Fall 2020
Math 300: Foundations of Mathematics   Sections 902 and 905.

Daily Schedule

August 20 25 27   September 1 3 8 10 15 17 22 24 29 October 1 6 8 13 15 20 22 27 29 November 3 5 10 12 17 19 24
Day 1 : August 20   We will hit the ground running.
Section 1.1 : Statements and Conditional Statements
The zeroeth written assignment is due before class.
Also, read Section 1.1 of our book and watch the following videos, and complete the concept quiz 1 in Gradescope, before class.
Watch   Welcome message
Statements and Non-Statements (5:38) GVSU Screencast 1.1.1
Conditional Statements (5:33) GVSU Screencast 1.1.3
Worksheet for Today
Start to look at math videos for the first written assignment, which is due on Monday, before the next class meeting.
August 24   First written assignment is due. Please use LaTeX template.
Day 2 : August 25  
Section 1.2 : Constructing direct proofs
Read Section 1.2 of our book and watch the following videos, and complete the concept quiz 2 in Gradescope, before class. Today has more videos that typical. The first two on definitions are fairly key to the rest of the course as definitions are how we make sure that we know what we are talking about in mathematics. The next two, while fundamental as they dissect a proof, are good to watch, but you are not expected to be able to construct proofs on your own after watching them. We will spend a lot of time this semester on that task.
Watch   Working With Definitions (9:25) GVSU Screencast 1.2.1
Working With Definitions, part 2 (8:42) GVSU Screencast 1.2.1b
Direct Proofs of Conditional Statements using Know-Show Tables (8:24) GVSU Screencast 1.2.2
Direct Proofs of Conditional Statements using Know-Show Tables, Part 2 (8:31) GVSU Screencast 1.2.3 This is optional, but could be useful.
Worksheet for Today
Day 3 : August 27   Section 2.1 : Statements and Logical operations
This section is basic to what comes next in introduces many basic concepts in propositional logic. Read it, and complete the concept quiz. The five GVSU videos on truth tables (2.1.2—2.1.6) concentrate on two tasks; first parsing statements in English into symbols and logical operators, and then analyzing them in terms of truth tables. I will only assign the first, but you may find the remaining useful, particularly when you start to work on your next homework. They will also fill in what I do not cover in class. The first video on tautologies is also a review of truth tables.
Watch   Negations of simple statements (3:34) GVSU Screencast 2.1.1
Truth tables, part 1 (6:07) GVSU Screencast 2.1.2
Tautologies and Contradictions I (6:44) GVSU Screencast 2.1.7
Worksheet for today Concept Quiz.
August 31   First homework due. The homework assignment, in .pdf, and the same file in .tex.
I have a one-page palette of colors defined using the dvipsnames option in xcolor (which is invoked in the above .tex file).
You should let Gradescope know which page has which problem. Placing a \newpage command just before each question statement should put each question on a separate page and facilitate this process. Here is a link to more instructions on this process, including a video showing the submission process. (Whoever does this early and successfully should post a note to Piazza about this.)
Day 4 : September 1   This day will be remote yet synchronous; Sottile is organizing a conference August 31–September 2.
Sections 2.2-2.3 : Substitute teacher Prof. Sarah Witherspoon. The zoom link is the same as for all other class meetings.
The last class in August ended with the beginnings of logical equivalence, which is the first topic of Section 2.2; while we will not review this in class, you may want to watch the GVSU video. Section 2.2 introduces the notions of converse and contrapositive, which are 'operations' on an implication, and are fundamental. It also discusses how to negate conditional statements (disjunctions and conjunctions came earlier with De Morgan's Laws), which will give us the tools to understand how to negate logical statements, and the section ends with more on logical equivalence. Section 2.3 gives some information on sets and how to form them, this is needed for the next section and beyond. Read Sections 2.2 and 2.3 and watch the videos before class.
