Homework and Announcements for Logic and Proof

Day 27  (4-22-2009)

1)    Extra Credit: In section 8.3 do problem #1

2)    Submit the FINAL Proof Portfolio

3)    Submit the Number Theory Portfolio (Finish Section IV Linear Diophantine Equations)

4)     In preparation for the Final Exam, please review the proof of the Divisibility Rule for 9.  Be able to produce this proof on your own without looking.

5)    Take a look at TEST 3 Review as your prepare for the Final Exam.

Day 26  (4-20-2009)

1)    In section 8.2 do problem #2

2)    Finish-up Section III (Prime Numbers) in the Number Theory Portfolio.

3)    Put the finishing touches on your Proof Portfolio.

Day 25  (4-15-2009)   

1)     In the Number Theory Portfolio do #6, 7 from section I (Divisibility).  Also, do #3, 4, 5, 7 from section II (Congruence).  You will have to do some investigative work on your own to finish off section II.  You will need to look up a statement of Fermat's Little Theorem on the internet.

2)    In section 8.1 do problem #7 (all parts)

3)    Work on your Proof Portfolio

4)    Do Preview Activity 1 in section 8.2 and read the subsection entitled "Greatest Common Divisors and Relatively Prime Integers" also in section 8.2

Day 24  (4-13)

1)     Reminder:  The FINAL Proof Portfolio is due on Wednesday April 22, 2009 (last day of class).  You may continue to submit problems for review until Monday April 20.

2)     The Number Theory Portfolio is due on Monday April 27, 2009 (final exam day)

Day 23  (4-09)

1)    Look over section 8.1

2)    In section 8.1 do exercises #1 (parts a, b, and c only), #3 (part a only)

3)    How many natural number divisors of 72 are there?  List them out in prime factored form.

4)    Work on your Proof Portfolio.  We only have a couple of weeks left in the semester!!

Day 22  (4-06)

1)    STUDY FOR TEST 2   

Day 21  (4-01)

1)    Do exercise #5 (both parts) in section 7.3.

2)    In section 7.4, read the subsection entitled "The Integers Modulo n" and the definition at the bottom of page 387.

3)    Do Progress Check 7.19 on page 388.  (Solution at bottom of page 396)

4)    Do Activity 7.22 (parts 1 and 2 only) on page 391

5)    Do Activity 7.23 (all parts) starting at the bottom of page 391.

6)    Do exercises #1 (parts a, c), #2 (parts a, b, e, f), #13.  The arithmetic table for Z5 is on page 388.  Also #13c should ring a bell.  You did this one before as #15 in section 3.5 (remember the nine cases?)  Now with the power of modular arithmetic, cases are no longer needed.

7)    I made up a formal proof for the Divisibility Rule for 9.  We basically just gave an example in class to see why it worked but never gave a formal proof.   

8)    Continue to work on the Proof Portfolio.
 

Day 20  (3-30)

     1)   Do Preview Activity 1 in section 7.2 and review the definition on top of page 361.

2)  2)   Read the sections entitled “Directed Graphs and Properties of Relations” and “Definition of an Equivalence Relation” on pages 362 – 364.

3)  3)   Do Activity 7.9 on pages 367 – 368

4)  4)   Do exercises #3, 10, 14a in section 7.2

5)  5)   Review the definition at the bottom of page 374 and read the subsection “Congruence Modulo n and Congruence Classes” as well as “Properties of Equivalence     Classes” in section 7.3.

6)  6)   Read the proof of Theorem 7.11 on pages 376 – 377.

Day 19  (3-25)

1)    I suggest you start working on your Number Theory Portfolio if you have nor done so already.  At this point in the semester you should be able to complete problems #1, 2, 3, 4, 5 from the Divisibility section and #1, 2, 6 from the Congruence section and #1, 3 from the Prime number section.

2)    We have TEST 2 on Wednesday April 8, 2009.  Click on the Review Sheet to start your preparations for this test.

3)    I gave out the solution to problem #13 on page 303.  Use this solution as a help and reference as you complete Problem Eight in the Proof Portfolio.  You may hand-in a draft of this problem for Monday (see due dates on the first page of the Proof Portfolio).

4)    Read Theorem 6.22 and its proof on page 313.

5)    Do activity 6.23 on page 314 - 315.

6)    Know the two definitions on page 335 and prove Theorem 6.39 part 2 on page 339.

7)    In section 6.6 do problems #2a, c, e, f, g, h and #6    

Day 18  (3-23)

1)    Read all of section 6.3

2)    Do exercise #1 in section 6.2.

3)    Do exercises #1, 3, 10, 13, 14 (part a) in section 6.3.  Important: Bring to class YOUR solution to #13 (both parts) to share.

4)    Read the handout I gave in class on the relationship between Induction and the Well-Ordering Principle and bring questions about it to class on Wednesday.

