MAT 108: Introduction to Abstract Mathematics


Finals week office hours, study guide posted

Go to [Final Exam Information] to see the study guide for finals week, as well as the office hours during finals week. You should also have received an email through Canvas about the virtual office hours.

Nugget hopes you'll fill out the course evaluations for this class.
Nugget says, "May I pretty please have a treat your course evaluation?"

(Pretty) Please complete the ACT Course Evaluation!

You should hopefully have also received an email asking you to complete a course evaluation for MAT108.

Please fill this out; it is mostly multiple choice and feedback is very important to my career, as well as Hans' career! Thanks in advance for your time; I know you're very busy right now.

Your responses are also very important to the math department, as we use these for evaluating personnel.

All MAT 108 Review Session

You should hopefully have received the following email (appended below) from the math department's Undergraduate Program Coordinator, Viviana Gonzalez. If you have the time, I encourage you to attend, as it is an opportunity to ask questions and review the various topics we covered this quarter.

I am friends with the other MAT108 instructors, and we have talked about what topics we've covered in our classes. The other two instructors are using the old book, so their topics might be slightly different. For for instance, we didn't spend a lot of time talking about formal logic. However, we did cover cardinality. With this in mind, I have a few tips for you:

On that last note, I will be posting the final exam study guide by this Tuesday, and there will be five exam-style practice problems you can work on.

Email from Viviana:
    Dear MAT 108 Students,

    I hope that this email finds you well.

    I am reaching out to let you all know that one of our amazing MAT108 tutors, Brian, has decided to host a MAT108 review session before finals week. 

    This review session will be held on Saturday, March 18, 2023, in the Mathematical Sciences Building, room 1147 (MSB 1147). This review session will be held from 12:00 PM to 4:00 PM.

    I would like to thank Brian for being gracious in offering this to you all, as I am sure it is going to be very helpful for those who decide to attend. I will be there as well to assist with any room/organizational concerns.

    Please feel free to contact me if our department may be of any further assistance.

    Thank you all for your time, and I wish you all the best of luck as we begin gearing up for Finals Week.


Exam 2 grades are up

Exam 2 grades are now posted on Canvas. Your posted grades are out of 100&percnt. The highest score was 100% (60/60 points); the average score was 77%. ] Solutions have been posted below in [Lectures and Materials].

Mock Exam 2

Here is a mock exam for Exam 2: [MockExam2.pdf]
It includes a list of important topics we covered in February, as well as sample problems in the same style as the exam problems. You will go over these problems in discussion on Tuesday before the exam; I will not be uploading solutions for these.

Issue with PS4 Exercise 1abc grading

Multiple students have notified me of an issue with the grading of PS4 Exercise 1abc; this will be fixed by next week. Since you won't be able to do revisions on this Exercise without your adjusted grade, I am giving everyone full credit on PS4 Revisions Exercise 1abc; in other words, you don't need to do revisions on this exercise. The solutions for PS4 will be posted this Saturday as usual.

No instructor office hours on Monday, 2/13

TA office hours + Thursday office hours will happen as usual.

Exam 1 grades are up

Exam 1 grades are now posted on Canvas. Your posted grades are out of 100% and were determined from your raw score out of 60 points using the function

GRADE = [(POINTS EARNED/60)*75 +25] %.
The highest score was 100% (60/60 points); the average score was 76%. ] Solutions have been posted below in [Lectures and Materials].

Future K-12 STEM teachers: see [CalTeach's brochure] and [Internship Opportunities].

Math Club Advertisement!

winter quarter 2023 math club poster

If you find mathematics interesting and want to find similarly interested friends, I encourage you to go to some math club meetings! Click the poster to view a larger version.

The Math Club at UC Davis is a space for people interested in Mathematics or curious about Mathematics. We are open to all backgrounds and majors. In doing so, we make mathematics accessible and engaging. The club strives for a supportive community by engaging members in fun social activities, and allowing our members to express themselves. We provide opportunities for advancement in one's career through various seminars and professional development workshops. Regardless of where one is in their math journey, we help our members pursue their passion or interest for mathematics and let them grow as mathematicians. They would learn to make meaningful and interesting connections in mathematics to their chosen careers. We also put our efforts in destigmatizing mathematics as a boring or inapplicable field or that only a specific group can do mathematics. While math may not be easy, it can be fun, and anyone can do it for any reason, and Math Club makes efforts that demonstrate that.

