Instructor
Office Hours
TA
References
Resources
Exams\
Homework
Robert Chang (rchang@reed.edu)
M 2:30–4:00, F 12:00–1:30 at Lib 390
Olivia McGough (mcgougho@) will hold discussion sessions M 5:30 p.m.–7:30 p.m. at Library 389
[P] David Perkinson, Math 112 lecture notes
[H] Hammack, Book of Proof
Math Help Center/Drop-in Tutoring: SuMTWTh 7:00–9:00 p.m. at Library 204
Individual Tutoring: Reed offers one hour per week of free, one-to-one tutoring
One in-class midterm
One in-person final exam
Homework solutions must be typeset with LaTeX (see getting started) and submitted on GradeScope (see Moodle page for the entry code).
Working together is encouraged, as collaboration is a nontrivial part of learning and of scholarship in general. But, in accordance with the honor principle and basic human decency, you must submit your own write-up with an acknowledgment of collaborators.
Week
1
Date
1/23
1/25
1/27
Topics and Reading
Set theory: basic notations and operations
Read [P, pp. 16-19], [H, pp. 3-8, 12-15, 18-19]
More set theory; some sample proofs involving sets
Read [P, pp. 20-22], [H, pp. 8-11]
Irrationality of √2
Read [H, Proposition on p. 139]
Click for additional comments on the proof that √2 is irrational
The proof is meant as a demonstration of "good" writing style (e.g., use of lemmas, "without loss of generality," "for the sake of a contradiction, assume...") and of common proof techniques (direct, contrapositive, contradiction).
With that said, I will not make a spectacle of the underlying logical structure (e.g., P → Q ⇔ ¬Q → ¬P) of an argument. Those interested may consult [H, pp. 34-64, 113, 128, 138] for formal logic and templates of the common proof techniques.
Homework 1 (pdf, tex, solutions) due on Monday, January 30
2
1/30
2/1
2/3
Functions: domain/codomain, image/range, injective/surjective, composition
Read [P, pp. 29-31], [H, pp. 225-232, 235-238]
Bijections, inverse functions
Read [P, pp. 33-36], [H, pp. 238-241]
Cardinality, Cantor's diagonal argument
Read [P, pp. 33-36], [H, pp. 238-241, 269-283]
Give [H, pp. 269–283] a read if you are interested in sizes of infinities. The pictorial representations of the various bijections (e.g., on p. 277) are very nice.
Homework 2 (pdf, tex, solutions) due on Monday, February 6
3
2/6
2/8
2/10
More examples involving cardinality
Read [P, pp. 33-36], [H, pp. 238-241, 269-283]
Field axioms
Read [P, pp. 41-47]
I will use the labels (A1)-(A4), (M1)-(M4), (D) from [P, pp. 41–42] when referring to the field axioms.
Consequences of field axioms
Read [P, pp. 41-47]
Homework 3 (pdf, tex, solutions) due on Monday, February 13
4
2/13
2/15
2/17
Ordered fields
Read [P, pp. 48-51]
I will use the labels (O1)-(O4) from [P, p. 48] when referring to the order axioms.
Least upper bounds
Read [P, pp. 52-54]
Least upper bound property, Archimedean property
Read [P, pp. 52-59]
Homework 4 (pdf, tex, solutions) due on Monday, February 20
5
2/20
2/22
2/24
Consequences of the Archimedean property
Read [P, pp. 52-59]
The field of complex numbers
Read [P, pp. 59-72]
Snow day; class canceled
Click for additional comments on properties of R
There is a lot to unpack in the last few lectures; I recommend carefully going over the relevant sections of [P, pp. 52-59].
First, make sure you are comfortable with the basic definitions on p. 52. Next, examine the examples given on pp. 52-53. Third, review the least upper bound property (aka completeness) on pp. 53-54. Finally, go over Propositions 1-4 on pp. 56-58 (we did similar, but not identical, things in class).
Homework 5 (pdf, tex, solutions) due on Monday, Wednesday, March 1
Some notes on what we covered in class
6
2/27
3/1
3/3
Arithmetic and geometry of complex numbers
Read [P, pp. 59-72]
Topology
Read [P, pp. 73-77]
Mathematical induction
Read [P, pp. 11-15], [H, pp. 180-197]
In-class midterm on Wednesday, March 8
Notes on field axioms. Practice problems for the midterm.
7
3/6
3/8
3/10
Review
In-class midterm
Sequences and limits
Read [P, pp. 78-82]
Spring break
9
3/20
3/22
3/24
Sequences and limits
Read [P, pp. 78-82]
Computations using limit laws
Read [P, pp. 83-94]
Proofs of limit laws
Read [P, pp. 83-94]
Next Homework due Monday, April 3
10
3/27
3/29
3/31
Some special sequences
Monotone convergence theorem
Read [P, pp. 95-96]
Cauchy sequences, Cauchy completeness of R
Read [P, pp. 98-100]
Homework 6 (pdf, tex, solutions) due on Monday, April 3
11
4/3
4/5
4/7
Series: notation and examples, geometric series
Read [P, pp. 101-106]
Test for divergence, comparison test
Read [P, pp. 107-112]
Comparison test, Cauchy condensation test, p-series
Read [P, pp. 107-112]
Homework 7 (pdf, tex, solutions) due on Monday, April 10
12
4/10
4/12
4/14
More examples of the comparison/condensation test, the number e
Read [P, pp. 107-112]
Ratio test, root test
Read [P, pp. 118-120]
Absolute convergence, alternating series test
Read [P, pp. 112-117]
Homework 8 (pdf, tex, solutions) due on Monday, April 17
13
4/17
4/19
4/21
Riemann's theorem on rearrangements, limits and continuity
Read [P, pp. 112-117, 126-232]
Continuity
Read [P, pp. 126-132]
Differentiability
Read [P, pp. 132-134]
Homework 9 (pdf, tex, solutions) due on Monday, April 24
14
4/24
4/26
4/28
Power series
Read [P, pp. 135-143]
Taylor's theorem
Read [P, pp. 144-151]
The complex exponential and Euler's formula
Read [P, pp. 151-154]
Final Exam on Thursday, May 11, 1-4 p.m., Elliot 314
Final exam practice problems