MATH 220: Mathematical Proof, Section 105, Winter term I, 2016/2017.
- Instructor: Julia Gordon
- office: Math 217. Office hours: Mondays 10:30-11:30am and Fridays
- Common website .
In particular, here you can find the course outline, marking scheme, and policies.
- Other Resources:
Drop-in tutorials begin during the second week of term. The schedule is available
Homework and quizzes
- A new list of optional fun problems
to hand in)
- All homework assignments will be posted on the
Common website and collected on Fridays. The hard deadline on homeork is 5pm on Friday; extensions beyond that are not possible.
Collaboration on homework: You are allowed to discuss homework with
other students, but you have to first think about it yourself, and then
write it alone in your own words.
If you discussed homework solutions with someone, please acknowledge this on your submitted work, that is, put "discussed with Jane Smith" next to your name (and Jane Smith then has to put the similar acknowledgment of collaboration with you on her work).
Please also see the notes on academic integrity .
- You are strongly encouraged (and will receive 1 points bonus at the end f the term if you do it) to type all your homework solutions using LaTeX.
Here are some LaTeX resources .
- In Section 105, you need to pre-read before each class. There will be a short quiz on the pre-reading almost every class. These quizzes (and other occasional group activities) are worth 2% of the mark in total. This is a very small number, but the point is that if you do the pre-reading and participate in the class actively, you will do much better on exams and will get a lot more from this course. Reading assignments, supplements, etc. will be posted on this page regularly.
- The sign-up link for Piazza forum . Please sign up for Piazza as soon as possible, and do ask and answer questions here!
Our class link
Five most active participants get a 1% bonus at the end of the class!
Office hours after the end of class, and review:
- CANCELLED BECAUSE OF THE SNOW: the planned office hours Monday
December 19 5-7pm are cancelled, but I will be available on skype 4-6pm,
my skype name is julygord1.
- NEW: last-minute office hours Tuesday (the day of the exam),
- Review session Friday December 16, 1-3pm, in GEOG 212
We will go over some
problems (based on your requests) from old final exams,
which can be found at Math Dept
(In these exams, ignore problems about limits of sequences, and problems
about "graphs" (which inlcude problems about cities connected by
roads/airlines, etc. from last year's Term 2).
Notes from the review session (with
thanks to Caitlin!)
Review materials for the final exam
Some old midterms with solutions, for practice
midterm 1 with
Midterm 1 from 2010 ;
Midterm 2 from last year with solutions .
(has good problems about sets and functions).
Please see also the review materials for the midterm (below) and the notes
posted at the corresponding lecture descriptions.
Review materials for the midterm
The Pre-reading assignments and lecture plans:
- September 7-9:
Sets. We have covered Sections 1.1, 1.3 and most of 1.4.
A handout about "subset" vs. "an element of"
-- please take a look if you are confused about the set notation and
Pre-reading for Monday Sept. 12:
Sections 1.2 and 1.4
- Monday September 12:
Cartesian products; cardinality of the power set. Sections 1.2, 1.3-1.4,
started 1.5. Mentioned Venn diagrams (1.7).
Pre-reading for Wednesday Sept.
Sections 1.5, 1.6, 1.7, 1.8
- Wednesday September 14:
Unions and intersections of sets, complement, Venn diagrams, indexed
collections of sets. Will finish everything in Chapter 1 up to the end of
1.8 in the book. Please look at 1.9 and 1.10.
- Friday September 16:
The quiz and discussion were on 1.8 (indexed collections); covered 2.1
Pre-reading for Monday Sept.
19: Sections 2.1-2.3.
- Monday September 19-Friday
Logic -- all of Chapter 2.
A supplementary note on double
Another note on the use of quantifiers, the
expression 'such that", and related questions Check out the problems
at the end of this handout!
Answers to the questions from Friday class
(Pre-)reading for Monday September
26: all of Chapter 2 (especially Sections 2.9-2.11), and Section 4.1
- Monday September 26:
Discussion and quiz on logic (including negating statements with double
- Wednesday September 28:
Definition of divisibility; division with remainder. Even and odd numbers.
Direct proof (Sections 4.1-4.2)
- Friday September 30:
The definitions of the greatest common divisor and the least common
multiple. The division algorithm.
