 12/8
 5.3
 Turn in on 12/8: 5.2.6

The problem of the week: You can earn extra credit by submitting a correct and carefully justified solution to the problem of the week by the deadline.
  1. Posted 9/8, due 9/20. In the distant country of Logica, there are three kinds of people: square shooters, who always tell the truth; liars, who always lie; and normal people, who sometimes tell the truth and sometimes lie. One time, I was sitting next to a Logican (that's what residents of Logica call themselves) and asked him "which kind of Logican are you?" Instead of answering, he handed me a note. After reading the note, I was certain that he was not a square shooter, but could not tell if he was normal or a liar. Then he passed me a second note. After reading this note, I knew I was dealing with a normal. What might he have written on the two notes?

  2. Posted 9/21, due 9/29. There is a well-known puzzle about two doors guarded by two guards one of whom is a liar and one of whom is a square shooter. Behind one of the doors is a treasure chest, behind the other is a bloodthirsty monster. You are supposed to find which door leads to the treasure by asking only one yes/no question to the guard of your choice. But you don't know which guard is which. You have probably heard this problem before; and if not, I trust you can figure out (or google) a solution in less than 5 min. Suppose that you go treasure hunting in Logica and manage to find the two doors in the above puzzle. To your surprise, one of the two guards is on furlough that day--budgets are tight in Logica too. Of course, you don't know which one. You are still allowed to ask the remaining guard only one yes/no question. What question do you ask to find out which door leads to the treasure?

  3. Posted 9/27, due 10/4. On one of your explorations in Logica, you go treasure hunting in the northwestern corner of the country. The people who live there all understand English just fine, but because of a strange historic accident--which no one quite remembers--instead of yes and no, they say boo and bah. Unfortunely, we don't know which of these means yes and which means no. Once again, you come across a two-door-situation: one door leads to the treasure, the other to a bloodthirtsy monster. One guard is guarding the doors. He is either a square shooter or a liar. He will  answer one boo/bah question. What question do you ask to find out which door leads to the treasure?

  4. Posted 10/4, due 10/11. Three men named Antonio, Beto, and Chuy went to the Fiestas del Sol with their wives. The ladies' names are Dolores, Eli, and Fernanda. Each of the six bought some trinkets. In fact, each bought exactly one kind of trinket, and of that kind as many as the number of pesos he (or she) paid for one trinket. (So for example, if the trinket costs 5 pesos, then (s)he bought 5 such trinkets. BTW, this implies that the price of each trinket is an integer number.) Each woman spent 45 pesos more than her husband. Dolores bought 17 trinkets more than Antonio, and Eli bought 7 trinkets more than Chuy. Who is married to whom?

  5. Posted 10/11, due 10/18. As we already stated, there are three kinds of people in Logica: square shooters, liars, and normal people. Suppose you meet a Logican, and you would like to know if they are normal. Is there a question you could ask them such that the answer is guaranteed to reveal if they are normal?

  6. Posted 10/18, due 10/25. An evil magician has cast a spell on about half of the population of Logica that made them lose their senses. Those that are under the spell are confused about the truth and believe all true statements to be false and all false statements to be true. So whatever a confused square shooter says is actually false, but whatever a confused liar says it true! For example, a sensible square shooter would say 1+1=2; a confused square shooter would say 1+1 is not 2 (because he/she believes 1+1 is not 2); a sensible liar would say 1+1 is not 2 (because he/she knows 1+1=2, but lies); and a confused liar would say 1+1=2 (because he/she believes 1+1 is not 2, but lies about it). Let P be some statement. Suppose a Logican does not believe that (s)he believes P. Does this tell you anything about whether P is true or false?

  7. Posted 10/25, due 11/15 (deadline extended). All my friends in Logica are square shooters or liars. They are not necessarily all the same type. For every friend A, there is a friend B such that B says both A and B are liars. How many friends do I have in Logica?

  8. Posted 11/15, due 11/22. Imagine a town, whose inhabitants are numbered 1, 2, 3, ..., n instead of having names. Each number between 1 and some finite number n belongs to one and only one person. The townies have a number of social clubs, which are also numbered instead of having names. Curiously, the number of clubs is the same as the number of inhabitants. Now, each townie can belong to one club, several clubs, or no club at all. To complicate matters a bit, someone who belongs to a club, can do so either openly or secretly. It is considered a good thing to belong to the club numbered the same as you. So a person who openly belongs to the club that has the same number as him/her is considered righteous. A person who secretly belongs to the club numbered the same as him/her is called sneaky. Now one more curious thing is that all the unrighteous--note that unrighteous is not the same as sneaky!--people form a club too. What can you say about the number of sneaky people in town?

  9. Posted 11/22, due 11/29. Suppose we have a chess club--that's a a club whose members play chess against each other--of which the following is known:
    1. Every club member has defeated at least one other member in chess.
    2. No club member has been defeated by more than one other member.
    3. There is a champion in the club who has never been defeated by anybody else.
    Based on the above information, can you tell if the chess club has more than 100 members? Why or why not?

  10. Posted 11/29, due 12/6. Imagine you are in a dark room with a table and 100 casino chips on the table. Each chip has one red side and one green side. 24 of the chips are turned so that the red side is up. The remaining chips have the green side up. Unfortunately, it's dark so you can't see the chips and can't tell which side is up. The two sides of the chips are identical to the touch. Your job is to separate the chips into two piles (not necessarily of equal size) so that each pile has the same number of chips with the red side up. How can you do this? You are allowed to flip chips if you wish.

  11. Posted 12/6, due 12/13. Solve any of the problems of the week you haven't already solved.
Exam solutions:
 The Putnam Competition: This is not strictly related to this course. The William Lowell Putnam Mathematical Competition will be held on Sat, Dec 4 this year. It is administered locally at the Calexico campus. The exercises primarily test your ability to construct rigorous mathematical arguments to solve (difficult) unfamiliar problems and not lexical knowledge. Here is an archive of past exercises and solutions.

Educational links:
Math links: The links below lead to sites with encyclopedias of math terms.  You can use them to find definitions, examples, and some theorems.
Some scholarship opportunities for prospective teachers: