Times & Places | Lectures are in Skiles 170 on Tuesdays and Thursdays from 12:05 -- 1:25. | |

Office hours are currently scheduled for Monday from 2 - 3pm and Tuesday/Thursday from 1:30 - 2pm in Skiles 211B. (Meeting at other times is possible by appointment.) | ||

Office hours are now scheduled for Monday from 4:30 - 5:30pm, Wednesday from 2 - 3pm, and Tuesday/Thursday from 1:30 - 2pm in Skiles 211B. | ||

Announcements | Information about Permits and Overloads | |

Handout from Tuesday, August 23rd: Course Syllabus | ||

Extra office hour on Wednesday, August 24 from 2 - 3pm. | ||

Logic review on Tuesday, August 30th from 5:30 - 6:30pm. | ||

Extra office hour on Wednesday, Sept. 7th from 2 - 3pm. | ||

Assignment 1 is now due on Thursday, Sept. 8th. | ||

Typos in #4(c) and (d) of Assignment 1. #4(c) should read "f f^{-1} is the identity on P(Y) if f is onto" and #4(d) should read "f^{-1} f is the identity on P(X) if f is one-to-one." | ||

A clarification/hint for #4(c) and (d) of Assignment 1. For #4(c), consider the map h: P(Y) -> P(Y) defined by h (C) = f(f^{-1}(C)) for C a subset of Y. Show that h is the identity function. #4(d) is similar. | ||

REMINDER: Bring a copy of Assignment 1 (as well as the original!) to class on Thursday, Sept. 8th. | ||

Second round for Assignment 1, #4, 5, 6. Choose exactly one of the following: Option 1 -- Submit a sheet of paper (with your name) that says that you want the original solutions (from last week) graded. Option 2 -- Submit (possibly revised) solutions for all three problems (#4, 5, 6). Remember that solutions must be easily readable to be graded. | ||

Correction to Assignment 2, #6. The last line should read, "Also, give examples to show that each inequality can be strict." That is, "is" -> "can be." | ||

The first midterm exam will be on Thursday, October 6th. | ||

Some problems from the textbook to consider when preparing for the exam: pages 61 - 62, #5, 8, 9, 10, 14, 15, 16. | ||

Solutions for Exam 1 from Thursday, Oct. 6th. | ||

Alternative grading scheme: 0% Midterm 1, 20% Midterm 2, 50% Final exam, and 30% Homework. | ||

Midterm 1 grades have been adjusted by adding 15 points to the original score. | ||

Correction to Assignment 4, #1. The first line should read "Let (E,d) be a metric space with subsets S and T." | ||

Assignment 4 is due on *Tuesday, Oct. 25th.* (Right day, wrong date on assignment.) | ||

Hint for #1(a) and others on Assignment 4: Let T be the union of all open sets contained in S. (Compare with the definition of closure of S.) Can you prove that T equals the interior of S? If so, then you can use this alternative definition of the interior of S when helpful. | ||

Office hours on Monday, Oct. 24th (only) will be 2 - 3pm, not the usual 4:30 - 5:30pm. | ||

First hint for Assignment 5, #2. Let E be the real numbers with the usual metric. S = (0,1] and A = (0,1/2]. You should see (if not, then prove it!) that A is both open and closed in S, but A is neither open nor closed in E. | ||

Second hint for Assignment 5, #2. CORRECTION FROM LECTURE. Show that if A is open (respectively closed) in (E,d), then A \cap S (read "A intersect S") is open (resp. closed) in (S,d). The first hint shows that the converse is not true! | ||

Hint for Assignment 5, #3. These problems should remind you of proofs from the lectures/textbook. Compare and contrast. | ||

The second midterm exam will be on Tuesday, Nov. 15th. It will cover up to and including section 4.3. | ||

Correction to Assignment 6, #1(c): the limit should be h -> 0, not infinity. | ||

Clarification to Assignment 6, #2: use the Euclidean metric on r^2 and R. | ||

A reminder about the practice problems below when preparing for the exam. | ||

Solutions for Exam 2 from Tuesday, Nov. 15th. | ||

Correction to Assignment 7, #3(b): "Prove that if h is another continuous extension of f...". | ||

Email all questions, requests, suggestions, etc. for topics, problems, etc. for next week by 11pm on Sunday, Dec. 4th. | ||

There will be no office hours on Monday, Dec. 12th or Tuesday, Dec. 13th. | ||

Due to unexpected scheduling changes, office hours on Wednesday, Dec 7th have been moved from 2 - 3pm to 4:30 - 5:30pm. Please email if you were planning to come during the original time and cannot come during the new time instead. | ||

Correction to Assignment 6, #4. Assume that the limit function f is also continuous. | ||

Final exam is on Thursday, Dec. 15th from 11:30am - 2:20pm in Skiles 170. | ||

You may bring one (1) sheet (8.5" x 11" paper) of notes to the final exam. Typed is OK, writing on front and back is OK. Assume nothing else is OK, unless you check with me first. | ||

There will be office hours on Wednesday, Dec 14 from 3pm - 5pm. | ||

Scores on the final exam are available in T-Square. The maximum score was 149/150, and the average was 97/150 = 65%. | ||

Recall that the adjusted average on Midterm 1 was 71% and on Midterm 1 was 71%. The homework average was 80%. | ||

Numerical grades were calculated as the maximum of three possible numbers: the original grading scheme given in the course syllabus, the alternative grading scheme given above, and a third scheme (M1 25%, M2 25%, Final 25%, Homework 25%). | ||

Readings | Week 1: Review propositional logic and basic set theory. See links below and/or library reserves. | |

Week 2: Chapt. 1 of the textbook (_Introduction to Analysis_ by M. Rosenlicht). | ||

Week 3: Chapt. 2 of the textbook. | ||

Week 4: Chapt. 2 of the textbook. | ||

Week 5: Chapt. 3, Sections 1 & 2 of the textbook. | ||

Week 6: Chapt. 3, Sections 2 & 3 of the textbook. | ||

Week 7: Chapt. 3, Section 3 of the textbook. | ||

Week 8: Chapt. 3, Section 4 of the textbook. | ||

Week 9: Chapt. 3, Section 5 of the textbook. | ||

Week 10: Chapt. 3, Sections 5, 6 of the textbook. | ||

Week 11: Chapt. 4, Sections 1, 2 of the textbook. | ||

Week 12: Chapt. 4, Sections 2, 3 of the textbook. | ||

Week 13: Chapt. 4, Section 4 of the textbook. | ||

Week 14: Chapt. 4, Sections 4, 5 of the textbook. | ||

Week 15: Chapt. 4, Section 6 of the textbook. | ||

Week 16: Review. | ||

Homework | Assignment 0, Part 1 due on Tuesday, August 30th. | |

Assignment 0, Part 2, will be completed in class on Thursday, August 25th. | ||

Assignment 0, Part 3, will be completed in class on Tuesday, August 30th. (Actually, collected in class on Thursday, Sept. 1st. | ||

Assignment 1 originally due on Tuesday, Sept. 6th, now due on Thursday, Sept. 8th. | ||

Assignment 1, #4, 5, 6 are due on Thursday, Sept. 13th. See announcements above for more information. | ||

Solutions for Assignment 1. | ||

Assignment 2 due on Thursday, Sept. 22nd. (Revised version fixes a typo in #4.) | ||

Solutions for Assignment 2. | ||

Assignment 3 due on Thursday, Sept. 29th. | ||

Solutions for Assignment 3. | ||

Assignment 4 due on TUESDAY, Oct. 24th. CORRECTION: Oct. 25th. | ||

Solutions for Assignment 4. | ||

Assignment 5 due on Thursday, Nov. 3rd. | ||

Solutions for Assignment 5. | ||

Assignment 6 due on Thursday, Nov. 10th. CORRECTION for 1(c): the limit should be h -> 0, not infinity. | ||

