Math 2110 (Section 410) Homepage
News & Announcements
Assignments:
Class Topics: (at & after the end of the book)
- Some function facts [Source: (.tex)]
Zermelo-Frankel Axiom of Choice Set Theory [Source: (.tex)]
Cantor-Schroder-Bernstein Theorem [Source: (.tex)]
Types of Infinity Preprint
Video: Cardinality (60 mins)
This video supplements the above handouts. It is most closely related to the
types of infinity preprint.
- Equivalence Relations and Partial Orders (.pdf) [Source: (.tex)]
Video: Equivalence Relations, Partition, and Partial Orders (75 mins)
- The Euclidean Algorithm and Basic Number Theory (.pdf) [Source: (.tex)]
I wrote a couple of SageMath interactive webpages to automate the Extended Euclidean Algorithm.
They can be found here.
Video: Divisbility and the Extended Euclidean Algorithm (part 1 of 3) (33 mins)
[On part 1: SORRY! The camera went out of focus for a few minutes.
You can follow along by looking at the proof in the handout.]
Divisbility and the Extended Euclidean Algorithm (part 2 of 3) (56 mins)
Divisbility and the Extended Euclidean Algorithm (part 3 of 3) (55 mins)
Part 1 covers the Well Ordering Principle and how it relates to Math Induction.
Then the Division Algorithm is proven.
Part 2 covers some basic number theory, definitions of gcd/lcm, the extended Euclidean algorithm,
and some results about facotoring into primes.
Part 3 covers "basics" about modular arithmetic and some examples of computing multiplicative
inverse mod n. Bonus: how to rationalize a denominator using the Euclidean algorithm.
- Factoring Functions: A set theoretic version of the first isomorphism theorem.
Video: Factoring Functions (72 mins)
This is a review of one-to-one, onto, and other function stuff. Then we show that
every function can be pulled apart into an onto, invertible, and one-to-one piece
(i.e., the first isomorphism theorem for sets). Then we have a brief discussion of
this theorem related to group theory and linear algebra.
- Defining Continuity
Video: Continuity: Analysis Style (48 mins)
Continuity: Metric Space Style (52 mins)
In the first video, we give a quick review of the definition of the limit of a sequence.
Then we define limits of functions and followed by the definition of continuity. Then we
have an example (i.e., proving that $f(x)=x^2+3$ is continuous at $x=-1$). This is followed
by some sample theorem proofs: If $f$ and $g$ are continuous, then $f+g$ is continuous, and
composing continuous functions yields a continuous function.
In the second video, we look at generalizing continuity to $\mathbb{R}^n$ and we then we
generalize to metric spaces. Definitions of metric, normed, and inner product spaces are
given along with some basics concepts like completeness.
- Some Topology. The first two videos are related to the paper:
The Simplicity and Beauty of Topology: Connecting with the Intermediate Value Theorem
The third video is related to:
Fitting the Extreme Value Theorem into a Very Compact Paper
The last video takes a tour of this preprint:
Crossing Through and Bouncing Off Infinity: Graphing Rational Functions
Video: Continuity: Topology Style (50 mins)
Connectedness and the Intermediate Value Theorem (60 mins)
Compactness and the Extreme Value Theorem (66 mins)
Graphing Rational Functions (20 mins)
In the first video, we show how to reshape the definition of continuity into a statement
about open sets. We give the definition of a topology and discuss some basic topological
notions.
In the second video, we introduce the topological notion of being connected. We characterize
connected subsets of the reals as being intervals. Then we prove the Intermediate Value Theorem
and discuss a little about path connectedness.
In the third video, we introduce the topological notion of compactness. After giving examples
and non-examples, we characterize compact subsets of the reals as closed and bounded subsets.
We also discuss a little about homeomorphisms and one and two point compactifications of the
real numbers.
In the fourth video, we take a tour of the Crossing Through and Bouncing Off Infinity preprint.
Here we discuss treating infinity as a number. A discuss how local behavior near zeros of rational
functions is very much like local behavoir near vertical asymptotes (i.e., poles).
- Some Group Theory.
Video: Binary Operations & Group Axioms (56 mins)
Exponents, Orders, & Cyclic Groups (53 mins)
Subgroups, Cosets, & Lagrange's Theorem (64 mins)
Group Actions & The Orbit-Stabilizer Theorem (52 mins)
The first video covers the basics about binary operations and some popular properties. We then
look at the definition of a group and look at examples and non-examples.
The second video contrasts multiplicative and additive notations. We look at exponents and their
laws. Then we define the notion of a cyclic group, the order of an element, and look at some examples.
The third video covers basics about subgroups. We then define and prove basic facts about cosets of
subgroups and work up to Lagrange's Theorem plus an application showing groups of prime order are cyclic.
These topics are all accompanied by some examples.
The fourth video covers the definition of a group action, stablizer of an element, and orbit of an
element. After some basic examples, we show that stabilizers are subgroups, orbits partition, and
prove the Orbit-Stablizer Theorem. The video ends with a sketch of the classification of Platonic
solids and deriving the sizes of their rotational symmetry groups.
