# 18.S097 Special Subject in Mathematics:Introduction to Proofs

## IAP 2015

 Syllabus.Instructor: Eric Baer, ebaer@math.mit.eduOffice: 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. Initialdiscussion of "working with integers".Lecture Notes 2, In-class discussion problems,Solution to first in-class discussion problem(note: the second problem is assigned asHomework 2) (none assigned) 3 Working with integers (continued discussionworking from Lecture Notes 2). Irrationality of √2 and e. Introduction to mathematicalinduction.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 tomathematical induction. In-class discussion problems (note: we only discussedproblem 7 today -- we will treat countability andProblem 8 tomorrow). Solution to Problem 7. (none assigned) 5 Countability and uncountability: definitions; countabilityof the rationals, uncountability of the reals.Lecture Notes 3In-class discussion problems (note: Problem 8 repeatedfrom Day 4, we only discussed Problem 8 today, and willmove 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 LectureNotes 3).In-class discussion problems: Problems 9 and 10from Day 5.Solution to Problem 9. Homework 4 (due Wed 1/14) HW4 Solution 7 Continued discussion and practice with limitsand continuity. Taking negation of statements(with applications to Problem 10 and HW4).Open sets: definition and basic notions. Practiceworking with the definition (via in-class problems).In-class discussion problems (note: Problems 10and 11 repeated from prior days) (none assigned) 8 Continued discussion of examples related toworking with open sets. Definitionof closed sets.Sets of measure zero: definition and examples.Countable sets have measure zero.Construction of the Cantor set (an uncountableset 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 Hausdorffdimensions (definitions and examples). (none assigned) 10 Brief continued discussion of notions of dimension.Inequalities: some examples and practice.In-class discussion problemsTwo proofs of the Cauchy-Schwarz inequality[to be added momentarily] (none assigned)

Solutions to in-class discussion problems 10--17
(many of these were discussed in class, but printed solutions