18.S097 Special Subject in Mathematics:
Introduction to Proofs

IAP 2015


Instructor: Eric Baer, ebaer@math.mit.edu
Office: Room E18-308.
Office Hours: by appointment.

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 MeetingTopicHomeworkSolutions
1Introduction, overview of methods of proof, sets,
Lecture Notes 1, Example Sheet 1
Homework 1 (due Wed 1/7)HW1 solution
2Continued 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) 
3Working with integers (continued discussion
working from Lecture Notes 2). Irrationality
of √2 and e. Introduction to mathematical

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) 
5Countability 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
6Working 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
7Continued 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) 
8Continued 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
9Continued discussion of the Cantor set construction.

Notions of dimension: Minkowski and Hausdorff
dimensions (definitions and examples).
(none assigned) 
10Brief 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) 

Solutions to in-class discussion problems 10--17
(many of these were discussed in class, but printed solutions
were not previously handed out). [this link will be uploaded shortly]

Bibliographic Notes