Math 302, Transition to Higher Mathematics
(7:25-8:40 MW in C-10)


Grades are posted. You may see your scores on Blackboard and final grade on Webportal.

Syllabus: Here is the syllabus.

Online resources:
 Homework:

Assigned
 Due date Read
 Exercises
1/28
 2/4
 Your syllabus
1.1-3
 Study your syllabus. There will be a quiz on it on Mon, 2/2.
1.2.1, 4, 6, 12-15
2/4
 2/11
 1.4
 1.3.1, 5-7, 9, 12
2/11
 2/18
 1.4
 1.3.4, 8, 11
1.4.1, 2
2/18
 2/25
 1.5
 1.4.3, 4
2/25
 3/4
 1.5, 2.1
 1.5.2-4, 6-8
3/4
 3/11
 1.5, 2.1 1.5.5, 9

Translate the following sentences into statements that Block World understands. Manipulate the objects on the board to test if your statement really does what it's supposed to.
(a) There is a triangle that is between a square and a pentagon.
(b) Every medium pentagon is in the same column as a large triangle.
(c) The only medium triangle is b.
(d) There is only one large pentagon.
(e) There are at least two medium squares.
3/11
 3/18
 2.1 1.5.1, 10-12
3/23
 4/6
 2.2
 2.1.1
2.2.2-5
4/6
 4/13
 2.3, 4
 2.3.2, 5-8
4/13
 4/20
 2.5, 6
3.1, 2
 2.4.2, 3, 9, 10
2.5.2, 4
4/20
 4/27
 3.3, 4
 2.5.6, 10
2.6.1
3.2.7, 11, 13, 15
4/27
 5/4
 5.1, 2
 3.3.6-8, 11, 20
3.4.2
5/4
 5/11
 5.2, 3
 5.1.2-5, 9, 11
5/11
 5/18
 5.3
 5.2.2, 6-8
5.3.1

The problem of the fortnight: The Mathematics Department in San Diego posts a new fun problem periodically.  If you'd like to submit a solution, give it or e-mail it to me, so that I can fax them all together to San Diego. Winners receive prizes.

The problem of the week:
  1. Posted 2/2, due 2/11. 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?

  2. Posted 2/13, due 2/23. In the distant country of Logica, there are two kinds of people: square shooters, who always tell the truth, and liars, who always lie. On one of your travels to Logica, you come across three of the locals--let's call them A, B, and C--in a busy inn. You ask A, "How many of the three of you are square shooters?" Unfortunately, you can't understand his answer from the noise. "What did A say?" you ask turning to B. "He said one of us is a square shooter." replies B. Finally, C says, "Don't trust B, she is a liar."
    What are B and C? Can you tell if A is a square shooter or a liar? Did A really say that one of them was a square shooter? If not, can you tell what he said?

  3. Posted 2/23, due 3/2. There is a small island off the coast of Logica, where there are two types of people: one type lie on Mon, Tue, and Wed and tell the truth on other days. The other type lie on Thu, Fri, and Sat and tell the truth on other days. According to ancient custom, people of one type marry only people of  the other type. One time, when I visited this island, I ran into a young couple. The woman said "If I told the truth yesterday or lied the day before yesterday then my husband tells the truth today." The man said "If I tell the truth today, then I my wife will tell the truth tomorrow." Can you tell what day of the week it was? Can you tell what types the man and the woman were?

  4. Posted 3/3, due 3/9. 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. In fact, every child in Logica knows how to solve that problem because it is a popular nursery rhyme in that country. But I digress. Suppose that you go treasure hunting in Logica and manage to find the two doors in the above puzzle. To your dismay, one of the two guards has been laid off as a result of corporate downsizing. Of course, you don't know which one. You are still allowed to ask the guard only one yes/no question. What question do you ask to find out which door leads to the treasure?

  5. Posted 3/9, due 3/16. Rumor has it there is a place in Calexico where every drink sold costs only $0.01. Anytime, not only during happy hour. Find this place.

  6. Posted 3/16, due 4/6. Since I am not going to be around next week, I will post two problems this time. Each is worth the usual 1% extra credit. 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 there is a person in Logica who believes that he believes P. Does this imply P must be true? Now suppose that there is a person in Logica who says "I believe P." Does this imply P must be true?

  7. Posted 3/16, due 4/6. Here is the other problem. Same setup as problem 6, with half the population of Logica under the spell. Find two conditional statements P and Q such that
    1. P and Q are converses of each other.
    2. Neither statement follows from the other.
    3. If a person from Logica states either P or Q then it follows that the other statement must be true.

  8. Posted 4/8, due 4/15. Remember that every native of Logica is either a square shooter or a liar. Among the natives I know in Logica, the following is true. For every native A, there is a native B such that B says both A and B are liars. How many natives do I know?

  9. Posted 4/16, due 4/27. 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 people form a club too. Can you say anything about the number of sneaky people in town?

  10. Posted 5/5, due 5/11. 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?

  11. Posted 5/5, due 5/11. Given a nonempty set of people S, let * be the relation defined by x*y if x is a friend of y. Let * be symmetric (i.e. if x is a friend of y, then y is also a friend of x) but not reflexive (i.e. nobody is his/her own friend). Now, let n be any positive integer. Prove that in any group of n people, there is either someone who has no friends, or there are two different people who have the same number of friends.

  12. Posted 5/11, due 5/18. Solve any one of the problems of the week you haven't already solved, except problem 5.
Exam solutions:
 Educational links:
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