Solutions for Assignment 6. | ||

Assignment 7 due on Thursday, Dec. 1st. | ||

Solutions for Assignment 7. | ||

Assignment 8 due on Thursday, Dec. 8th. | ||

Solutions for Assignment 8. | ||

Practice Problems | From Discrete Mathematics and Its Applications by K. H. Rosen, Fourth Edition. See links below and/or library reserves: | |

Sec 1.1, pgs 11 - 14, #4, 6, 8, 16, 18, 20, 21, 23, 25. | ||

Sec 1.2, pgs 19 - 20, #6, 8, 12, 14, 16, 18. | ||

Sec 1.3, pgs 33 - 37, #6, 8, 10, 12, 14, 20, 22, 24, 33. | ||

Sec 1.4, pg 45, #7, 11, 12, 16, 21, 24. | ||

Sec 1.5, pg 54 - 56, #6, 7, 9, 10, 12, 13, 14, 15, 19, 20, 21, 25, 26, 27, 28. | ||

Sec 1.6, pg 67 - 69, #3, 4, 5, 12, 13, 22, 23, 28, 29. | ||

From the textbook: | ||

Chapt. 1, pg 12 - 13, #2, 3, 4, 5, 6, 9, 10. | ||

Chapt. 2, pg 29 - 31, #2, 3, 10, 13, 14. | ||

Prove that there does not exist a rational number s such that s^2 = 6. | ||

Trying proving some of the algebraic or order properties of the real numbers on your own. | ||

Prove that a non-empty finite set of real numbers has a l.u.b. and a g.l.b. | ||

Give an example of a set of rational numbers that is bounded but does not have a rational supremum. | ||

Prove that the union of two bounded sets is bounded. | ||

Chapt. 3, pg 61 - 65, #1, 2, 3, 4, 6. | ||

Chapt. 3, pg 61 - 65, #9, 10 11, 12, 13, 25. | ||

Chapt. 3, pg 61 - 65, #24, 26, 27, 30, 32, 33. | ||

Chapt. 4, pg 90 - 95, #1, 3, 4, 6. | ||

Chapt. 4, pg 90 - 95, #7, 8, 9, 10, 11. | ||

Chapt. 4, pg 90 - 95, #14, 17, 18, 19, 20, 26, 27, 28. If feeling ambitious, try #13, 24, and 25 also. | ||

Chapt. 4, pg 90 - 95, #33, 34, 35, 36, 39, 40. If feeling ambitious, try #44, 45, 46 also. | ||

Challenge Problems | From Discrete Mathematics and Its Applications by K. H. Rosen, Fourth Edition. See links below and/or library reserves: | |

Sec 1.2, pgs 19 - 20, #28, 29, 35. | ||

Sec 1.3, pgs 33 - 37, #42, 43. | ||

Sec 1.4, pg 45, #25, 26. | ||

Sec 1.5, pg 54 - 56, #30, 31. | ||

Sec 1.6, pg 67 - 69, #57. | ||

Chapt. 3, pg 61 - 65, #6. | ||

Links | Check your grades on T-Square. | |

Student are expected to follow the Georgia Tech Academic Honor Code and Student Code of Conduct. | ||

Additional Resources | Writing Proofs by Prof. Chris Heil, Georgia Tech and How to write proofs: a quick guide by Dr. Eugenia Cheng, U Sheffield. | |

Library Reserves | ||

Logic | ||

Propositional Equivalences | ||

Predicates and Quantifiers | ||

Solutions to Odd-Numbered Exercises (1.1 - 1.3) | ||

Sets | ||

Set Operations | ||

Functions | ||

Solutions to Odd-Numbered Exercises (1.4 - 1.6) | ||

Countable Sets from Set Theory and Metric Spaces by Irving Kaplansky |

Fall 2011