Homework for Logic and Proof (Mhf 2300)
Day 1 (1/8/07) Section 1.1
1) Do exercises #2, #3, #5, #7.
Day 2 (1/10/07) Section 1.2
1) Read preview activities 1 and 2 on pages 13 and 14.
2) Do exercise #2 parts (b) and (c) by writing out a formal proof for each part using the definition of odd and even integers.
3) Do #3 part (a) by writing out a proof similar to the proof for Theorem 1.6 found at the bottom of page 19.
4) Do all of exercise #9
5) Do problem #10 parts (a) and (b) only
6) Read the article given out in class and be prepared to discuss it next time we meet.
Day 3 (1/17/07)
1) Do preview activity 1 on pages 76 - 77
2) Read pages 79 - 83
3) Do exercise #1 part a, b, and d on page 89
4) Read "Additional Writing Guidelines 1-4" on page 87
Day 4 (1/22/07)
1) Read pgs. 129-133 (The Division Algorithm material). Pay special attention to the proof of Proposition 3.28 and bring questions about it to discuss in class on Wed.
2) Do Preview Activities 1 and 2 of section 2.1
3) Do exercise 3 from section 3.4 on page 125.
4) Print out and read the directions for the Proof Portfolio and the Number Theory Portfolio.
5) Start working on ONE problem for the portfolio.
Day 5 (1/24/07)
1) Do preview activity 1 on pg. 120
2) Do exercise #5 on pg. 125 (this exercise uses preview activity 1 on page 120)
3) Do preview activity 3 on page 38
4) Read examples 2.6 and 2.7 on pages 40 - 41
5) Do exercises #1, 2, 3, 8, 11 in section 2.2
6) Submit a problem for the proof portfolio (be sure to use an equation editor).
Day 6 (1/29/07)
1) Read pages 41-43 (Do Progress Check 2.8 AND Activity 2.10 on page 43)
2) Do Preview Activities 1 AND 2 on pages 93 - 94
3) Read Theorem 3.6 (and its proof) on page 96 - 97.
4) Do # 10 on page 105. You may find it helpful to APPLY the contra-positive of problem #3b on page 24 when working this problem.
5) Prove the following statement: If n is an integer then n(n+1)(n+2) is divisible by 3. Hint: apply The Division Algorithm on the integers n and 3 and use cases on the remainder.
Day 7 (1/31/07)
1) Read ALL of section 3.3 EXCEPT for (Progress check 3.17 and Activity 3.19)
2) In section 3.2 do HW exercises #2d, #8, and #9.
3) In section 3.3 do HW exercises #1 (very important), #5, and #7c,d. Note: for your proof of #5 in section 3.3 you will need to use the results from #2d in section 3.2. Also you will want to use the proof of Theorem 3.18 on pg. 113 as a "template" and guide when you write up the proof for #5.
4) Submit a 2nd problem for the Proof Portfolio.
Day 8 (2/5/07)
1) Do preview activity 3 on page 77 - 78
2) Read the section on "Congruence" on pages 84 - 85. Pay special attention to the definition on page 85 and commit this to memory.
3) Do progress check 3.3 at the bottom of page 85.
4) Come to class with questions about the proof of theorem 3.4 starting on page 86.
5) Do exercise #7 (both parts) in section 3.1
6) Use proof by contradiction to prove that any integer n cannot be both even and odd. We proved this result before using direct proof in conjunction with the division algorithm. Recall that the unique remainder guaranteed by the division algorithm upon division of a given integer by 2 is either 0 or 1. If the remainder is 0, the given integer is even. If the remainder is 1 the given integer is odd. We can use proof by contradiction (also known as indirect proof) to prove the same result without resorting to the division algorithm. To get you started, we begin such an indirect proof with the statement: "By way of contradiction, assume that n is an integer that is both even and odd." Next apply the definitions of both even and odd integers and work towards a contradiction.
Day 9 (02/7/07)
1) Do exercise #9a in section 3.1
2) Do exercise #3 (all parts) in section 3.5
3) Read the sections entitled "Another Look at Congruence" and "Congruence and the Division Algorithm" on pages 133 - 137. Note: Pay special attention to Proposition 3.33 on page 136. Next, compare this with Proposition 3.28 at the bottom of page 132. You will notice that the two propositions are identical. Finally, compare the proofs of both propositions and be prepared to come to class to discuss the similarities and differences of the respective proofs. Which proof do you like best and why? What advantages does the proof of the former proposition have over the latter?