MAT108 Tutoring starts Thursday, January 26!

The math department has organized an additional MAT108 drop-in tutoring service, which are like office hours:

We have updated our MAT108 tutoring schedule. All MAT 108 tutoring drop-in hours are held in our calculus room (in the Mathematical Sciences Building). The Calculus Room is located near the front entrance of the building, right after the elevators (there is a 'Calculus Room' sign above), room 1118 MSB.

Please see below for our updated Winter 2023 MAT 108 Tutoring schedule:

Tuesdays: 6-8 PM
Wednesdays: 6-8 PM
Thursdays: 6-8 PM

If you joined the class late:

If you joined the class on or after Monday, Jan 16 and were unable to submit your PS1 solutions to Canvas by Jan 17, you may submit your PS1 by Friday, Jan 27 at 11:59 pm to the PS1 assignment on Canvas, for full credit. Note that for this course, each PS is usually graded twice; see the syllabus for more details. If you joined late, your PS1 will only be graded once, for full credit.

See the Lectures and Materials section below for a summary of what we covered in previous classes. I will not be posting my lecture notes; the book sections are also listed below, and you should make full use of the textbook as your main written materials for this course.

Course information

[Course syllabus]

[Class calendar]

Course instructor: Dr. Melissa Zhang, MSB 2145

Instructor office hours: MSB 2145, Mondays 1-2 pm and Thursdays 1-3 pm

Teaching assistant (TA): Hans Oberschelp, MSB 2202

TA office hours: MSB 2202, Mondays 2-4 pm and Thursdays 10-11 am

Lectures VEIMYR 116 MWF 11:00 am -- 11:50 am
Section D01 45823 Discussion WELLMN 207 T 5:10 pm -- 6:00 pm
Section D02 45824 Discussion WELLMN 201 T 6:10 pm -- 7:00 pm

Textbook: The Art of Proof by Matthias Beck and Ross Geoghegan, available digitally [here].

Course drop date: February 06, 2023 (20 Day Drop)

Final Exam Information

Important information:

[Study Guide Version 2]: This contains both a Mock Final and some additional practice problems.

Each problem in the Mock Final is at the upper end of difficulty for problems on the related topic. If you are having trouble with a given problem, feel free to take a look at the hints below:

Here are my solutions to the Mock Final Exam: [MockFinalSolutions.pdf]
This is a handwritten PDF, so the size is quite large.

I will not post solutions to the additional practice problems, but some select problems will be covered during Discussion.

Solutions to Exams 1 and 2 are available in [Lectures and Materials].

Finals week office hours:

Instead of the usual office hours, Hans and I will be holding finals week office hours as follows: The Zoom link is the same for all the virtual office hours. You do need a ~secret password~ to join; this will be sent to you via an email through Canvas.


Problem Set 9: Due Tuesday, 3/14/23 at 11:59 pm on Canvas



This (last!) problem set will be graded for completion; see the beginning of the problem set for more details.

Here are my solutions: [PS9solns.pdf]

Problem Set 8: Due Tuesday, 3/7/23 at 11:59 pm on Canvas



There was on Exercise 3b: It should be 0.346127127127... (note the 6). The files are updated on the website, but your computer might cache the old versions.

Here are my solutions: [PS8solns.pdf]

Problem Set 7: Due Tuesday, 2/28/23 at 11:59 pm on Canvas



Here are my solutions: [PS7solns.pdf]

Problem Set 6: Due Tuesday, 2/21/23 at 11:59 pm on Canvas



If you are having trouble with Exercises 1 and 2, look back at your notes from class last Monday, 2/13 about functions between sets, and Section 9.1 in the book. Here's an email response to a student that might offer some hints as well:

Here are my solutions: [PS6solns.pdf]

Problem Set 5: Due Tuesday, 2/14/23 at 11:59 pm on Canvas



Here are my solutions: [PS5solns.pdf]

For Exercise 1(a), inverse means multiplicative inverse (as opposed to additive inverse, which would not help you define division!).

Problem Set 4: Due Tuesday, 2/7/23 at 11:59 pm on Canvas



Here are my solutions: [PS4solns.pdf]

For future revisions: Keep your original solution in the document. Under your original solution, add

\paragraph{Revised solution.}
and write down your revised solution. In particular, if you aren't revising a solution, just don't include this new paragraph. For PS4 and beyond, a new command called
will be included in the preamble of the provided TeX file.

