Mathematics 239
: Introduction to Mathematical Proof
Fall 2023
Introduction
Professor: Jeff Johannes
Section 2
MWF 1:30 - 2:20p Fraser 119
Office: South
326A
Telephone: 245-5403
Office Hours: Monday 5-6p in South 309, Tuesday 1-2p in
South 309. Wednesday 12:30-1:20p in South 309, Thursday 8-9p in
South 336, Friday 2:30-3:30p in South 309, and by
appointment or visit.
Email Address: Johannes@Geneseo.edu
Web-page:
http://www.geneseo.edu/~johannes
Textbooks
Mathematics:
A Discrete Introduction, Third Edition, Edward R.
Scheinerman (link to chapter 1)
Purposes
- to develop familiarity and comfort with the language of more
formal mathematics and proofs before taking upper division
mathematics coursework
- to learn and justify several techniques for counting
Overview
It is often said that mathematics is a
language. In this class you will begin to learn to speak this
language. Just like in an introductory language course, we
will start with the most fundamental concepts and grammar
rules. After we have some familiarity with the language of
formal mathematics, we will practice this language in the setting of
counting problems of different types. More like an advanced
language class, merely memorizing the vocabulary will not
suffice (in fact, hopefully we can keep vocabulary to a minimum),
but rather you will be required to understand and speak clearly in
this language. The material learned here will help you
understand the mathematics you read and clarify the mathematics you
write. Because we are learning how to write mathematics,
exposition will also be a component in your evaluation.
I have intentionally chosen a very readable
text. In addition to planning time to do homework, please take
time to carefully read the sections in the book. Notice use of
the words “time" and “carefully". Read the sections
slowly. As the author indicates in the preface, read
actively. If you do not understand some statement reread it,
think of some potential meanings and see if they are consistent, and
if all else fails, ask me. If you do not believe a statement,
check it with your own examples. Finally, if you understand
and believe the statements, consider how you would convince someone
else that they are true, in other words, how would you prove them?
Because the text is exceptionally accessible, we
will structure class-time more as an interactive discussion of the
reading than lecture. For each class
day there is an assigned reading. Read and take notes on
the section before coming to class. In addition to the
reading, there are also indicated exercises to check that you
understood the reading. When we complete questions from the
reading we will discuss those indicated exercises during the class
discussion.
Learning Outcomes
Upon successful completion of Math 239 a student
will be able to
- Apply the logical structure of proofs and work symbolically
with connectives and quantifiers to produce logically valid,
correct and clear arguments,
- Perform set operations on finite and infinite collections of
sets and be familiar with properties of set operations,
- Determine equivalence relations on sets and equivalence
classes,
- Work with functions and in particular bijections, direct and
inverse images and inverse functions,
- Construct direct and indirect proofs and proofs by induction
and determine the appropriateness of each type in a particular
setting. Analyze and critique proofs with respect to logic and
correctness, and
- Unravel abstract definitions, create intuition-forming
examples or counterexamples, and prove conjectures.
- Write solutions to problems and proofs of theorems that meet
rigorous standards based on content, organization and coherence,
argument and support, and style and mechanics.
Grading
Your grade in this course will be based upon your
performance on homework, quizzes, one colloquium report, and three
exams. The weight assigned to each is designated below:
Homework (6)
5% each
Quizzes (3)
5% each
Colloquium Report (1)
5%
In-class exams (2)
15% each
Final exam
(1) 20%
Problem Sets
There are problem sets for each chapter.
They will be due shortly after each chapter ends. You are
encouraged to consult with me outside of class on any questions
toward completing the homework. You are also encouraged to
work together on homework assignments, but each must write up their
own well-written solutions. A good rule for this is it is
encouraged to speak to each other about the problem, but you should
not read each other's solutions. A violation of this policy
will result in a zero for the entire assignment and reporting to the
Dean of Students for a violation of academic integrity. Each
question will be counted in the following manner:
0 - missing or plagiarised question
1 - question copied
2 - partial question
3 - completed question (with some solution)
4 - completed question correctly and well-written
Each entire homework set will then be graded on a 90-80-70-60%
(decile) scale. Late items will not be accepted, as solutions
will be posted immediately. Homework will be returned on the
following class day. Please feel free to discuss any homework
with me outside of class or during review.
Solutions and Plagiarism
There are plenty of places that one can find all
kinds of solutions to problems in this class. Reading them and
not referencing them in your work is plagiarism, and will be
reported as an academic integrity violation. Reading them and
referencing them is not quite plagiarism, but does undermine the
intent of the problems. Therefore, if you reference solutions
you will receive 0 points, but you will *not* be reported for an
academic integrity. Simply - please do not read any solutions
for problems in this class. Any work written, developed, or
created, in whole or in part, by generative artificial intelligence
(AI) is considered plagiarism and will not be tolerated. While the
ever-changing developments with AI will find their place in our
workforces and personal lives, in the realm of education and
learning, this kind of technology does not help us achieve our
educational goals. The use of AI prevents the opportunity to learn
from our experiences and from each other, to play with our creative
freedoms, to problem-solve, and to contribute our ideas in authentic
ways. Geneseo is a place for learning, and this class is
specifically a space for learning how to advance our thinking and
professional practice. AI cannot do that learning for us.
Opening Meeting
Students will earn two extra
points on the first problem set by visiting office hours during
the first two weeks of classes, i.e. no later than 11 September.
Presentations
When
discussing the new material for each section, each day will be
begin with an opportunity for student presentation. There
are presentation problems listed on the reading schedule.