4) Read example 3.34 and Do activity 3.35 on page 137
5) Submit your 3rd proof portfolio problem
Day 10 (02/12/07)
1) Do #14 (part a only), 18 (both parts) from section 3.5
2) Read "Negations of Quantified Statements" on page 60 and "Counterexamples and Negations of Conditional Statements" on page 62.
3) Read "Statements with More than One Quantifier" and "Writing Guideline" on pages 64 - 67.
4) Do #3a,b,c,d AND #4a,b,c,d AND #6 in section 2.4
5) Start reviewing for TEST 1 (see announcement section of website).
Day 11 (02/14/07)
1) Submit your 4th portfolio problem
2) Do preview activity 1 on page 151 - 152.
3) Read the subsections entitled "Set Equality, Subsets, and Proper Subsets" and "Operations on Sets" on pages 154 - 155 and 157 - 158
4) Study for TEST 1
Day 12 (02/19/07)
1) Do exercise #4 in section 3.5
2) Read the subsections entitled "The Power Set" and "Venn Diagrams" on pages 155 - 156 and 159 - 161 in section 4.1
3) Do exercises #1, 2, 3, and 7 in section 4.1
4) Do preview activities 2 and 3 in section 4.2 and Read "The Choose-an-Element Method" subsection on pages 169 - 170
Day 13 (02/21/07)
1) Do progress check 2.17 on page 53
2) Read the subsections entitled "Proving Set Equality" and "Disjoint Sets" on pages 172 - 174 (pay special attention to the proof of proposition 4.14). Also read the section entitled "Using the Choose-an-Element Method in a Different Setting" on page 174 - 175.
3) Do activities 4.16 and 4.17 on pages 175 - 176
4) In section 4.2 do exercises 1, 2, 3, 10, 11 (parts a and b), and 13a
5) Do preview activity 1 on page 180.
6) Read Theorem 4.18 on page 182 and read the selected proofs on page 183 - 184. Pay special attention to the proof on page 184 as this will help you in doing Problem 5 in the Proof Portfolio.
7) Memorize the results of Theorem 4.21 on page 185 and read the proof of one of DeMorgan's Law.
8) Do activity 4.23 on page 187.
9) In section 4.3 do exercises 4 and 7 on page 188 - 189.
10) Submit your 5th portfolio problem.
Day 14 (02/26/07)
1) Do exercises #1 (parts a, b, c, d only), and #2 (parts a, c, e, g only) in section 4.4 You may need to read some of the more important highlights of section 4.4.
2) Read ALL of section 6.1
3) Do exercises #1, 4, 5 (parts a, b, c only), and 6 in section 6.1
Day 15 (02/28/07)
1) Do preview activity 1 in section 6.2
2) Read the subsections entitled "Examples of Functions Using Verbal Descriptions" and "Functions Involving Congruences" and "Equality of Functions" and "Sequences as Functions" in section 6.2
3) Do Progress Checks 6.8, 6.9, and 6.10 in section 6.2.
4) Do exercises #1, 6 (parts a, b, c, g, and h) in section 6.2
5) Read the subsections entitled "Consequences of the Definition of a Function" and "Injections" and "Surjections" and "The Importance of the Domain and Codomain" and also "Bijections" in section 6.3. Pay careful attention to any DEFINITONS given in this section.
6) Do exercises #1, 3, 10, 13 (part a), and 14 (part a) in section 6.3.
7) Submit your 6th Portfolio Problem.
Day 16 (03/05/07)
1) Do Activity 6.6 on page 273
2) Do Preview Activity 2 on page 219
3) Read the "Procedure for a Proof by Mathematical Induction" on pages 222 - 223
4) Read Proposition 5.2 on pages 223 - 224 and carefully read its proof. Bring questions next time to class regarding this proof. Also read the "Writing Guideline" at the bottom of page 224.
5) Read "Summation Notation" on page 225 and "Some Comments about Mathematical Induction" on page 226.
6) Start/Continue working on your Number Theory Portfolio.
Day 17 (03/07/07)
1) Do exercises #3 (part a only), 6 (all parts), 7 (all parts), 8 (part a only), and 9 in section 5.1.
