Introduction to Analysis
MAT 3749
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Welcome to the course pages for Brian Gill's
Autumn 2005 Introduction to Analysis class! Instructor: Dr. Brian Gill Autumn 2005 Office Hours: Course syllabus |
Daily Information and Homework
Current Assignments:
Tentative Future Assignments
- exercises 3 and 7 from section 8 of Galovich -- after we cover induction
Past Assignments
- Homework for Wednesday, Sept. 28: Turn
in personal information sheet and read Section 1 of Galovich.
- Homework for Friday, Sept. 30:
Solutions
Read Section 2 of Galovich and turn in exercises 5, 6, 7a, 10, 12, 14, and 15 from Section 1 of Galovich.
- Homework for Monday, Oct. 3:
Solutions
Turn in the exercises from the following worksheet: Homework #2
- Homework for Friday, Oct. 7:
Solutions
Turn in the following exercises:- Exercises 1 and 10ab from section 10 of Galovich (in exercise 1, U represents the "universe" to be used when finding complements).
- Exercises 5, 6, 7,and 9 from Section 2 of Galovich.
- Let A and B be sets and explain as clearly and concisely as you can why the following sets must be equal (you do not need to do a formal proof).
Read section 3 of Galovich.
- Homework for Monday, Oct. 10:
Solutions
Turn in exercises 1, 2, 3, 4, 8, 12, 13, 14, 16, 17, and 21 from Section 3 of Galovich. For exercises 12-17, in part (a) try to write the sentence entirely in symbols (no words at all!). In part (b), write the negation in TWO ways: (i) in symbols and (ii) in words. Also, when writing the negation in words, avoid using words of negation as much as possible (particularly phrases like "it is not the case that...").
Read Sections 4 and 5 of Galovich.
- Homework for Wednesday, Oct. 12:
Solutions
Turn in exercises 2, 3, 5, 6, and 7 from Section 4 of Galovich.
- Homework for Friday, Oct. 14:
Solutions
Turn in exercises 1, 2, 4a, 5, 7, and 8 from Section 5 of Galovich.
Read Sections 8 and 9 of Galovich. (Note, however, that the proof on p. 73 uses mathematical induction, a topic which we have temporarily skipped over, so some of you may not be able to fully understand that proof.)
- Homework for Monday, Oct. 17:
1. Complete Lab 1: Boundedness of Sets. Turn in one copy for yourself and your partner. You should include- a detailed response for questions 7, 10, and 13 from Section 1.3 (critical Thinking Questions). You do not need to turn in questions 1-6, 8, 9, 11, or 12, but you will need to carefully think through all of those questions to be able to answer 7, 10, and 13.
- answers for questions 1, 3, 4, and 5 from the questions for reflection (you do not need to do 2 or 6)
2. Also turn in exercises 1, 2, 3, and 4 from Section 9 of Galovich. (This is separate from the lab -- each of you should turn in your own copy of this assignment.) Solutions
NOTE: For exercise 4, see example 3 on p. 72 for a definition of the "additive property".
- Homework for Friday, Oct. 21:
Turn in Homework #8.
- Homework for Monday, Oct. 24:
Solutions
Turn in exercises 3acd, 4bd, and 5(iii-v only) from section 10 of Galovich.
Some special instructions:- Provide careful "element chasing" proofs for 3a, 3c, and 3d.
- For 4bd and 5(iii), provide careful proofs, but whenever possible you can use earlier results from our list to try to avoid the need for "element chasing". (For 4bd, you may use any of the results from the list. 5(iii) is #27 from the list, so you may use anything above that on the list.
- For 5(iv) and 5(v), use Venn diagrams along with a brief verbal explanation to explain why the equalities should hold (you don't need to do a formal proof).
There will also be a quiz on Monday covering the terminology of relations (relation, reflexive, symmetric, transitive, antisymmetric, equivalence relation, partial ordering)
- Homework for Friday, Oct. 28:
Solutions
There will be a quiz on terminology of functions. You should be able to state the definitions of function, domain, codomain, range, injective, surjective, and bijective.
Also, turn in exercises 1, 3, and 4 from Section 11 and exercises 5, 10, 8a, and 12 from section 13 of Galovich. In addition, prove that the following relation on the positive integers is a partial ordering:
For exercise 3, make a complete list of all relations on the set and make a table showing whether each is reflexive, symmetric, transitive
- Homework for Monday, Oct. 31:
Solutions
Turn in exercises 1, 2, 8, 3, 4, 5, 14, 11bcd, and 15 from Section 12 of Galovich. Also turn in the extra proof that I mentioned in class.
(For #2, no formal proofs are needed -- just answer each question and give a brief explanation of your answer).
- Friday, Nov. 4:
Take-home portion of midterm collected.
In-class portion of midterm taken in class.
- No homework for Wednesday, Nov. 2
- For Wednesday, Nov. 9:
Extra credit assignment!
- Homework for Monday, Nov. 14:
- Homework for Wednesday, Nov. 16:
- Prove part 2 of Proposition 1 from the "Definition for the Real Numbers" handout
- Exercises 1, 2acd, 6, 7a, 9, and 14 from Section 6 of Galovich
Solutions
(solutions for 2 exercises are missing; I'll update later)
- Homework for Monday, Nov. 21: Turn
in exercises 1, 5, 6, 8, 13, and 18 from Section 7 of Galovich.
Solutions
- Homework for Wednesday, Nov. 23:
- Turn in proof for the sequence exercise given in class.
- Turn in Lab 2, one copy for your group of 2 (or 3).
- Prove the equivalence of the two definitions and answer the question at the end of 2.4.
- Answer questions 1, 3, 4, 6, and 7 from the questions for
reflection.
- Homework for Wednesday, Nov. 30:
Turn in exercises 1.3.2, 1.3.6, 1.3.7, 1.3.9, and 1.4.4 from Abbott. Partial Solutions
- Homework for Friday, Dec. 2:
Turn in Lab 3 (1 copy for each group). You ONLY need to turn in the following parts:- responses to the questions in section 3.3.2
- answers for #2, 3, and 4 from the questions for reflection (section 3.4)
- Homework for Monday, Dec. 5:
Solutions
Turn in exercises 2.2.4, 2.2.6, 2.2.1, 2.2.2, and 2.2.8 from Abbott.
(For each part of 2.2.6, you need to select one of "larger" or "smaller" to make the statement true, and briefly explain your choice)