Students will earn one extra point for the corresponding problem
set by attempting a presentation and two extra points by
presenting well. I will have a priority list at all times
for presentations. Students may present more than once per
chapter, but representations have lowest priority. Students
may earn no more than two extra points in this fashion, and may
only earn two points by doing a presentation well, not by
presenting poorly twice.
Quizzes
There will be short quizzes after the homework
has been returned, covering the material in the chapter from the
homework. For chapters immediately preceding exams, there will
be no quiz. Quizzes will consist of routine questions, and
will have limited opportunity for partial credit. Because quizzes
will consist of routine questions, they will be graded on a decile
scale. There will be no makeup quizzes.
Attend one of the department colloquium talks.
Write a report. In the report, describe the content of
the talk (including a detailed discussion of the mathematics).
In addition to your description of the talk, also write how
this talk added to your understanding of the nature of
mathematics. Papers are due within a classweek of the
colloquium presentation. I will gladly look at papers before
they are due to provide comments.
Exams
There will be two exams during the semester and a
final exam during finals week. If you must miss an exam, it is
necessary that you contact me before the exam begins. Exams
require that you show ability to solve unfamiliar problems and to
understand and explain mathematical concepts clearly. The bulk
of the exam questions will involve problem solving and written
explanations of mathematical ideas. The final exam will be
half an exam focused on the final two chapters, and half a
cumulative exam. Exams will be graded on a scale approximately
(to be precisely determined by the content of each individual exam)
given by
100 - 80% A
79 - 60% B
59 - 40% C
39 - 20% D
below 20% E
For your interpretive convenience, I will also give you an exam
grade converted into the decile scale. The exams will be
challenging and will require thought and creativity. They will
not include filler questions (hence the full usage of the grading
scale).
Feedback
Occasionally you will be given anonymous feedback
forms. Please use them to share any thoughts or concerns for
how the course is running. Remember, the sooner you tell me
your concerns, the more I can do about them. I have also
created a web-site
which
accepts anonymous comments. If we have not yet discussed
this in class, please encourage me to create a class code.
This site may also be accessed via our
course page on a link entitled anonymous
feedback. Of course, you are always welcome to approach
me outside of class to discuss these issues as well.
Social Psychology
Wrong answers are important. We as
individuals learn from mistakes, and as a class we learn from
mistakes. You may not enjoy being wrong, but it is valuable to
the class as a whole - and to you personally. We frequently
will build correct answers through a sequence of mistakes. I
am more impressed with wrong answers in class than with correct
answers on paper. I may not say this often, but it is
essential and true. Think at all times - do things for
reasons. Your reasons are usually more interesting than your
choices. Be prepared to share your thoughts and ideas.
Perhaps most importantly "No, that's wrong." does not mean that your
comment is not valuable or that you need to censor yourself.
Learn from the experience, and always try again. Don't give
up.
Accessibility Accommodations
SUNY Geneseo is dedicated to providing an
equitable and inclusive educational experience for all students. The
Office of Accessibility (OAS) will coordinate reasonable
accommodations for persons with disabilities to ensure equal access
to academic programs, activities, and services at Geneseo.
Students with approved accommodations may submit a semester
request to
renew their academic accommodations. Please visit the OAS website
for information on the process for requesting
academic accommodations. Contact the OAS by email, phone, or in-person:
Office of Accessibility Services
Erwin Hall 22 585-245-5112
access@geneseo.edu
Religious Holidays
It is my policy to give students who miss class
because of observance of religious holidays the opportunity to make
up missed work. You are responsible for notifying me no later
than September 10 of plans to observe the holiday.
Schedule (subject to change)
August 28
- September 11 Chapter I reading discussions
September 13 Chapter I homework due
September 18 Chapter I quiz taken
September 13-22 Chapter II reading discussions
September 25 Chapter II homework due
September 25-27 review Chapters I and II
September 29 In class
exam covering Chapters I and II
October 2-16 Chapter III reading discussions
October 18 Chapter III homework due
October 23 Chapter III quiz taken
October 18-25 Chapter IV reading discussions
October 27 Chapter IV homework due
October 27 - 30 review Chapters III and IV
November 1 In class exam covering Chapters III and IV
November 3 -15 Chapter V reading discussions
November 17 Chapter V homework due
November 27 Chapter V quiz taken
November 17
- December 6 Chapter VII reading discussions
December 8 Chapter VII homework due
December 8-11 review Chapters V and VII, and course as
whole
Friday, December 15 3:30-6:30p Final exam, first half covering
chapters V and VII
second half covering course
Problem Sets:
Assignment for Chapter 1: To the student.1, 3.2, 3.6, 4.3,
4.10, 5.2, 5.12, 5.15, 5.23, 6.2, 6.4, 6.11, 7.8, 7.11
For example, this means that you must complete the second exercise
in section 3. There is also a required exercise at the end of
the "to the student" section.
Assignment for Chapter 2: 8.5, 8.12, 8.18, 9.2, 9.8, 10.3,
10.11, 11.5, 12.18, 12.21
Assignment for Chapter 3: 14.6, 14.13, 14.16, 15.6, 15.12,
15.15, 16.11, 16.13, 17.14, 17.23, 17.24, 17.33
Assignment for Chapter 4: 20.5, 20.7, 20.14, 22.9, 22.16
(valued as 6 questions)
Assignment for Chapter 5: 24.4, 24.8, 24.14, 24.16, 24.17,
24.20, 24.22, 25.9, 25.16, 26.4, 26.12, 27.2, 27.13
Assignment for Chapter 7: 35.4, 36.11, 36.15, 37.3, 37.14
(valued as 5 questions), supplemental E-primes.