Problem Set 3: Due Tuesday, 1/31/23 at 11:59 pm on Canvas



Here are my solutions: [PS3solns.pdf]

Problem Set 2: Due Tuesday, 1/24/23 at 11:59 pm on Canvas



Hint for Exercise 2: First show that for all natural numbers k, k is greater than or equal to 1. For this problem, you may use the fact that the relations < and > are transitive.

For Exercise 4, you may use the fact that every natural number has a prime decomposition. (This is a weaker version of what's called the "Fundamental Theorem of Algebra".)

Here are my solutions: [PS2solns.pdf]

Problem Set 1: Due Tuesday, 1/17/23 at 11:59 pm in TeX'd and typeset PDF form on Canvas


Upload the TeX file to your own Overleaf account. For example, you could create an Overleaf project called "MAT108 Homeworks" and upload the PS1.tex file there. Then, press the green "Compile" or "Recompile" button to see the PDF. For your convenience, here's what the rendered PDF should look like: [PS1.pdf]

Here are the solutions.: [PS1solns.pdf]
Compare my writing style to your style and the style of the textbook. Remember that writing mathematics is an important component of this course. You will be graded on your writing style!

Lectures and materials

Lecture 27: Final Review 3

Lecture 26: The complex numbers

We introduced the complex numbers as a closed field coming from including i, the square root of -1, to the real numbers. Specifically, we showed that addition, subtraction, multiplication, and division are defined for the complex numbers. We discussed the modulus of a complex number, and how to convert between rectangular and polar coordinates. Finally, we used Taylor series to see why the exponential function appears in the polar representation of complex numbers.

Lecture 25: Groups

We defined groups axiomatically. It turns out, I disagree with the axioms the book gives! Here are my notes, with Proposition D.2 stated as axioms in the definition of a group:
  [Group Axioms and Uniqueness Proofs]
(This is why we weren't able to prove Proposition D.1 during class.) From there, we can use the axioms to prove that the identity element is unique, and also that given any element g in G, the inverse is also unique.

We gave familiar examples of abelian (i.e. commutative) groups, as well as an example of a non-abelian group, the group symmetries of the square (a dihedral group). We finished by defining rings and fields in terms of groups. We will talk about a very important field, the complex numbers, next class.

Lecture 24: Powersets and an infinite hierarchy of infinities (Section 13.3, 13.4)

We defined the symbols ≤ and < for cardinalities, and defined the powerset of a set to be the set of all subsets. We proved Theorem 13.30, which states that the powerset of any set has a strictly greater cardinality than the original set.
We also discussed the Cantor middle-thirds set, which we might revisit later during the last week of class. We also mentioned the Cantor-Schroeder-Bernstein Theorem, as an example of how proofs about cardinality are generally quite difficult.

Lecture 23: Cardinality for infinite sets (Section 13.1, 13.2)

We defined countably infinite sets to be those with the same cardinality as the natural numbers. We used Cantor's diagonalization argument to show that the set of real numbers is not countable.

Lecture 22: Cardinality for finite sets (Section 13.1)

We defined cardinality for finite sets by discussing bijections to (or from!) the sets [n] = {1,2,...,n}. We proved Theorem 13.4, which states that there is no bijection between the sets [m] and [n] when m and n are not equal.

[Exam 2 solutions]

Lecture 21: Geometric series, decimal expansions (Chapter 12)

We defined geometric series (with leading term a and common ratio r), and noted that they converge iff |r| < 1. We gave a formula for the sum of such a series. We then defined decimal expansions for real numbers. It turns out that every real number has a decimal expansion, and every rational number has exactly two infinitely repeating decimal expansions. For the remainder of class, we discussed how to turn an (eventually) repeating decimal expansion into the fraction (i.e. rational number) that it represents, and vice versa.

Lecture 20: Decimal and Binary, infinite series (Section 12.1, and some of Chapter 4)

We discussed writing natural numbers in base 10 (decimal) and base 2 (binary). We then discussed writing fractions as decimals, leading us to define "infinite series" rigorously. We talked about two divergent series to keep in mind. In the first, the terms alternate between -1 and 1; since the sequence of terms diverges, the series diverges as well. We then talked about the harmonic series, whose sequence of terms is 1, 1/2, 1/3, 1/4, ... . We needed to compute the integral of 1/x from 1 to infinity to show that this series diverges, even though its sequence of terms converges. We concluded by stating the Comparison Test.