5)    Work on the Number Theory Portfolio.

Day 17  (3-18)

1)    Read all of section 6.1

2)    Do exercises #1, 4, 5(parts a, b, c only), and 6 in section 6.1

3)    Do Activity 6.6 on page 273.   

4)    Do Preview Activity 1 in section 6.2.

5)    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.

6)    Do progress checks 6.8 , 6.9, 6.10 in section 6.2

7)    Submit another Proof Portfolio problem.  I suggest problem 5 or 6.

Day 16  (3-16)

1)    Prove that 7^n - 4^n is divisible by 3 for all natural numbers n by induction.  Note: This is a rather difficult induction problem so finish the rest of the HW first and come back to this one later.  One hint is to use the trick of adding and subtracting the "same amount" in the induction step.

2)    Finish the proof that every non-empty finite subset of the natural numbers has a least element using weak induction and the ideas given in class.

3)    Do exercises #3 (part a only), 6 (all parts) in section 5.1.

4)    Do preview activities 1 and 2 in section 5.2

5)    Read the "Domino Theory" on pages 237 - 238 as well as "Using the Extended Principle of Induction" and review proposition 5.7 on page 239.

6)    Read "The Second Principle of Induction" on pages 240 - 242.

7)    Do exercises #1 (part a only) in section 5.2.

Day 15  (3-4)

1)    Do activity 4.23 on page 187.

2)    In section 4.3 do exercises 4 and 7 on pages 188 - 189.

3)    Read section 4.4 pages 190 - 197.

4)    In section 4.4 do exercises #1 (parts a, b, c, d only), and #2 (parts a, c, e, f, g only), #3.

5)    Do Preview Activity 2 on page 219   

6)    We covered Mathematical Induction today in class.  Review the "Procedure for a proof by Mathematical Induction" on pages 222 - 223.

7)    Read Proposition 5.2 on pages 223 - 224 and carefully go over its proof.  Be sure to read the "Writing Guideline" at the bottom of page 224.  Bring questions to class next time about the process of Induction.

8)    Read "Summation Notation" on page 225 and "Some Comments about Mathematical Induction" on page 226.

9)    In section 5.1 do exercises #7 (all parts), #8 (part a only), and #9.

10)   Submit a 6th proof from your proof portfolio for the Monday following Spring Break.  If you did not turn in proof #5 you may turn that one in as well without penalty.  Problems 7 and 10 from the proof portfolio are suggested.

11)    Now would be a great time to start working on your Number Theory Portfolio so you won't have to worry about it all at once at the end of the semester.

Day 14  (3-2)   

1)    Do Problem #7 in the Proof Portfolio for Wednesday.

2)    Be prepared to discuss how 3) implies 1) in your notes today.  Do both a direct proof and a proof by contradiction.

Day 13  (2-25)

1)   The deadline to submit your fifth portfolio problem has been postponed to Wednesday March 4, 2009.  Problem #7 is suggested.  We will learn the final operation on sets (cross product on Monday)

2)   Do Progress Check 2.17 on page 53.

3)   Read the subsections entitled "Proving Set Equality" and "Disjoint Sets" on pages 172 - 174 (pay special attention to the proof of Proposition 4.14).

4)   Do Activity 4.17 on page 176

5)   In section 4.2 do exercises #1, 2, 3, 10, 11 (parts a and b).

6)   Do Preview Activity 1 on pages 180 - 181

7)   Read section 4.4   

Day 12  (2-23)

1)  Study for TEST on Wednesday.

Day 11  (2-18)

1)    Submit your 4th proof portfolio problem for Monday.

2)    Start to review for the TEST next Wednesday.  The review sheet is posted in the class handouts section of the website.

3)    Read pages 154 - 161.

4)    Do preview activities 2 and 3 in section 4.2 and read "The Choose an Element Method" subsection on pages 169 - 170.

Day 10  (2-16)   

1)    In section 4.1 do problems #1, 2, 3, 7.

Day 9  (2-11)

1)    In section 3.5 do exercise #3 (all parts), 4, 15, 18 (part a).  Hint: for #4 argue by contradiction and assume that there exists a natural number n such that sqrt(3n + 2) is a natural number.  Call this number p and consider the three cases (for congruence modulo 3) provided by Corollary 3.32 on page 136.  For #15 consider nine cases where only ONE of which will not lead to a contradiction.  I gave a big hint in class for #18.  Be prepared to share your answer to #18 (part a) in class on Monday.

2)    Submit your 3rd proof portfolio problem on Monday.

3)    Do preview activity 1 on page 151 - 152.

4)    Get caught up with everything because we start chapter 4 on Monday.

Day 8  (2-09)

1)    Review the section on Congruence on pages 84 - 87 including progress check 3.3.  Commit the definition on page 85 to memory.

2)    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?

