Here is a copy of the final exam.

Here are some counting problems for review: part 1 with answers and part 2 ( with answers). Try picking some of these problems at random, and compare answers.

Tablet notes from recent classes:

• further discussion of functions
• Counting Examples, bit strings with odd numbers of 0's.
• inverse functions
• P.I.E.
• Counting Examples - 3
• Counting Examples - 1
• Permutations with repetition
• Combinatorial Proofs
• from 12/4
• from 12/4

Here is a copy of Exam 2 and brief answers.

The six classes following Thanksgiving break will be held on zoom at the usual class times, MWF 9:45 to 10:35pm. The zoom links for the classes will be posted on the course canvas calendar.

Notes from the zoom classes will be posted here.

• from Novenber 30
  • Combinatorial proof example. Read the discussion in section 3.10 of Hammack's book (page 108-110).
  • Permutations and combinations with repetition. These techniques are discussed in section 3.8 of Hammack's book.
  • Technique of stars and bars.

Notes (part 1, part 2, part 3 ) from the zoom review session on Saturday, November 21.

The second midterm exam will take place in class on Monday, November 23. It will cover topics that have been discussed since the last test:

• definition of congruence modulo m
• working in the arithmetic system of the integers modulo m
• equivalence relations
• working with the definitions of injective/surjective/bijective functions
• proof by mathematical induction
• working with recursively defined sequences
• basic principles of counting
• counting functions between finite sets
• permutations and combinations

For some of these topics I can recommend Hammack's discussions:

• integers modulo m: Exercises #1-7 page 221. [See also Day's notes.]
• equivalence relations: section 11.3.
• injective/surjective/bijective functions: section 12.2. Exercises: #1-18 page 232
• math induction: section 10.1. Exercises: #1-12 page 214
• principles of counting: section 3.2-3.3. Exercises: #1-4, 5-8 page 73; #1-6, 8-10
• permutations and combinations: section 3.4-3.5. Exercises: #7-10 page 84; #1-8, 13, 15

Class notes from Friday, November 20:

• injective and surjective functions 1
• injective and surjective functions 2
• injective and surjective functions: examples
• on modular arithmetic

Here are some comments on permutations and combinations in a finite set, and on counting the number of functions between two sets: part 1: functions and strings and part 2: numbers of functions/injections/surjections.

For students that haven't yet earned the bonus points for Exam 1, the next iteration of the bonus assignment is due by November 15. (Carefully study the documents posted below to see some examples and discussion of strategy to use in element-wise proofs. It is especially recommended to study the "more detailed perspective".

In the next portion of the course the main focus will be on solving counting problems. Most of the results will stem from elementary counting principles known as the product rule, the sum rule and the relabelling rule. These rules are discussed in sections 3.2 and 3.3 of Hammack's book.

Here are some comments on counting functions. (Note: the original posting had an error. It should have indicated that the ith entry in the string is f(a_i) not a_i.)

Here are some notes from recent classes:

• about equivalence relations. We will discuss relations and equivalence relations some more next week. You can read about them in section 11.3 (page 210) of Hammack's book.
• the Fibonacci Numbers. Further discussions can be found in Wikipedia, and in section 10.5 on page 193 of Hammack's book.

Here are some notes that discuss multiplicative inverses in modular arithmetic. (Based on class discussions on Monday, October 26.) Some other references are:

• Here is an excerpt from the (online) book "An Introduction to Proofs and the Mathematical Vernacular" by Martin Day which discusses the integers modulo m.
• There is a Wikipedia page on modular arithmetic which includes many detailed discussions. The introductory material may be helpful to read through if you are having confusions. Also there is a section entitled "Applications" which will give you a sense of the importance of modular arithmetic in many diverse fields of study.

Part of the next group assignment will involve reading the document Writing Proofs by Christopher Heil. This is the type of document that many math professors might share with students at the beginning of an upper division course for math majors, such as "Real Analysis", "Abstract Algebra", "Abstract Linear Algebra" or "Introduction to Topology". It suggests the expectations and the importance for students in those courses to be able to analyze and write mathematical proofs.

This week we will discuss Fibonacci numbers, modular arithmetic, and counting functions between two finite sets.

The first midterm exam took place on October 14. Here is a sample exam and some review problems from this exam.

ON ELEMENT_WISE PROOFS:
Here is a simple example of an elementwise proof, and a basic description of the element-wise proof technique used to show that one set is a subset of another set. For further discussion read a more detailed perspective.
It may be useful to remember how to phrase the most basic set theory definitions from an element-wise perspective.

Regular weekly problem session/office hours will take place on Wednesday afternoons between 4pm and 5pm on zoom. Please feel free to enter at any time during this hour. The zoom link will be available in announcements at the canvas course page, and posted on the canvas calendar.

Overheads and outlines of class presentations will be posted here:

• October 19, basic definitions for surjective and injective functions
• Sept 30, basic set theory definitions from an element-wise perspective
• Sept 23, example of an element-wise proof
• Sept 23, Propositional Logic 3
• Sept 21, Propositional Logic 2
• Sept 21, Propositional Logic 1
• Sept 14, Project 3
• Sept 14
• Sept 9
• Sept 2

Here is the proof that the square of two is irrational as discussed in class.

The various group projects are available here:

• Group project #11 for Wednesday, November 11.
• Group project #10 for Wednesday, November 4.
• Group project #9 for Friday, October 23.
• Group project #8 for Friday, October 16.
• Group project #7 for Friday, October 9.
• Group project #6 for Friday, October 2.
• Here is Group project #5 for Friday, September 25.
• Here is Group project #4 for the zoom class meeting on Friday, Sept 18.
• Here is the problem sheet for group project #3 from September 11.
• Here is the problem sheet for group project #2 from September 4, and the team constituencies. More integer grid paper may be useful for this project.
• This is group project #1 from August 28 and the student team constituencies.

The "Book of Proof" by Richard Hammack will be the principal textbook for this course. It is available on-line at "www.people.vcu.edu/~rhammack/BookOfProof".

The Course Syllabus describes the course subject, policies and expectations for students.

Math 2513, Fall 2019