MAT 3749
Introduction to Analysis
Autumn 2009
 

Current Assignments

• Due Friday, December 4: Solutions (16.7 is currently missing)
  Exercises 16.6cde, 16.7bcde (you may also use the algebraic limit theorem), 16.9, 16.10, 16.13. (I removed 16.8 and 17.4)

Tentative Future Assignments

Past Assignments and Solutions

• Due Wednesday, September 30:
  Turn in your solution to the "Liars and Truth Tellers" exercise distributed by e-mail on Monday. 
  Read Section 1 of the main textbook (which I'll refer to as "Lay" from here onward on this page), Logical Connectives, p. 1-8.
   

• Due Friday, October 2:

  • Read Section 2 of Lay (Quantifiers, p. 11-14)

  • Turn in your personal information sheet

  • Turn in exercises 4, 6, 8, 10, 12, 13, and 14e from Section 1 (unless otherwise noted, homework exercises will be from Lay).  Solutions

  • With the small group that was assigned on Wednesday, work on the "checkerboard" problems from class and write up your solutions as a group.  This should be turned in Friday, separately from your individual homework assignment.  Two of the groups will also be asked to present solutions on Friday.

• Due Monday, October 5:
  Read Section 3 of Lay (Techniques of Proof I, p. 17-23)
  Turn in exercises 2.4, 2.10, 2.12, 2.14, 2.16, 2.17, and 2.19 from Section 2 as well as exercises 3.3, 3.4, and 3.5 from Section 3.  Solutions
  Note that most of the odd numbered exercises in this section of the text have answers in the back of the book.  The even numbered exercises tend to be very similar to the preceding odd numbered exercise, so you may find it helpful to work through the odd exercise first as practice to make sure you are on the right track.
   
• Due Wednesday, October 7:
  Read Section 4 of Lay (Techniques of Proof II, p. 26-32)
  Submit your table of answers from the exploratory examples from p. 2 of the lab from Monday.  I'll check over your answers quickly to make sure that they are all correct before you complete the assignment for Friday.  Please try to submit them by early Wednesday morning at the latest so that I can check them and return them in class.
   

• Due Friday, October 9:
  Two separate items to turn in, one individual assignment from the textbook and one group assignment (completing the lab from Monday):

  • Turn in exercises 3.6 parts abeghik, 3.7 parts abefg, 3.8, 4.11, and 4.15 from Lay.  Solutions
    Note that there may be a typo in the last sentence of the "Proof" in 4.15.  The last sentence should read "We conclude that xy must be irrational."  (The contradiction implies that the original assumption that xy was rational must be false, thus proving the "Theorem".  However, the point of the exercise is that there is some flaw in the "Proof" other than this typo; what is the flaw?)

  • Turn in answers to all of the questions for reflection (#1-13) on boundedness of sets.  You may turn in a single copy for your entire group.  You do not need to turn in the proofs at this time.
     

• Due Monday, October 12:
  Read Section 5 of Lay (Basic Set Operations, p. 36-45)
  Turn in exercises 4.8, 4.10, 4.13bc, 4.14acd, 4.16, 4.21, 4.22, and 4.9.  Solutions
  Some Hints:
  • On exercises 4.14-4.22, you are asked to "prove or give a counterexample" for six statements; 3 of the 6 are false. 
  • Also, there may be a time or two on this assignment where the result of a previous exercise can be used to help prove a later exercise -- you are welcome to use the previous result without reproving the same thing again!
  • Exercise 4.9 is tricky, which is why I listed it last.  I think that everyone last year had the right general approach, but nobody wrote a completely correct proof that addressed all of the issues well...so be careful!
     

• Due Monday, October 19:
  Turn in exercises 5.4, 5.5b, 5.6, 5.7, 5.10, 5.11, 5.20, 5.21, 5.24 (parts beg), and 5.25.  Solutions
  For 5.5b, see the answer in the back for 5.5a to get an idea of the type of answer I'm looking for.  Provide a brief justification for each of your answers for 5.6.
  Read Section 6: Relations.
   

• Due Wednesday, October 21: Solutions

  • Complete and turn in example 1 from class.  That is, create a table listing all possible relations on the set A = {1, 2}.  Then determine whether or not each relation is (a) reflexive, (b) symmetric, (c) transitive, (d) antisymmetric, (e) an equivalence relation, (f) a partial ordering, (g) a total ordering.

  • Turn in exercises 6.4, 6.5, 6.6, 6.7, 6.8, 6.9, and 6.11abefh. On 6.11, also determine whether or not each relations is antisymmetric, an equivalnce relation, a partial ordering, and/or a total ordering.  You do not need to provide formal proofs for 6.11, but please provide explanations of each of your answers.

  • In addition, there will be a quiz on terminology relating to relations. 

  • Read Section 7: Functions.

• Due Friday, October 23: Solutions
  Turn in exercises 6.16, 6.17, 6.18, 6.20, 6.22, 6.24, 6.25, and 6.26.  In addition, prove that the relation shown below is a partial ordering on the positive integers.  Is it also a total ordering?  Explain.

• Due Wednesday, October 28: Solutions
  Turn in exercises 7.4, 7.6, 7.7, 7.8, 7.9, 7.10, 7.13, 7.14, and 7.22.
   

• Due Friday, October 30: Homework #10
   

• Due Wednesday, November 4: Solutions
  Turn in exercises 10.4, 10.14, 10.17, 10.19, 10.21, and 10.30.
   

• The take-home mid-term exam will be distributed on Monday, Nov. 2.  It will be collected at the start of class on Monday, Nov. 9.  The in-class midterm exam will be Monday, Nov. 9.
   

• Due Monday, November 16:
  Exercises 8.1, 8.2, 8.15, 8.16, and 9.13.
   

• Due Friday, November 20:
  Prove part 2 of proposition 1 (uniqueness of multiplicative inverses) from the "definition of real numbers" handout from class. 
  Exercises 11.3(parts bcdef), 11.5, 11.6, 11.7, 11.8, and 11.11.  Solutions (still need to add a couple of solutions)
   

• Due Wednesday, November 23:
  Exercises 12.3, 12.4, 12.5, 12.6, 12.8, and 12.16. Solutions
  Also prove the following two claims:
  a. Every nonempty subset of the real numbers that is bounded below has a greatest lower bound.
  b. The irrational numbers are dense in the real numbers.  That is, if x and y are real numbers with x < y, prove that there exists an irrational number w such that x < w < y.
   

• Due Wednesday, December 2:
  Turn in exercises 13.3, 13.4, 13.5, 13.6, 13.7, 13.15, 13.17, 13.18, and 13.20.  Solutions
  In addition, turn in your report on Lab 3.