Lecture 19: Square root of 2, rationals and irrationals (Sections 10.5, 11.1)

We proved that the "square root of 2" really squares to 2. Then we defined the rational and irrational numbers and discussed properties of rational numbers (Propositions 11.1 -- 11.6). We proved Theorem 11.8, which tells us that the rationals are dense in the real numbers. We concluded by proving that the "square root of 2" must be an irrational number.

Lecture 18: Monotone Convergence Theorem, Square Roots (Sections 10.4, 10.5)

We stated the Monotone Convergence Theorem and proved half of it. We saw an example where the Monotone Convergence Theorem tells us immediately that a sequence must converge. Then, we defined square roots of positive real numbers. I optimistically started a proof of why our "square root of 2" really does square to 2... We will begin Friday with this proof.

Lecture 17: Limits (Section 10.4)

We defined the limit of a convergent sequence of real numbers and saw a few examples. Then, we stated Proposition 10.23, which is a list of properties of limits, and worked through the proof of part (iv). This is our first serious example of a proof involving the definition of a limit, convergence, epsilons, and unbounded natural numbers. This type of proof is fundamental to understand if you plan on studying anything related to the field of mathematics called analysis, and it's worth really understanding the structure of Case 1 of this proof, before moving on to understanding Case 2.

Lecture 16: Absolute value and distance functions (Sections 10.2, 10.3)

We defined absolute value as a norm on the real numbers. Equivalently, we can think of absolute value as a distance function. In general, norms (distance from a special point 0) and distance functions (distance between two points) must satisfy three criteria, including the triangle inequality. We also mentioned the Euclidean norm on the real plane, which is the Cartesian product of two copies of R.

Lecture 15: More vocabulary about functions, embeddings, how to embed Z in R (Parts of Chapter 9)

We introduced the terms image, preimage, injective (one-to-one), and embedding in the context of functions. We then defined a function which we claim to be a very reasonable embedding of Z into R. This injective function allows us to view our axiomatically defined set Z as a subset of our axiomatically defined set R.

Lecture 14: Proving that the integers are unbounded (Section 8.4, 10.1)

We proved that the set of positive real numbers doesn't have a smallest element by constructing the element 1/2 and using it to derive a proof by contradicition. We sketched a proof of the fact that between any two real numbers, there is always another real number. We used the Completeness Axiom to show that the natural numbers are unbounded; therefore, the integers are unbounded as well.

Lecture 13: Properties of the real numbers, upper bounds (Sections 8.3-8.4, 6.3)

We introduced the integers mod n, focusing on the integers mod 7 (which is a prime, making the number system very special). We compared the real numbers to the integers, the integers mod p, and the rationals. Finally, we added the Completeness Axiom to our list of axioms for the real numbers; this is an axiom we didn't need to have when we defined the integers.

Lecture 12: Well-ordering principle, functions, the real numbers (Sections 2.4, 5.4, 8.1, 8.2)

We defined the smallest element (or minimum) of a subset of integers, and discussed the well-ordering principle for the natural numbers. We then discussed notation and vocabulary for discussing functions. We defined the real numbers using axioms similar to those we used for defining the integers, and defined multiplicative inverses and division. Finally, we showed that there is no analogue for the well-ordering principle for the positive real numbers, by giving a counterexample.

Lecture 11: Cartesian products, equivalence relations (Sections 5.3, 6.1)

We discussed Cartesian products of sets. We then briefly described relations, and moved on to equivalence relations, which are relations that satisfy three defining properties. We saw two examples of equivalence relations on the integers that are not the same.

[Exam 1 Solutions]

Lecture 10: Exam 1

Midterm Exam 1 will be taken in class on Wednesday, Feb 1 and will cover material from Lectures 1-8 and Problem Sets 1-3. The cover of the exam (available [here]) contains information about the exam format. Discussion on Tuesday will cover some mock exam problems.

Lecture 9

We didn't cover any new material. We covered some announcements, discussed the structure of Exams in this class, and talked about PS3. In particular, I emphasized the importance of writing and typesetting in this course.

If you got a lower score on a revised problem on PS1 than your original score, email me to let me know, stating exactly which problem is relevant. I don't want there to be a penalty for revising, but it is also impractical for the graders to consider your previous submission while grading the revision, with the current setup.

