# Introduction to Mathematical Thinking

### CS 198-087 @ UC Berkeley

Access the notes for this class here.

## Announcements

This course was discontinued after Spring 2019. However, it appears that some students have found the material useful, so we will keep all content public on this site. If you’ve used the material on this site for any reason, please fill out this brief survey to help us understand how this site is being used. Thanks!

## Content and Schedule

• Homework generally follows a Friday-Friday schedule.
• Notes written specifically for this course can be found at notes.imt-decal.org. Other readings will be linked in the table below.
• The schedule is tentative.
Week Date Topic Resources Homework
1 Tue. 01/29 Course Overview Slides
Notebook
Video
Thu. 01/31 Set Theory, Functions Slides
Video
Note: Sets and Set Operations
Note: Functions and Bijections
HW 1
Solutions
Video
2 Tue. 02/05 Bijections, Number Sets Slides
Video
Note: Sets of Numbers
Thu. 02/07 Number Sets, Propositional Logic Slides
Video
Note: Propositional Logic
HW 2
Solutions
3 Tue. 02/12 Propositional Logic Slides
Video
Note: Notation Cheat Sheet
Thu. 02/14 Basic Proof Techniques
Quiz 1 in class
Slides
Video
Note: Foundational Proof Techniques
HW 3
Solutions
4 Tue. 02/19 Basic Proof Techniques Slides
Video
Thu. 02/21 Induction Slides
Video
Note: Mathematical Induction
HW 4
Solutions
5 Tue. 02/26 Strong Induction, Series and Sequences Slides
Video
Thu. 02/28 Series and Sequences
Quiz 2 in class
Slides
Video
Note: Series and Sequences
HW 5
Solutions
6 Tue. 03/05 Division Algorithm and Primality Slides
Video
Note: Primality and Divisibility
Thu. 03/07 Modular Arithmetic Slides
Video
Note: Modular Arithmetic
7 Tue. 03/12 Finding Modular Inverses
(Ani Nrusimha*)
Slides (from Fall 2018)
Video (from Fall 2018)
Thu. 03/14 Review of NT/MA
(Lecture by TAs Adel, Jai, Sagnik)
Quiz 3 online
Refer to Homework 5 solutions
8 Tue. 03/19 Counting
(Ani Nrusimha)
Slides by Ani
Slides (from Fall 2018)
Note: Principle of Inclusion-Exclusion
Note: Key Examples in Counting
Note: Counting (by Jerry Huang)
Thu. 03/21 Counting
(Ani Nrusimha)
Slides by Ani
Video (from Fall 2018)
Note: Stars and Bars
HW 6
Solutions
9 Tue. 03/26 No Class (Spring Break)
Thu. 03/28 No Class (Spring Break)
10 Tue. 04/02 Counting Review Slides
Video
Thu. 04/04 Counting Review, Combinatorial Proofs Slides
Video
11 Tue. 04/09 Binomial Theorem
Quiz 4 in class
Slides
Video
Note: Binomial Theorem
Thu. 04/11 Binomial Theorem, Vieta’s Formulas Slides
Video
Note: Vieta’s Formulas
HW 7
Solutions
12 Tue. 04/16 Vieta’s Formulas Slides
Video
Thu. 04/18 No Class
13 Tue. 04/23 Review Slides
Video
HW 8
Solutions
Thu. 04/25 Review Slides
Video
14 Tue. 04/30 Quiz 5 in class
Thu. 05/02 Probability, Closing Thoughts Slides
Notebook
Video
Extra Credit

* Ani Nrusimha, `aninrusimha@berkeley.edu`, will be covering these lectures. Feel free to reach out to him with any questions.

Spring 2019 quizzes:

From previous semesters:

• How to Prove It: A Structured Approach, by Velleman (2nd edition) covers the material in our class through mathematical induction, albeit in a slightly different order. link
• Discrete Math and Its Applications, by Rosen (7th edition) is the textbook that Math 55 at UC Berkeley uses. It also covers most of the material in the course, including counting (which the above textbook does not cover). link
• Art of Problem Solving is an online community centered around preparing for math competitions, with several wikis on various topics. These wikis are especially relevant towards the latter part of our course, with some excellent articles on combinatorics, the Binomial Theorem and Vieta’s Formulas. link
• Mathematical Reasoning is a set of lecture notes by Hermish Mehta on some of the earlier topics in the course. link

## Description

Berkeley’s highly theoretical Computer Science curriculum demands a high level of mathematical maturity. While those with extracurricular math experience from high school are familiar with dense notation, complex mathematical objects, and proof techniques, many students find foundational courses like CS 70, CS 170, and Math 55 confusing and inaccessible.