We proved that every common divisior of a and b divides gcd(a,b).
- Monday October 3:
Congruence. The relationship between congruence and remainders.
We learned how to figure out the last digit of any power of any number,
using congruence (the example done in class was the last digit of 7^2016).
(Sections 4.4-4.5 and 5.2).
We started talking about contrapositive proof (Section 5.1).
Home reading: the rest of Chapter 5.
- Wednesday October 5:
The contrapositive proof and proof by contradiction.
Finished Chapter 5 and did two classic proofs by contradiction: that
square root of two is irrational, and that there are infinitely many prime
numbers (Sections 6.1-6.2).
- Friday October 7:
Finished Chapter 6 and did examples from 7.1.
- Wednesday October 12:
- Friday October 14:
Review workshop. We will review and work on problems for Chapters 1-2 and
4-6. (See Probelm set 6 on the common website for the problems).
- Monday October 17:
- Wednesday October 19:
Section 7.3. Please pay special attention to Proposition 7.1 on p.126.
Notes on Euclidean algorithm (this is
optional, we did not discuss it in class and it's not on the exams).
- Friday October 21:
Finished Chapter 7. Started 8.1
- Monday October 24:
Proofs involving sets. Sections 8.1-8.2. We discussed the connection
between set operations and logic (it is not emphasized in the book).
Notes on the relation between sets
and open sentences .
- Wednesday October 26:
More on proofs involving sets. Finished Section 8.3.
- Friday October 28:
Perfect numbers (Theorem 1 from Section 8.4; we will skip Theorem 2).
Chapter 9: disproof.
- Monday October 31:
Induction. (Chapter 10). We discussed Section 10.1, but also a topic not
quite covered in the book: how the pronciple of mathematical induction is
equivalent to the well-ordering axiom of natural numbers.
Notes on well-ordering (by Prof. Chen).
- Wednesday November 2:
Mathematical induction, continued. Fibonacci numbers.
We covered all of Chapter 10 except graphs. The graphs will not appear on
- Friday November 4:
Induction, continued. Strong induction.
- Monday November 7:
Induction finished: proof of the Fundamental theorem of Arithmetic using
strong induction. (Theorem 10.1 in Section 10.2).
Stared Relations: Section 11.1
- Wednesday November 9:
Relations, continued. Sections 11.2 and 11.3: equivalence relations and
- Friday November 10:
- Monday November 14:
Finish Chapter 11: congruence of integers (Section 11.4); start Chapter
12: functions. (Sections 12.1-12.2)
- Wednesday November 16:
Functions (12.1). Domain, codomain and range.
Injective and surjective functions (Section 12.2).
There was a quiz on functions and relations.
- Friday November 18:
Injective/surjective functions, continued. Pigeonhole
principle. Composition and inverse functions. (Sections 12.2-12.5)
(Pre)reading for Monday:
all of Chapter 12, especially 12.6.
- Monday November 21:
Continued with composition, injective, surjective, bijective functions.
and preimage of sets (12.6).
Notes on images and preimages
- Wednesday November 23:
Images and preimages, continued.
Started chapter 13: cardinality of sets. The definition of "same
- Friday November 25:
Cardinality of sets, sontinued. Defined countable and uncountable sets.
Survey of Chapter 13.
We discussed specifically: Section 13.1, Theorem 13.1 with proof.
Stated that |R| is not equal to |N|, but will prove it next class.
Proved that "being of the same cardinality" is an equivalence relation on
the set of subsets of a given universal set U (see p.222)
Discussed continnum hypothesis (see end of 13.4).
Covered Section 13.2 up to Theorem 13.3 with proof.
- Last week of class:
We proved some of the remaining results about cardinality announced last
in particular, that Q
is countable, R is not.
The product of finitely many countable sets is countable.
Next class: will prove that taking power sets makes cardinality go up.
We will cover *everything* in Sections 13.1-13.3, and some of Section
13.4 (for now you can skip the proof of Cantor-Schroeder-Bernstein
theorem, but you need to understand its statement).
Workshop that we
started working on.
Solutions to the
first two problems
We will continue working on the workshop at the last class on Friday.
The workshop with full solutions