If needed:   Logical Equivalence (8:04) GVSU Screencast 2.2.1
Watch   Converse and Contrapositive (7:00) GVSU Screencast 2.2.2
Negations of Conditional Statements (4:29) GVSU Screencast 2.2.3
Logical Equivalence Without Truth Tables (7:08) GVSU Screencast 2.2.4
Sets and Set Notation (5:35) GVSU Screencast 2.3.1
Open Sentences and Truth Sets (7:24) GVSU Screencast 2.3.2
Elements, Subsets, and Set Equality (7:01) GVSU Screencast 2.3.3
Set-Builder Notation (9:25) GVSU Screencast 2.3.4
Worksheet for today
Day 5 : September 3   Section 2.4 : Quantifiers and Negation, and the start of Section 3.1.
Before class meets, read Section 2.4, and watch the following videos. There is also a concept quiz for today. Please complete that, as well.
We will start class by discussing the last two problems on sets from the worksheet from Tuesday.
Watch   Quantified Statements (10:06) GVSU Screencast 2.4.1
Negating Quantified Statements (10:01) GVSU Screencast 2.4.2
Integer Divisibility (8:53) GVSU Screencast 2.3.1
Worksheet for today.   Concept Quiz.
September 7   Second homework due.     The homework assignment, in .pdf, and the same file in .tex.
You should let Gradescope know which page has which problem. Placing a \newpage command just before each question statement should put each question on a separate page and facilitate this process.
Day 6 : September 8   Section 3.1 : Direct proofs
Before class meets: Read Appendix A, as well as Section 3.1. We begin a more complete study of direct proofs in mathematics. To have more to discuss, the text introduces notions of divisibility and congruence for integers. You likely are familiar with the first, but not the second.
Watch   Direct Proof Involving Divisibility (8:50) GVSU Screencast 3.1.2
Integer Congruence (9:28) GVSU Screencast 3.1.3
Proofs Involving Integer Congruence (8:01) GVSU Screencast 3.1.5
Worksheet for today.   Concept Quiz.
Day 7 : September 10   Section 3.2 More Methods of Proof :
We first finish Section 3.1, and then go on to Section 3.2. We will likely only get through using the contrapositive
Before class meets: Read Section 3.2, watch the following videos, and do the Concept quiz.
Watch   Proof by Contraposition (6:50) GVSU Screencast 3.2.1
Proof of biconditional statements (7:13) GVSU Screencast 3.2.3
Constructive proofs (7:09) GVSU Screencast 3.2.5
Worksheet for today   Concept Quiz.
September 14   Third homework due at 23:59. The homework assignment, in .pdf, and the same file in .tex.
I recommend that you check that my .tex compiles in your LaTeX environment and that the output matches (mutatis mutandis) the .pdf I provide. Also, as you answer problems in the .tex, continue to compile it to check for errors.
You should let Gradescope know which page has which problem.
Day 8 : September 15   Sections 3.2 More Methods of Proof : We will discuss how to prove biconditionals, proofs by construction, and talk about the test. Today there are no videos, and no concept quiz.
Problems from old exams of Sottile   This is a sample of the types of questions Sottile may ask, nothing more, and nothing less.
September 16   Term Paper topic due. Please submit it to Gradescope, there is an assignment "Term Paper Topic", and you are asked to type in your topic, and then a short description of what you intend to write into dialog boxes. I'd recommend formulating this first in a plain text editor and then cutting and pasting (and revising) what you write, as I reserve the right to deduct points for exceptionally poor writing.
Day 9 : September 17   First Test. Chapters 1–3.   This will be administered remotely. The Mathematics Department is using Zoom proctoring, and here are its instructions for the student. Our exam will be administered through Gradescope, so you will have to make the necessary changes.
Day 10 : September 22   Section 3.3 Proof by Contradiction and Section 3.4 Proof by Cases (We skip Section 3.5):
Proof by contradiction Reductio as Absurdum is perhaps the the most daring and powerful method of proof. Here is a famous quote about it:
"It (proof by contradiction) is a far finer gambit than any chess gambit: a chess player may
offer the sacrifice of a pawn or even a piece, but a mathematician offers the game."
