Syllabus. Instructor: Eric Baer, ebaer@math.mit.edu Office: Room E18-308. Office Hours: by appointment. |
Description: An introduction to writing mathematical proofs, including discussion of mathematical notation, methods of proof, and strategies for formulating and communicating mathematical arguments. Topics include: introduction to logic and sets, rational numbers and proofs of irrationality, quantifiers, mathematical induction, limits and working with real numbers, countability and uncountability, introduction to the notions of open and closed sets. Additional topics may be discussed according to student interest. There will be some assigned homework problems -- there is no textbook. |
Course meetings: Jan. 5-9 (10am to 12noon) Jan. 12-16 (10am to 12noon). Location: Room 4-149. |
3 units (U level), graded P/D/F. To register: pre-register on WebSIS and attend first class. Listeners allowed, space permitting (Space may be limited; please email ebaer@math.mit.edu to reserve a spot.) |
Course Meeting | Topic | Homework | Solutions |
1 | Introduction, overview of methods of proof, sets, quantifiers. Lecture Notes 1, Example Sheet 1 | Homework 1 (due Wed 1/7) | HW1 solution |
2 | Continued discussion of sets, functions. Initial discussion of "working with integers". Lecture Notes 2, In-class discussion problems, Solution to first in-class discussion problem (note: the second problem is assigned as Homework 2) | (none assigned) | |
3 | Working with integers (continued discussion working from Lecture Notes 2). Irrationality of √2 and e. Introduction to mathematical induction. In-class discussion problems (solutions at link below) | Homework 2 (due Fri 1/9) | HW2 solution |
4 |
Discussion of in-class problems from Class 3 (solutions). Continued discussion and examples related to mathematical induction. In-class discussion problems (note: we only discussed problem 7 today -- we will treat countability and Problem 8 tomorrow). Solution to Problem 7. | (none assigned) | |
5 | Countability and uncountability: definitions; countability of the rationals, uncountability of the reals. Lecture Notes 3 In-class discussion problems (note: Problem 8 repeated from Day 4, we only discussed Problem 8 today, and will move into subsequent material next week) Solution to Problem 8. | Homework 3 (due Mon 1/12) | HW3 solution |
6 | Working with real numbers: completeness, limits, continuity (notes for this material are in Lecture Notes 3). In-class discussion problems: Problems 9 and 10 from Day 5. Solution to Problem 9. | Homework 4 (due Wed 1/14) | HW4 Solution |
7 | Continued discussion and practice with limits and continuity. Taking negation of statements (with applications to Problem 10 and HW4). Open sets: definition and basic notions. Practice working with the definition (via in-class problems). In-class discussion problems (note: Problems 10 and 11 repeated from prior days) | (none assigned) | |
8 | Continued discussion of examples related to working with open sets. Definition of closed sets. Sets of measure zero: definition and examples. Countable sets have measure zero. Construction of the Cantor set (an uncountable set having measure zero). In-class discussion problems | Homework 5 (due Fri 1/16) | HW5 Solution |
9 | Continued discussion of the Cantor set construction. Notions of dimension: Minkowski and Hausdorff dimensions (definitions and examples). | (none assigned) | |
10 | Brief continued discussion of notions of dimension. Inequalities: some examples and practice. In-class discussion problems Two proofs of the Cauchy-Schwarz inequality [to be added momentarily] | (none assigned) |