Last modified Mar 11, 2008 9:07 AM

For the solutions of Exam 2, see Solutions, 0-3 and Solutions, 4-7 (pdf documents).

The comments below address general issues with the mistakes which were made in the attempts to solve these problems. Refer to the links above for the original problems.

0. The fact that “IF p, THEN q OR r” is equivalent to “IF p AND NOT q, THEN r” was part of Lesson 5 and was the technique you were asked to use to solve Exercise 4.8.
   The fact that this is also equivalent to “IF p AND NOT r, THEN q” was mentioned in class and should be obvious from the symmetry of the original statement in the statement variables q and r.
   The fact that each of these equivalent statements is equivalent to the conjunction of the latter two (i.e., that (C) on the exam is equivalent to (A) and to (B)) is a consequence of the fact, which you should be able to understand, that if two statements,  a and b, are equivalent, then each is equivalent to the conjunction  “a AND b”.
   
   
1. This problem, dealing with the inequality  2x/(x - 1) ≥ 1,  checks to see if (i) you really knew how to do Exercise 4.8 (which is best done by using Problem 0(A)) and, most important, (ii) if you really understand the Common Error 4 in Project I (Common Errors).
   
   Some students preferred to do this problem “by cases” instead of using the equivalence suggested by Problem 0(A) above and suggested in connection with Exercise 4.8 in the homework. In this context, this is a waste of time. For a discussion of this, see By Cases.
   
   Another error was to look at only INTEGERS less than 1, instead of considering all real numbers less than 1. This error was discussed in the Common Error 10 in Project I (Common Errors).
   
   
2. This problem is about the properties of the “divides” relation, which was discussed in Exercise 6.11(a). The problem is modified a little here since the number 0 (zero) is included in the set on which the relation is defined. The important issues (and common errors) here are:
   1. As pointed out in the Lesson in which Ex. 6.11(a) was assigned, you don't need to use division to prove things about the “divides” relation; it is purely a multiplicative property.
      
   2. Most important, if you nevertheless want to do this problem by using division, you can't divide by zero, and you can't divide by a variable unless you're sure the variable is not zero.
      
   3. You have to investigate the properties of the relation by using the definition given, and one of the results of the definition given is that zero divides zero (remember, “divides” is a MULTIPLICATIVE property).
      
   4. To prove that a universally quantified statement is false, usually the best way is to GIVE A COUNTEREXAMPLE.  To prove that a universally quantified statement is false, you do NOT try to prove that it is true, show that the proof breaks down, and then claim that the statement is false.
   5. From the first day of class, the importance of not doing proofs backwards has been emphasized. Try hard to avoid doing proofs backwards! See also the Common Error 15 in Project I (Common Errors).
   
   
   
3. Do you know the definition of a relation, and the definition of a relation between two sets?
   Do you understand set notation well enough to know that, although 0 and 1 are both integers and real numbers, they are NOT elements of the set E given in Problem 4?
   
   
4. Do you know what an equivalence relation is and what an equivalence class is in the context of an equivalence relation on a set?
   
   
5. A proof of the fact that the product of odd numbers is odd was posted on the course web site. Part (c) emphasizes the fact that if you already know (from part (b)) that the product of two odd numbers is odd, then essentially no more work is required to prove that the square of an odd number is odd.
   
   
6. Do you really understand the definition of “transitive” ?  Examples like this were done in class.
   
   
7. Do you know how to start the proof that a relation on a set — ANY relation on ANY set, even if you have never heard of it before — is reflexive?

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