Question: To whom is this quote attributed?
Proof by contradiction is powerful and fundamental, which is why I am assigning all three videos. The second one has, in passing, a discussion of the fundamentally flawed method of "backwards proofs", which are popular with high school teachers. Keep this in mind.
While I am only assigning the first video on cases, you may want to eventually watch all of them. Do read both Section 3.3 and 3.4; you will be responsible for the material in both. Do not forget to do the concept quiz on Gradescope, after watching the assigned videos and reading the text.
Watch   Proof by Contradiction (6:58) GVSU Screencast 3.3.1
Proof by Contradiction, part 2. (5:09) GVSU Screencast 3.3.2
Proof by Contradiction, part 3. (3:45) GVSU Screencast 3.3.3
Proof Using Cases, part 1. (4:48) GVSU Screencast 3.4.1
Worksheet for today   Concept Quiz.
Second Written Assignment   Your second written assignment involves researching Klein Bottles, and then writing a short essay/discussion about Klein bottles. Here are some sources to get you started, you should look up several other sources about Klein bottles. Above all, enjoy these. The assignment is due Friday 2 October.
Day 11 : September 24   Section 4.1 Mathematical Induction:
Mathematical Induction is one of the premiere methods of proof that we teach. It is a departure from previous methods, but builds on the maturity that you amassed in handling proofs using reductio ad absurdum. You will learn it as a method for proving statements about the positive integers, but its appearance in advanced classes goes far beyond the integers.
Do not forget your new homework assignment.
Watch:   The traveler and the strange staircase. (3:19) GVSU Screencast 4.4.1
Mathematical Induction: Part I (5:23) GVSU Screencast 4.4.2
Mathematical Induction: Part II (6:15) GVSU Screencast 4.4.3
Worksheet for today   Concept Quiz.
September 28   Fourth homework due.
Day 12 : September 29   Section 4.1 Mathematical Induction:
We are continuing to study mathematical induction, in Section 4.1. I recommend that you watch two more videos on mathematical induction, and re-read the book.
Watch:   Mathematical Induction: Integer division (7:05) GVSU Screencast 4.1.4.
Mathematical Induction: Inequalities (4:45) GVSU Screencast 4.1.5.
Mathematical Induction: Calculus Example (8:27) GVSU Screencast 4.1.6.
Worksheet for today   Concept Quiz
Day 13 : October 1   Section 4.2 Other Forms of Mathematical Induction:
We will discuss extensions of mathematical induction, which make the method more useful.
Do not forget your new homework assignment.
Watch:   Mathematical Induction: Domino toppling (3:07) (University of Durham)
The extended principle of Mathematical Induction (9:16) GVSU Screencast 4.2.1
There is no worksheet for today
October 2   Second Written Assignment due. Submit it both to Gradescope (where I will mark it) and to turnitin.com, using Class ID: 25616912 and Enrollment key 314159265.
October 5   Fifth homework due.
Day 14 : October 6   Section 4.3 Recursively defined sequences:
Today, we will work with recursively defined sequences, mostly with the Fibonacci numbers. There was a recording error on Thursday in both classes, the first 25 minutes were lost.
There will be a concept quiz for this assignment.
Watch:   Second Principle of Mathematical Induction (8:49) GVSU Screencast 4.2.3
Recursively Defined Sequences (9:22) GVSU Screencast 4.3.1
Fibonacci Numbers (9:07) GVSU Screencast 4.3.2
Proofs involving Fibonacci numbers (11.43) GVSU Screencast
  Concept Quiz for today
Day 15 : October 8   Section 5.1 Sets and operations on sets, and the start of Section 5.2:
Read Section 5.1 and the first part of Section 5.2 on subsets.
Watch:   Sets and set operations (8:18) GVSU Screencast 5.1.1
Operations using infinite sets (9:40) GVSU Screencast 5.1.2
Subsets and set equality (9:41) GVSU Screencast 5.1.3
Proving subset inclusion (7:06) GVSU Screencast 5.2.1
Concept Quiz for today
October 9   Paper outline due.