3)    In section 3.1 do exercise #9a.  Note that this exercise is part 1 of theorem 3.30 on page 134.

Day 7  (2-04)

1)    In section 3.2 do exercises #2d, 8, 9

2)    In section 3.3 do exercises #1 (very important), #5, and #7c,d.  Note: for your proof of #5 you will need to use the results from #2d in section 3.2.  You may want to use the proof of Theorem 3.18 on page 113 as a "template" and guide when you write up the proof for #5.

3)    In section 2.3 do exercises #2, 3, 4, 6j,k,l,m.  We did most of #6 in class.  Your HW is to finish the last few i.e. parts j through m.   

4)    Read section 2.4.

5)    In section 2.4 do exercises #1, #2a,c,e AND #3a,b,c,d AND #4a,b,c,d.

6)    In section 3.1 do exercises #3, 7.

7)    Submit the 2nd portfolio problem for Monday. 

Day 6  (2-02)

1)    Do Preview Activity 1 on page 120.

2)    Do exercises #3, and #5 from section 3.4 on page 125.  For #3, consider the cases where q is even or odd after you apply the definition of odd integer to n.  For #5 use Preview Activity 1 from page 120.

3)    Use the fact that the square root of 2 is irrational as a corollary in proving the square root of 18 is irrational.  As a review, you may want to read Theorem 3.18 (and its proof) on page 113.  Remember that, in class, we added more detail to the proof in the book by considering the cases where m and n do have a common factor greater than one.

4)    Do exercise #10 on page 105.  (Hint: use the contra positive of: If n is even then n squared is even).

5)    Write out the details of the proof that for any integer n the quantity n(n+1)(n+2) is always divisible by 3.  Hint: apply the division algorithm and consider the cases for the remainder.

Day 5  (1-28)

1)   Prove:  If  xy  is an even integer AND x is an odd integer then y is an even integer.

2)   Complete the proof that the square root of two is irrational by considering the case where we left off in class, namely where p and q do have a common factor, and arrive at a contradiction.

3)   Read the rest of section 2.2.

4)   At the end of section 2.2 do exercises #1, 2 (use DeMorgan's Law), 3, 8, 10, 11.

5)   Read ALL of section 2.3.

6)    Do Preview Activities 1 and 2 of section 2.4

7)    Be ready to hand in a 1st draft of a portfolio problem for Monday.  

Day 4  (1-26)

1)    Prove that if n is an integer then n^2-5n+7 is an odd integer.  We proved this proposition for the case where n is an even integer, now prove it for the case where n is an odd integer.

2)    Prove that n is even if and only if n^2 is even for every integer n.  This was a lemma given in class.

3)    Read pages 107 - 112 up to and including the section entitled "Important Note".  This is review of the proof by contradiction method.

4)    Read pages 129 - 133 (The Division Algorithm material).  Pay special attention to the proof of Proposition 3.28 and bring questions about it to class on Wednesday.

5)    Do Preview Activities 1 and 2 of section 2.1

6)    Read pages 33 - 35 on "Constructing Truth Tables" and "The Biconditional Statement".

7)    Do preview activities 1, 2, and 3 in section 2.2               

Day 3  (1-21)

1)    Do progress check 2.1 on pages 32 - 33.

2)    At the end of section 2.1 do exercises #9, 10.

3)    Read pages 79 - 83

4)    At the end of section 3.1 do exercise #1 (parts a, d, and h only)

5)    Write up the solution to #1 part a using the guidelines on pages 475 - 479.  Be prepared to share you work with your classmates next class meeting.

Day 2  (1-14)  The homework below may seem like a lot but keep in mind that next Monday is a holiday (no class) so you actually have a week to do it.

1)    At the end of section 1.1 do exercises #2, 3, 5, finish 7

2)    Read section 1.1 doing the included "progress checks" and "activities" as needed.  Note: solutions to the "progress checks" are found on page 26. 

3)    Read all of section section 1.2.

4)    At the end of section 1.2:

        a) do exercise #2 parts (b) and (c) by writing out a formal proof for each part using the definition of odd and even integers.

        b) do exercise #3 part (a) by writing out a proof similar to the proof for Theorem 1.6 found at the bottom of page 19.             

        c) do all of exercise #5

        d) do all of exercise #9  Note: There is a TYPO in this problem.  An integer a is said to be a type 0 integer if there exists an integer n such that a = 3n + 0.

        e) do exercise #10 parts (a) and (b) only.

5)    Enjoy the day off on Monday.

Day 1  (1-12)    Read the article entitled "On Proof and Progress in Mathematics" by William P. Thurston by clicking on the adjacent link and be prepared to discuss this article as a class on Wednesday.  In particular, pick out one or two things that really stood out or made an impression on you as you read the article.  Read the writing guidelines on pages 20 - 21.  These guidelines will be the basis for how you will write up your solutions to the problems in your proof portfolio.