- What is a Manifold?
Video: Cardinality vs. Dimension (46 mins)
In this video we contrast cardinality and dimension. We explore the definitions of topological and smooth manifolds
and briefly discuss some related concepts like tangent spaces.
- What are the Real Numbers? Some related handouts:
Axioms Characterizing the Real Numbers [Source: (.tex)]
Characterizing the Real & Complex Numbers [Source: (.tex)]
Also, check out John Baez's lovely exposition on the weird and wonderful Octonions.
Video: The Real Numbers Axioms and Characterization (60 mins)
This video looks at the axioms that characterize the real number system. Then we briefly look
at how one can construct the real numbers. We finish by looking into the real and complex number
systems' relations to geometry.
- Turing Machines & Computability
Random Wiki Entries: Turing Machines, Recursive Sets, and Recursively Enumerable Sets.
Jeff Hirst taught a great Computablity Theory course a few years ago. He used a
textbook by Rebecca Weber. It's a lovely little text.
Video: Turing Machines & Computability (59 mins)
In this video, I attempt to explain what a Turing machine is and how this relates to the idea of computability.
I also show that the halting problem is unsolvable and explain what recursive and recursively enumerable sets
are. At the end I briefly indicate the connection between computability and provability.
- Category Theory
There are many good resources out there. I especially like Steve Awodey's Category Theory text. The
introduction (approx first 50 pages) of Bell's Topos Theory book is another great run down of basic notions.
If you want videos that go into more detail, I highly recommend Steve Awodey's Series of Lectures on YouTube.
Also, anything by the "Catsters" on YouTube is worthwhile. The Catsters have dozens of videos which
explain all sorts of basic category theoretic definitions and ideas.
Video: Category Theory (63 mins)
This video runs through the definition of a category along with some examples. We then take a look at some
definitions like that of isomorphism, monic, epic, subobject, terminal, initial, product, and functor.
Suggested Schedule:
- By 03/23, finish function, set theory, & Cardinality related topics
- 03/24--03/25 = Relations
- 03/27--03/31 = Extended Euclidean algorithm related videos
- 04/01--04/03 = Factoring functions
- 04/06--04/07 = Continuity
- 04/08 = Continuity Topology style
- 04/13--04/15 Connected, Compact, and Bouncing Off Infinity
- 04/17 = Binary operations & group basics
- 04/20 = Exponents & cyclic groups
- 04/21--04/22 = Subgroups, cosets, Lagrange's Theorem, group actions, & Orbit-Stablizer Theorem.
- 04/24 = Cardinality vs. Dimension & What is a manifold?
- 04/27 = What are the real numbers?
- 04/28 = Computablity and Turing Machines
- 04/29 = Category Theory
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
03/27--03/31 The Euclidean Algorithm and Basic Number Theory (.pdf) [Source: (.tex)]
I wrote a couple of SageMath interactive webpages to automate the Extended Euclidean Algorithm.
They can be found here.
Video: Divisbility and the Extended Euclidean Algorithm (part 1 of 3) (33 mins)
[On part 1: SORRY! The camera went out of focus for a few minutes.
You can follow along by looking at the proof in the handout.]
Divisbility and the Extended Euclidean Algorithm (part 2 of 3) (56 mins)
Divisbility and the Extended Euclidean Algorithm (part 3 of 3) (55 mins)
Part 1 covers the Well Ordering Principle and how it relates to Math Induction.
Then the Division Algorithm is proven.
Part 2 covers some basic number theory, definitions of gcd/lcm, the extended Euclidean algorithm,
and some results about facotoring into primes.
Part 3 covers "basics" about modular arithmetic and some examples of computing multiplicative
inverse mod n. Bonus: how to rationalize a denominator using the Euclidean algorithm.
03/24--03/25 Equivalence Relations and Partial Orders (.pdf) [Source: (.tex)]
Video: Equivalence Relations, Partition, and Partial Orders (75 mins)
03/23 Let's meet via Zoom and then (and as a backup plan)...
Video for Monday: Cardinality (60 mins)
Join my Zoom Meeting
03/16-03/20 [Spring Break Part 2: Brought to you by Coronavirus Hysteria.]
03/09-03/13 [Spring Break]
03/06 Finishing function basics. A pair of short proofs regarding image and inverse images of sets.
Discussed cardinality and the Cantor-Schroder-Bernstein Theorem [Source: (.tex)]
Types of Infinity Preprint
Don't forget that...
Homework #5 [Source: (.tex)] is due Friday, March 20th 27th.
03/04 Short paper presentations.
Some function facts [Source: (.tex)]
03/03 Left/right/two-sided inverses vs. injective/surject/bijective.
A little bit of images and preimages.
03/02 Short paper discussion. Resources? Mathscinet, stack exchange, preprint archive, etc.
02/28 4.3 = functions, 1-1, and onto
Homework #5 [Source: (.tex)] is due Friday, March 20th.