October 12   Sixth homework due.
Day 16 : October 13   Section 5.2 Proving Set Relationships, and the start of Section 5.3 Properties of set operations:
Watch:   Proving subset inclusion (7:06) GVSU Screencast 5.2.1 Only if you did not yet do this
Proving Set Equality (14:59) GVSU Screencast 5.2.2.
Identities about Sets (9:18) GVSU Screencast 5.3.1.
Concept Quiz for today
Day 17 : October 15   6 :
Watch:   Identities about Sets (9:18) GVSU Screencast 5.3.1
Using Set Identities (8:57) GVSU Screencast 5.3.2
Using Set Identities, III (12:25) GVSU Screencast 5.3.4
Concept Quiz for today
October 19   Seventh homework due.
Day 18 : October 20   Sections 5.4 and 5.5 : Cartesian products and indexed Families of sets.
Read both Sections. I will not have a concept quiz, but you want to become familiar with the definitions, as they may end up on the test. In class, I will give a better way to prove results such as Theorem 5.25 than given in the book. This also applies to results in Section 5.5.
Problems from old exams of Sottile   This is a sample of the types of questions Sottile may ask, nothing more, and nothing less.
Watch   Cartesian Products (11:31) GVSU Screencast 5.4.1
Proofs Involving Cartesian Products (8:13) GVSU Screencast 5.4.2
Day 19 : October 22   Second test. Section 3.3, 3.4, Chapters 4 and 5. This will be administered remotely. The Mathematics Department is using Zoom proctoring, and here are its instructions for the student. Our exam will be administered through Gradescope, so you will have to make the necessary changes.
October 26   Completed draft of paper due. This is worth 20 points towards the written assignments. Submit through Gradescope. Late or incomplete work will lose points.
Day 20 : October 27   Section 6.1 : Functions
We turn our attention now to yet another foundational concept in mathematics, that of a function. The second video gives a "definition" of function in terms of a rule:

Inadequate Definition from Video:
A function from a set A to a set B is a rule that associates with every element x of the set A exactly one element of the set B.

This is also used in our book. We actually do not take this to be the definition of function in mathematics. The problem with this definition is that the word rule in the definition is completely imprecise (what is a rule)? and it is misleading (is the function the process or the result?) In Mathematics, we define a function to be what you would call in Calculus the graph of a function. I will test you on this. Here is the correct definition:

Definition:
A function f from a set A to a set B is a subset f of the Cartesian product A×B such that.
    (1) For every element x in A, there is an element y of B such that (x, y) lies in f.
    (2) For every element x in A and any elements y, z of B, if we have both that (x, y) lies in f and that (x, z) lies in f, then y=z.

Conditions (1) and (2) are often combined and restated as "For every element x in A, there is a unique element y of B such that (x, y) lies in f.

Because of this inadequacy in their definition of function, the emphasis on the process of computing a function is misplaced. It is OK as a rhetorical tool, but you need to remember that A function is not a process. This begins to be remedied in the definition of equality of two functions.
The videos below (as well as reading the book) are for you to do after we cover this material in class.

Watch   Functions: The Big Concept (6:12) GVSU Screencast 6.1.1
Watch Functions: Terminology (15:44) GVSU Screencast 6.1.2
Optional   Function Example: Names to Initials (7:57) GVSU Screencast 6.1.3
Optional   Function Example: Counting Primes (8:29) GVSU Screencast 6.1.4
Optional   Function Example: Integer Congruence (7:17) GVSU Screencast 6.1.5
Watch Function Example: Derivatives (6:56) GVSU Screencast 6.1.6
The focus on "process" causes this video to make another patronizing error:
By The Fundamental Theorem of Calculus, every continuous function has a derivative. Since e-x2 is continuous (why?), it has a derivative.
