MATH 220: Mathematical Proof, Section 202, Winter term II,
2019/2020.
 Instructor: Julia Gordon
 office: Math 217. Office hours: Tuesday 3:304:30pm and Friday 11am12noon.

Common website .
In particular, here you can find the course outline, marking scheme, and
policies.
 Other Resources:
Math Learning Centre. The schedule is available
here .
Homework
 All homework assignments will be posted on the
Common website on Thursdays and due Thursday the following
week. Individual extensions 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).
 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 .
 You are strongly encouraged to preread before each class.

The signup link for Piazza forum . Please sign up for Piazza as soon as possible, and do ask and answer
questions here!
Our class link
Three most active participants get a 1% bonus at the end of the class!
Announcements:
 Please take an informal
Survey about the course (it helps me improve something for the
second half).
 The course has moved online! Please keep checking both this website
and Canvas. And read the textbook.
Review materials for the midterm
 A collection of problems from old
midterms, etc.
Solutions
Disclaimer: as this is a
collection of problems from old exams, solutions are written by different
instructors at different times; "I" does not always mean the same person,
and there may be references to things that happened in the class that
year (and not now). Hopefully they give you enough information to see how
to solve the
problems and avoid most popular mistakes. Ignore the notes about marking
that are occasionally present.
The references to the notes on quantifiers,etc. are to the two notes that
are linked below.
 Blank worksheets from our classes:
 Some notes in case you are confused by the quantifiers (or other
topics):
The Prereading assignments and lecture notes:
 January 79:
Sets and logic.
We discussed: the notion of a set and being an element of a set. The empty set. Logical statements and beginnings of propositional calculus:
we defined conjunction, disjunction, negation, and conditional statements.
Read: Sections 1.1, 2.1, 2.2 (covered these sections on January 7, will continue discussing them on Thursday).
Notes from Lecture 1
Read for Thursday: 2.2, 2.3 and 1.3.
Notes for Thursday (Lecture 2) .
 Tuesday Jan 14: Conditional statements. Start direct proof.
Prereading for Tuesday:
Sections 2.3 and 4.1, 4.2, 4.3.
We continued discussing conditional statements and started using them to make proofs! Please read sections 4.14.4.
Defined the notion of divisibility for integers; discussed even and odd integers.
Notes for January 14 .

Thursday Jan 16: Finished Chapter 4 and
went back to Chapter 2.
Reading for Thursday: 4.24.5 and
2.4, 2.5, 2.6.
Notes for Thursday
 Tuesday January 21:
Reading for Tuesday: 2.4, 2.5, 2.6 (covered last week); Contrapositive
proof; congruence of integers  sections 5.1, 5.2, 5.3.
Notes
 Thursday January 23:
Please read the rest of Chapter 2. The real content we will discuss is
Sections 2.7, 2.8, 2.10.
We will also continue talking about congruence of integers (section 5.2).
Notes
 Tuesday January 28:
(Pre)reading:
all of Chapter 2 (especially Sections 2.92.11)
Quantifiers; negating complicated logical statements.
Corrected worksheet .
The lecture was centered on answering the questions from the worksheet.
Lecture notes .
Please make sure you know how to solve all the questions except maybe for
the
last two. (We will discuss the last two questions next class).
If you feel confused, you might find the supplementary note below
helpful:
A note on the use of quantifiers, the
expression "such that", and related questions
(This discusses out worksheet questions in some more detail).
 Thursday January 30:
Continued discussing statements with multiple quantifiers, in particular,
the notion of the function f(x) tending to infinity as x goes to
infinity. Notes .
 Tuesday February 4:
Existence proof; disproof.
(Pre)reading:
Sections 7.17.4 and Chapter 9.
(Some of section 7.3 may look very hard; you can skip that if needed).
We discussed Proposition 7.1 on p.126; then discussed the proof "The
following are equivalent" type of statements.
In the process, we reviewed operations with congruences of integers, in
particular, how to use congruences efficiently to figure out remainders of
large powers modulo a small integer.
Notes from the class .
You an also look at
Notes on Euclidean algorithm , which
supplement the discussion of Proposition 7.1 (this
is completely optional, we did not discuss it in class and it's not on the
exams).
 Thursday February 6:
Review (Proof and disproof: chapters 7 and 9).
Start Induction (Chapter 10).
Notes from class .
We will discuss Section 10.1 next class; please read the start of Chapter
10 and Section 10.1.
 Tuesday February 11:
Mathematical induction, continued.
We will eventually cover most of Chapter 10 except graphs. The graphs
will not appear
on
exams.
Notes
 Thursday February 13:
Midterm (1111:45). Then finished induction (with an example of string induction).
Quickly discussed the definitions of: subset, intersection and union of sets, indexed collections
of sets (Sections 1.3, 1.5, 1.8).
Notes
 Tuesday February 25:
More about sets: sections 1.31.8.
Power sets.
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.
Also will discuss the connection between set operations and
logic (it is not emphasized in the book).
Notes on the
relation between sets and open sentences.
Supplement:
A handout about "subset" vs. "an element of"
 please take a look if you are confused about the set notation and
subsets.
Finally, started talking about proofs involving sets. Sections 8.18.3.
(we proved one of the DeMorgan laws for sets).
Notes from the class .
 Thursday February 27:
More on proofs involving sets. Finish Sections 8.18.3 and will do
Section 1.2 (Catesian products of sets).
Notes form the class
 Tuesday March 3 :
Relations: Sections 11.1  11.4; in particular, equivalence
relations and
partitions.
Notes from class
 Thursday Marh 5:
Finish Chapter 11: congruence of integers (Section 11.4); start Chapter
12: functions. (Sections 12.112.2)
Notes from class
 Tuesday March 10:
Functions (12.1). Domain, codomain and range.
Notes from class
 Thursday March 12:
Injective/surjective functions. Pigeonhole
principle. Composition of functions. (Sections 12.212.4)
Notes from class
(Pre)reading for Tuesday:
all of Chapter 12, especially 12.6.
 Tuesday March 17:
Continue with composition, injective, surjective, bijective functions.
Inverse functions. Image
and preimage of sets
Sections 12.5  12.6.
Notes on images and preimages
(based on notes by Prof. A. Rechnitzer ).
 Thursday March 19:
Images and preimages, continued.
Finsihed Chapter 12, started Proof by contradiction (chapter 6): went through
irrationality of the square root of 2.
Did Worksheet 16 (see worksheets section above, where solutions are posted, too).
 From here on this list is not going to be
maintained; I will continue posting worksheets at the top of this page, but all
information about the lectures has moved to Canvas! Please follow the Canvas
website!