MAT 660, Autumn 2004, Dr. S. Epp
Discrete Structures for Teachers

ASSIGNMENT 1

1. Read the following sections:

1.1: Focus on understanding the concept of logical form and De Morgan's laws. Also learn how to test for logical equivalence using a truth table.
1.2: Focus on learning alternate ways to express if-then statements and being able to formulate negations for them.
1.3: Focus on learning the concept of validity, on knowing the meaning of modus ponens, modus tollens, converse error, and inverse error, and on recognizing forms of valid arguments (but except for modus ponens and modus tollens you don't need to learn their names). Also learn how to test for validity using a truth table.
2.1: Focus on understanding statements that contain quantifiers, learning to express them in a variety of ways and being able to write negations for them.
2.2: Focus on understanding the meaning of statements containing more than one quantifier and being able to write negations for them.
2.3: Focus on understanding the meaning of universal instantiation and recognizing universal modus ponens, universal modus tollens, and converse and inverse errors for quantified arguments.
3.1-3.4: Focus on learning to use definitions to write short proofs of universal statements and on learning how to show that a universal statement is false.
3.6: Focus on understanding the ideas behind proof by contradiction and proof by contraposition.
3.7: Learn the proofs of the irrationality of the square root of 2 and the infinitude of the set of prime numbers.
4.1: Focus on learning how to manipulate terms of sequences and work with summation notation.
4.2: Focus on learning how to use mathematical induction to prove formulas and on being able to apply the formula for the sum of the first n integers and formula for the sum of the terms of a geometric sequence. Be sure to write your answers in a way that does not assume what needs to be proved.

2. Do the following exercises. Check your work against the answers given but not before you've done the exercise without looking at the answer! You will not hand in these exercises, but you will be held responsible for knowing how to do them. It is important not to get bogged down in the first two chapters. You should be starting to work the exercises in Chapter 3 by September 26, even if you have not finished all the exercises in the first two chapters. It is okay to work on Chapter 3 and Chapters 1 and 2 simultaneously.
1.1 #1, 3, 6, 8ad, 14, 25, 29, 31, 35, 37
1.2 #3, 14a, 16, 19, 20adf, 22adf, 23adf, 24, 28, 32, 35, 43, 45, 47
1.3 #1, 3, 6-8, 24-27, 39
2.1 #2, 3, 7ac, 9, 11, 13, 14, 16ace, 17a, 19, 21c, 22a, 23a, 25ab
2.2 #3ac, 4ac, 5ac, 6a, 9, 11, 13, 15ac, 17a, 18, 20, 22, 24, 27, 29, 31, 38, 40
2.3 #2ab, 3ab, 4a, 5, 7, 13a, 16, 18, 41 [Add 21]
2.4 #1bd, 2, 3, 5, 7-10, 16, 18
3.1 #1, 2a, 4, 11, 14, 17, 19, 20b, 22b, 24, 25, 29, 31, 34-36, 39-41
3.2 #1, 3, 4, 6, 8b, 9, 11-13, 20, 33, 34
3.3 #1, 4, 6-8, 10, 12, 14, 15, 29
3.4 #1, 3, 5, 13, 17, 27, 28a
3.6 #1, 3, 5, 8, 10, 16, 17, 19, 21, 28a, 29a
3.7 #1, 2, 3, 5, 7, 9, 16
4.1 #1,10-12, 14, 19, 20, 23, 27, 29, 32, 35
4.2 #3, 5, 6, 8, 10, 13

3. Hand in the following exercises on or before October 11. You may either (1) post them in the Blackboard “Drop Box” (having either scanned them to convert them to electronic form or typed them doing the best job you can with the mathematical symbols) or (2) you may send them by ordinary mail (postmarked no later than October 11). If you write your solutions using pencil and you scan your work, make sure it is dark enough to be readable. A third option is to fax them to me (847-256-6399), but in that case also be sure that your writing is dark enough to be read. These problems will be graded generously, with a significant amount of credit given for a serious attempt to solve them. I will post the solutions on the Blackboard website on October 12.
2.1 #21d, 22b, 29
2.2 #12, 14
[Add 2.3 #19, 22]
2.4 #17, 18
3.1 #26, 30, 56
3.3 #16
3.4 #29
3.6 #2, 9, 20, 28bc, 30
3.7 #25, 27
Extra Credit: 1.3 #38, 40; 2.4 #29, 32, 34

4. Hand in the following exercises at the beginning of class on October 16.
4.1 #33
4.2 #15

5. Start working on an idea for a lesson segment that incorporates material from this course.

6. Prepare for the midterm exam. Part I of the exam will consist of definitions questions; Part II will consist of problems similar to those that have been assigned as homework. You may bring a 3"x5" crib sheet to the exam which may contain any information you wish EXCEPT actual proofs or proof fragments. You will also need to be able to write the following definitions without using any books or notes: odd integer, even integer, prime number, rational number, divisibility of one integer by another. So while you will be not be allowed to use your crib sheet for the definitions questions, you may use it freely for the rest of the exam.