For future revisions: Keep your original solution in the document. Under your original solution, add

\paragraph{Revised solution.}
and write down your revised solution. In particular, if you aren't revising a solution, just don't include this new paragraph. For PS4 and beyond, a new command called
will be included in the preamble of the provided TeX file.

Aside on how Problem Sets and Revisions are weighted: Let PS be the PS average grade and R the revisions average grade, both out of 100%. Your HW grade (out of 100%) should be

    HW = PS + 0.75*(R-PS) = 0.25*PS + 0.75*R
with R ≥ PS, as previously discussed. This weighting should now be reflected on Canvas.

Lecture 8: Basic set theory (Sections 5.1-5.2)

We introduced unions and intersections of sets, and talked about De Morgan's laws.

Lecture 7: Notions in logic, continued (Sections 3.2-3.3)

We introduced concepts and terms such as the contrapositive of an implication, and showed how they can be used in proofs.

Lecture 6: Binomial coefficients continued; beginning of symbolic logic (Sections, 4.4, 3.1)

We finished the proof that the binomial coefficients are integers. Then we introduced basic notions in symbolic logic, including the "there exists ... such that" and "for every/all/any" symbols.

Lecture 5: Recursion, Binomial coefficients (portions of Section 4.1, 4.4)

We discussed sequences and how to define them using recurrence relations. We then discussed the binomial coefficients and began an induction proof to prove that they are integers.

Lecture 4: Mathematical induction (Section 2.2, 2.3)

PS2 covers material up to this lecture. We proved the principle of mathematical induction, and used it to prove the formula for the sum of the first n odd natural numbers.

Discussion on Tuesday, January 17 (Section 2.1)

We finished the uniqueness and existence proof from the previous lecture. We briefly reviewed set notation, including the subset symbol and curly bracket notation. We axiomatically defined the natural numbers, and used proof by contradiction to show that every integer is either in the natural numbers (i.e. is positive), is 0, or the additive inverse is in the natural number (i.e. the original integer was negative). The last 15 minutes of class were dedicated to answering questions about homework, as there were not office hours on Monday.

Lecture 3: If-then statements, existence and uniqueness (Section 1.2)

We compared two propositions, Prop A and Prop B, both of which are if-then statements. Since Prop A's hypothesis was more general (the if part is less stringent), it was actually the "weaker" if-then statement. In other words, Prop A follows immediately from Prop B.

Both Prop A and Prop B were statements about the uniqueness of the additive identity 0. Saw our first existence and uniqueness proof (Prop 1.23). There is one more line in the proof; read the text to see how to conclude the proof. (We will also talk briefly about this at the beginning of discussion on Tuesday.)

Lecture 2: Rigorous statement of axioms and immediate corollaries, proving if-then statements (Sections 1.2)

We wrote down all the axioms and propositions that have already proven, using their common names (e.g. "additive identity" or "identity element for addition"). We focused on the structure of if-then statements and emphasized how to prove statements involving equations.

Reminder: Next Monday is MLK Day, a university holiday. There will be no class or office hours. The following Tuesday, I will be teaching the discussion sections, and you can bring your homework questions to me then.

Lecture 1: Course structure, proof-based mathematics, axioms and the integers (Section 1.1)

We talked about the philosophy of proof-based mathematics. We start with axioms and deduce statements (propositions, theorems, lemmas, etc.). We saw how one version of the distributive property of the integers (what could be considered "right distributivity") can be deduced from (left) distributivity and the commutativity of multiplication.

Q: But can't I prove left distributivity if I assume right distributivity and commutativity as axioms?
A: Indeed! In this case you are starting with a different set of axioms, but arriving at the same set of "truths".

Q: For tests, do we need to memorize all the axioms?
A: The short answer is no. The reason why we are working with integers in this class is because, hopefully, you already know the axioms by heart! A main goal of this course is to practice writing mathematics, so I would focus more on practicing proving things clearly (e.g. trying to prove some of the exercises in the text), rather than "memorizing" anything. A more concrete answer: You should be able to cite properties of integers by their names (e.g. associativity, commutativity, distributivity, etc.), but you certainly don't need to remember which statements in Chapter 1 were axioms vs. propositions. (There is one exception: In Problem Set 1, you will be given a particular set of axioms which you are required to cite specifically when you use them.)