Optional   Function Example: Averages (5:07) GVSU Screencast 6.1.7
Watch Function Non-Example: (6.1.8) GVSU Screencast 6.1.8
Watch Equality of Functions (11:25) GVSU Screencast 6.2.1
Day 21 : October 29   Section 6.2 and Section 6.3:
First, watch the videos I assigned for Tuesday, October 27.
Next, read Section 6.3, on injections, surjections, and bijections. There is a concept quiz today.   Concept Quiz
November 2   Eighth homework due.
Day 22 : November 3   Sections 6.4 and 6.5 :
The first four videos review injective and surjective functions; you need not watch all of them, but should watch some of them if you need more review on the concepts of injections and surjections.
You have a concept quiz for Tuesday
Section 6.4 concerns compositions of functions and Section 6.5 is about inverse functions. A significant part of Section 6.5 is getting you used to the definition of functions as a set of ordered pairs, which we have already spent some time doing.
Maybe watch Injective Functions (6:49) GVSU Screencast 6.3.1
Advised to watch How to prove that a function is injective (9:05) GVSU Screencast 6.3.2
Maybe watch Surjective Functions (9:03) GVSU Screencast 6.3.3
Maybe watch How to prove that a function is surjective (15:49) GVSU Screencast 6.3.4
Watch Composition of Functions (8:42) GVSU Screencast 6.4.1
Maybe watch Proofs of results involving compositions of functions (7:54) GVSU Screencast 6.4.2
Watch Inverses of functions (10:41) GVSU Screencast 6.5.3
  Concept Quiz for today
Day 23 : November 5   Section 6.5 and Section 6.6:
Marking papers got me behind my schedule. There is a Concept Quiz for today, and we will complete discussions of functions.
Day 24 : November 10   Peer review of papers (remote, synchronous)
November 11   Ninth homework due.
It is Due at 17:00. The homework assignment, in .pdf, and the same file in .tex.
Day 25 : November 12   7.1 Relations and 7.2 Properties of Relations:
Relations are a mathematical concept/construct that underlies many, many concepts and constructions in mathematics, such as "is a subset of", 3<4, the construction of the integers, rational numbers, and real numbers. Functions are relations.
The minimalist definition of a relation is in fact useful, as a relation is then a very flexible notion that can model many mathematical ideas.
Watch   Relations (6:53) GVSU Screencast 7.1.1
Properties of Relations (9:55) GVSU Screencast 7.2.1
Concept Quiz for today
Day 26 : November 17   7.2 Properties of relations (equivalence relations) and 7.3 Equivalence Classes:
We had discussed that equivalence relations are a way to generalize the elementary notion of "=" (equality). This section (7.3) treats this in its entirety. I will be doing the material in GVSU Screencast 7.3.2: Properties of Equivalence Classes in our lecture (it is too important to leave for the videos), and it is also given in the book. The next concept, that of a partition of a set, and its relevance to equivalence relations, is the pièce de résistance of this section.
Watch:   Equivalence Relations (14:03) GVSU Screencast 7.2.2
Watch:   Equivalence Classes (5:54) GVSU Screencast 7.3.1
If needed Properties of Equivalence (15:14) GVSU Screencast 7.3.2
Watch:   Partitions (6:30) GVSU Screencast 7.3.3
Concept Quiz for today
November 19   Tenth homework due.
Day 27 : November 19   Today, on our last regular day of class, we finish up the discussion of partitions and equivalence relations, and then begin to discuss arithmetic modulo an integer m. There is no concept quiz today.
Course evaluations. Please go to the portal.
The integers modulo n (6:40) GVSU Screencast 7.4.1
Modular arithmetic (11:07) GVSU Screencast 7.4.2
Day 28 : November 24   Third test. Chapters 6 and 7.
Problems from old exams of Sottile   This is a sample of the types of questions Sottile has asked, nothing more, and nothing less.
Course evaluations. Please go to the portal.
November 30   Term paper due.
Submit to turnitin.com, using Class ID: 25616912 and Enrollment key 314159265.
Sample Rubric. This is to help you think about the different aspects of your paper.