Introduction to Mathematical Thinking bridges the gap. We teach mathematical maturity. Our curriculum exposes students to familiar concepts in a more precise, generalized way. By the end of our course, students will be able to:

• comfortably read mathematical language, including notation, definitions and proofs
• concisely and clearly express their ideas differentiate between a good proof and a proof with logical gaps

As a result, this course will prepare students for higher-level mathematics courses, such as CS 70 at Berkeley. However, students can enroll in the course even if they aren’t planning on taking these courses or are not in CS/EECS; these skills and concepts are highly transferrable.

There are no prerequisites for this course. We’re working really hard to make the material accessible for all backgrounds.

Disclaimer: This course is not a prerequisite for CS 70, nor is it affiliated with the CS 70 instructors or course staff in any way. The official prerequisites for CS 70 are specified in the course description. CS 70 staff makes no guarantees regarding the material covered in this course.

The course will be offered for 2 units, P/NP.

There will be weekly problem sets, which are graded on effort, not correctness. Attendance is mandatory, and NPs will be given to students who have more than 3 unexcused absenses.

The course is graded on a 100 point scale:

• 5 quizzes, each worth 12 points, for a total of 60 points
• Quizzes are on Feb. 14, Feb. 28, Mar. 14, Apr. 9 and Apr. 30, in class
• Weekly homeworks, worth a total of 40 points

A passing grade will be given to students with 65 points or more (note the new threshold). We reserve the right to change this threshold, but we would only decrease it (i.e. we will not make it any harder to pass).

• How do I know that this course is for me?
This course is designed for students without discrete math experience. In general, if you have significant math contest experience, or did well in a discrete math course prior to being at Berkeley, then this course probably isn’t for you. A good diagnostic is last semester’s midterm exam. If a lot of this seems unfamiliar to you, we’d be glad to have you!

• When should I take this course?
We think this course is best taken the semester before taking CS 70. Therefore, if you plan on taking CS 70 in Summer 2019 or Fall 2019, then Spring 2019 would be the time to take it. With that being said, you can take this course even if you don’t plan on taking CS 70, but due to constraints, we won’t be admitting many students from this category (perhaps in future semesters). This DeCal is not designed to be taken alongside or after CS 70 or Math 55.

• How much work will it be?
We know you have other courses in which your grade matters. However, the only way to actually develop and retain skills from this course is to put in the time into coming to lecture and discussion, reading the book/watching lecture videos, and (most importantly) doing the homework. Not including time spent in class, a rough estimate is 5 hours per week.

• Can I audit this course?
Sadly, we won’t have space for that. However, all materials – lecture notes, the textbook, lecture videos, assignments, exams – will be posted online.

## Staff

For all course related communications, please email `imt-decal@berkeley.edu`.

### Instructor

Suraj Rampure (`suraj.rampure@berkeley.edu`)

Hey, I’m a third year EECS major from Windsor, Ontario (right across the border from Detroit). I like cars, tech, teaching and rooting for LeBron (go Cavs Lakers!). This is my second semester teaching this course, fifth semester as a part of CSM, and fourth semester as a GSI; currently, I’m TA’ing Data 100, but have TA’d CS 61A and Data 8 in the past. I’m super excited that this course is finally a reality, and I’m hoping you are as well.

### Teaching Assistants

Jai Bansal (`jaibansal@berkeley.edu`)

Hey everyone! This is my first year at Cal hoping to major in CS and Applied Mathematics. In my free time, I love playing Ultimate Frisbee and watching basketball, football, soccer (or any sport you can name) and listening to way too much Logic. Feel free to talk to me about anything! Excited to TA for the first time!

Sagnik Bhattacharya (`sagnick@berkeley.edu`)

I’m a CS and stats-intended freshman. You can find me running on the streets of Berkeley when I’m not biking between classes or trying to hack the mainframe. Talk math and computers (and environmental science!) to me.

Alexia Colmenero (`acolmenero@berkeley.edu`)

I am a sophomore Computer Science major from San Diego. I am a gemini who likes video games, and is excited to work on this course! I use they/them pronouns.

Divya Mohan (`21dmohan@berkeley.edu`)

Heya! I’m a sophomore EECS major, originally from the Bay Area (specifically, Belmont). I am also meme trash; find me on UCBMFET any day, all day. I’m interested in data science, and I love helping my peers :)

Adel Setoodehnia (`asetoodehnia@berkeley.edu`)

Hello! I’m Adel, a third-year Mathematics and Computer Science student from Union, New Jersey. When I have free time you could probably find me playing soccer, listening to music, playing guitar, chilling with a book, or cooking/eating all types of good food. Looking forward to meeting you all!