MATH 220: Mathematical Proof, Section 201
Text: Mathematical proofs, a transition to advanced mathematics, by G.
Chartland, A. Polimeni, P. Zhang
Section 201, Instructor: Julia Gordon.
Where and when:
- Tuesday and Thursday, 11 -- 12:30 am, at
Leonard S. Klink (LSK) building, Room 200.
Instructor's office: Math 217.
e-mail: gor at math dot ubc dot ca
Office Hours: Wednesdays 10-11am; 3-4pm,
and by appointment.
The marking scheme, and all the course policies are found here.
The course outline.
(also from the common website).
Homework and quizzes
- All homework assignments will be posted here
on Wednesdays and collected on Thursday of the following week.
- Every other Thursday, the classes will be organized as workshops in
students in small groups will work problems. There will be a short quiz at the end.
The Homework assignments:
- Tips for
authors by James Milne .
Please consider these before writing homework (even though these are the
tips for authors of mathematical papers). These will become more relevant
as the course progresses.
- LaTeX resources if you want to type your
homework using LaTeX.
Einstein's riddle (the ultimate problem in a series of those similar
to Problem 7 from
Workshop 3 )
The final exam is Friday, April 12, 8:30am, in LSK 200 (our usual
Review session: Monday April 8, 12(noon)-2pm, in MATX 1100.
Office hours this week: Tuesday 3:30-5pm, Thursday 10-11:30am
(in my office Math 217).
Review materials for the final exam:
The Exam is based on material covered in the
lectures, workshops and HW's.
This is approximated by the following sections in the text: Ch. 1,
2.1-2.10, 3.2-3.4,Ch. 4, 5.1-5.3, 6.1, 6.2, 6.4, 8.1, 9.1-9.6, 10.1-10.4,
part of 10.5, 12.1. All material will be weighted equally.
Workshops with solutions
Older review handouts
Our midterm 1 with solutions
The other section's
midterm 1 (with solutions).
Midterm 1 from 2010 ; Solutions
Midterm 1 from 2012
Midterm 2 from last year
(Approximate) Course outline
Here I will post short summaries of each class and other relevant to our secion notes, as we go along.
- Thursday, January 3:
A few words about the course (please read the Preface and Chapter 0!); Introduction to sets; the notion of a subset.
(sections 1.1 and 1.2).
A handout about "subset" vs. "an element of"
-- please take a look if you are confused about the set notation and
- Tuesday, January 8:
Operations on sets; Venn diagrams: examples (in particular, intervals on
the real line). De Morgan laws. Examples of a proof of equality of sets.
Reading: Sections 1.3 -- 1.6. We covered only Section 1.3 (with some
extras), but please read everything till the end of Chapter 1 by
- Thursday, January 10:
Workshop 1 . Solutions
- Tuesday, January 15:
Stared Chapter 2: Mathematical logic. We discussed the notion of
statement, the truth tables, conjunction, disjunction and negation of
statements. Also, defined equivalent statements as statements having the
same truth tables. Discussed open sentences, and the connection between
statements and sets. Discussed that DeMorgan's laws amount to the
rules of negating conjunctions and disjunctions. This corresponds
approximately to Sections 2.1 -- 2.3 and 2.8, 2.9.
in the text; however, we have not yet discussed the implication. Will talk
about it on Thursday.
- Thursday, Jan. 17:
Chapter 2, continued. Implication and biconditional. Contrapositives.
Sections 2.4, 2.5, 2.6.
- Tuesday, Jan. 22:
Tautologies and contradictions (Section 2.7). Especially, please look at
Exercise 2.32 and 2.33, which were discussed).
Quantifiers. Section 2.10. We also discussed the use of quantifiers to
define the unions and intersections of indexed collections of sets --
in particular, covered Section 1.4 more rigorously.
There was also a brief discussion of 1.6.
Here is a handout about indexed
collections of sets and quantifiers.
- Thursday, Jan. 24:
Workshop 2 (this is the workshop 2
problems and quiz with solutions).
- Tuesday, Jan. 29:
We discussed the unions and intersections of indexed collections of sets;
and covered Sections 4.4, 3.1, 3.2.
Please read: Sections 4.4, 4.5, and 3.2, 3.3, 3.4
- Thursday, Jan. 31:
Proofs involving the integers. Sections to read: 3.2, 3.3, 3.4, 4.1, 4.2
We will continue with 4.1-4.2 on Thursday.
- Tuesday, February 5:
- Thursday, February 7:
Congruences of the integers. Sections 4.1, 4.2. Please read the rest of
Chapter 4. We will start Chapter 5 next class.
- Tuesday, February 12:
Counterexamples (Section 5.1); proof by contradiction (5.2-5.3)
- Thursday, February 14:
Workshop 3 and quiz (with solutions).
- February 19-21: Break
- Tuesday, February 26:
Well-ordering of the positive integers. Induction. Sections 6.1-6.2
- Thursday, February 28:
Induction, continued: 6.2, 6.4.
- Tuesday, March 5:
8.1, 9.1 - 9.4 Functions; domain and range. Injective, surjective,
- Thursday, March 7:
Workshop 4 .
- Tuesday, March 12:
Continuing 9.3-9.4, and the notions of image and pre-image (inverse
image). Unfortunately, the images and inverse images of sets under
functions are not discussed in the textbook, but it is a very important
topic. Notes on pre-images (inverse
images) of sets.
- Thursday, March 14: Midterm II.
- Tuesday, March 19:
Composition of functions (Section 9.5); the inverse function (9.6);
overview of Chapter 10 -- the notion of cardinality.
- Thursday March 21 -- Tuesday March
Cardinality (Sections 10.1-10.5). We did not go through all the proofs in
class. You need to know all the definitions from these sections (such as:
denumerable set, continuum; need to know what it means that the
cardinality of A is not greater than the cardinality of B. We proved that:
unions and finite products of denumerable sets are denumerable; that the
set of rational numbers is denumerable; that the interval (and therefore,
the set of all reals) is not denumerable. Stated Schroeder-Bernstein
Theorem (you have to know and understand the statement, but not the
proof). You are responsible for the proof that the cardinality of a power
set of A is strictly greater than the cardinality of A.
- Tuesday, March 28:
- Tuesday, April 2:
Defined sequences; the notion of a limit of a sequence; examples. Stated
arithmetic properties of the limits, and proved that the limit of a sum is
the sum of limits. This roughly corresponds to Section 12.1, but some of
what we did today is not in the book. Please see your calculus textbook
(the chapter on limits of sequences) for the properties of limits related
to taking sums, products, and quotients.
Scan of the lecture notes on limits from the doc
camera (this is unedited, and does not reflect all of the discussion
we had in class).
Here are Notes on
limits (adapted from the notes by Prof. Rechnitzer).
- Thursday, April 4:
Limits, continued. Will talk about upper bounds and least upper bounds,
and prove that an increasing sequence converges if and only if it is
bounded. Will define sum of a series. Please read 12.2 and the chapter on
sequences and series in your calculus book.
- Thanks to Prof. Rechnitzer for some of the notes (on limits and