2) Do Preview Activities 1 and 2 in section 5.2
3) Read the "Domino Theory" on page 237 - 238 as well as "Using the Extended Principle of Induction" and review proposition 5.7 on page 239.
4) Read "The Second Principle of Mathematical Induction" on pages 240 - 242.
5) Get caught up with your Number Theory Portfolio.
6) Submit your 7th Proof Portfolio Problem.
7) Start reviewing for TEST 2 (see announcement section of website)
Day 18 (03/19/2007)
1) Do problems #1a, 2 (all parts), and #5 from section 5.2. Problem #5 takes priority over the rest of the problems in this section for now. I say this because we will be discussing it in class next time so make sure you do a very nice job on it.
2) Continue working on your Number Theory Portfolio. Also, remind me next time to give you very important information about the number theory portfolio.
3) Start studying for the test next week. See the link for the study guide in the announcement section of the website.
Day 19 (03/21/2007)
1) Read all of section 7.4
2) Do problems #1 (parts a, b), #2 (parts a, b, c, and f), #4 (parts a, b), and #5 (both parts) in section 7.4
3) Work on more problems in your Number Theory Portfolio as we discussed last time in class.
4) Review for TEST 2.
5) Submit your 8th Proof Portfolio Problem.
Day 20 (03/26/2007)
1) Study for the TEST. Be sure you know the definitions by heart.
2) Last time in class we did not have sufficient time to go over one of your HW questions namely problem #13 on page 303. Please CLICK HERE to see a complete solution to this problem. Going through this problem with a "fine tooth comb" should aid in your understanding of injective and surjective functions.
Day 21 (03/28/2007)
1) Do activity 7.22 on page 391 AND activity 7.23 on pages 391 - 392. Use these activities to guide you in answering a previously assigned HW problem namely #5 on page 393.
2) Do exercises 9 and 10 on page 393 in section 7.4.
3) Do Preview Activities 1, 2, and 3 in section 8.1
4) Read the subsections entitled "The System of Integers" and "Remarks about the Greatest Common Divisor" in section 8.1
Day 22 (04/03/2007)
1) Get caught up with all you HW.
2) Work on both the Number Theory and Proof Portfolios.
Day 23 (04/04/2007)
1) Do exercises #1 (parts a, b, and c only), #3 (part a. only), #5 (parts a, b, and c only), #7 (parts a, b, and c only) in section 8.1.
2) How many natural number divisors of 72 are there? List them all out in prime factored form.
3) Use Theorem 8.3 on page 404 to prove that gcd(a, b) = gcd((a - b), b) for any pair of integers with a not equal to zero and b > 0. This is not as difficult as it looks but you do need to be very aware of what it is your are trying to prove and what is given to you.
4) Do Preview Activity 1 in section 8.2 and read the subsection entitled "Greatest Common Divisors and Relatively Prime Integers" in section 8.2.
Day 24 (04/09/2007)
1) Read the proof of Theorem 8.13 on page 413 - 414.
2) Do Progress Check 8.11 on page 412.
3) Read the subsection entitled "Comments about Prime Numbers" in section 8.2
4) Prove the second part of the Euclidean Algorithm as shown in class. (i.e. show that D2 is a subset of D1)
5) Do exercises 1 and 2 in section 8.2.
6) Read the proof to Theorem 8.17 on page 417. This is a proof that goes back thousands of years!
Day 25 (04/11/2007)
1) Get caught up with all previous HW!!!
2) Put the finishing touches on your Proof Portfolio. I will hand back problem #10 on Monday so you can rewrite it for Wednesday.
3) Work on Number Theory Portfolio.
Day 26 (04/16/2007)
1) Do exercises 1, 2 (parts a-d), and 5 in section 8.3
2) Make sure you know how to do exercises 1 and 2 from section 8.2 for the exam next Monday.
3) Review the Divisibility Rule for nine for the exam next Monday as well.
4) Continue working on both of your portfolios.
Day 27 (04/18/2007)
1) Check you ATLAS email. I sent everyone some important information.
2) Do Activity 8.28 on page 428. This activity will help you with a problem in the Number Theory Portfolio.
3) Complete BOTH the Number Theory Portfolio and the Proof Portfolio for Monday.
4) Study for the Final Test.