02/26 The Axiom of Choice
02/25 4.2 = intervals, unions, intersections, cartesian products, well ordering
02/24 4.1 = some set theory
Discussed all but AC today:
Zermelo-Frankel Axiom of Choice Set Theory [Source: (.tex)]
For later:
Cantor-Schroder-Bernstein Theorem [Source: (.tex)]
Everyone should put together a short "research paper" about an important theorem.
We will have short presentations (about 3 minutes each) about these theorems in
class on Wednesday, March 4th.
My short paper and slides [Source: (.zip)]
A few suggested theorems to discuss:
Rolle's Theorem and/or Mean Value Theorem [Aaron]
Intermediate Value Theorem [John]
Extreme Value Theorem [Abdel]
The Fundamental Theorem of Calculus
The Fundamental Theorem of Arithmetic [Gabbi]
The Fundamental Theorem of Algebra
Pick one and call "dibs".
Homework #4 [Source: (.tex)] is due Friday, February 28th.
02/21 Test #1 covering Chapters 1, 2, & 3.
Take home portion due Tuesday, February 25th.
02/19 Comments about Friday's test. Then a touch more "baby analysis".
Defined limit of a function, continuity at a point and gave an
example of each.
02/18 Review of logically valid, satisfiable, unsatisfiable (building models).
Showing a_n converges to A then 1/a_n converges to 1/A (for non-zero stuff).
and a few other theorems.
02/17 Counter-examples, theorems: conv implies bounded, lims unique, lim of sum
[4.2 & 4.3]
02/14 Showing n^2 is unbounded and more convergence examples.
[4.1 finished]
02/12 Test #1 will be Friday, February 21st. It will cover Chapters 1 through 3.
A couple old tests can be found here.
We did an example of a convergence proof and a boundedness example.
Examples of Divergent Sequences [Source: (.tex)]
Homework #4 [Source: (.tex)] is due ???Wednesday, February 19th???.
02/11 A bit about LaTeX (lists, labels, etc.)
4.1 Sequences, boundedness, convergence
02/10 Clean up 3.6 then Well Orderings and Division Algorithm.
02/07 3.5 & 3.6: Contradiction & other strategies.
We did a few more "divides" proofs. Showed sqrt(2) is irrational
and that there are infinitely many primes (contradiction).
Discussed some proof strategies.
02/05 More induction, strong induction, & divisibility.
02/04 3.3 & 3.4: Examples of induction and induction pitfalls.
02/03 3.1 & 3.2: Quick tour of E and PA.
Summary sheet for proofs in Systems E & PA [Source: (.tex)]
[Handed out photo copy of proofs of E theorems]
Discussed what system E = Equality is encoding and looked at a few proofs.
Discussed what system PA = Peano Arithmetic is encoding.
Grassmann = First proofs, von Neumann = set model
"Proved" 2+2=4 and 3*2 = 6
Discussed induction briefly and showed 0+...+n = n(n+1)/2
Homework #3 [Source: (.tex)] is due Wednesday, February 12th.
01/31 Finished System K examples. Proved K24, K26 (very tricky with lemma),
K28, & K29. Also, gave out big K19 hint.
01/29 More LaTeX. Proved K18 and Tex-ed up the proof.
Also, proved K22 and sketched K23.
01/28 More System K including Add there exists and Rule c.
Proved K11 - K15, & K20.
01/27 Today: Went over the LaTeX code in Deduction_Theorem.tex.
Want to install LaTeX on your computer? LaTeX Project has links for MAC, Windows, etc.
On a Mac, the typical LaTeX system is MacTeX.
On a PC, the typcial LaTeX system is MiKTeX.
Both of these are open source & free!
If you don't want to install LaTeX on your machine, you can typeset online through
a webbrowser. There are several options (you could Google "latex online editor"), but
the most popular is OverLeaf.
01/24 Discussed the deduction theorem (in System K). Proved K3 - K7, & K9.
Homework #2 [Source: (.tex)] is due Monday, February 3rd.
01/22 Summary sheet for proofs in System K [Source: (.tex)]
Finished discussing models started system K. Proved K1 & K2.
01/17 Discussed handout: The Deduction Theorem [Source: (.tex)]
Surveyed the rest of chapter 1 and proved Lemmas 7 to 11.
Homework #1 [Source: (.tex)] is due Friday, January 24th.
01/15 Proved Lemmas 1 to 6 and discussed the deduction theorem.
01/14 Summary sheet for proofs in System L [Source: (.tex)].
Adequacy of the Sheffer Stroke [Source: (.tex)] and the Wikipedia's article
01/13 Discussed the syllabus and class overview. Covered sections 1.1 - 1.3.
01/10 Syllabus, schedule, & homework pages updated.
ASULearn switched on.
Course Information:
MAT 2110 Section 410
MWF 9:00am - 9:50am in WA 302
T 10:00am - 10:50am in WA 105
Final Exam Date/Time: Monday, May 4 from 8:00am - 10:30am
Any questions about this class?
Send me an email at cookwj@appstate.edu