# Matemáticas Discrétas II

## Matemáticas Discretas II

This is a course offered by Facultad de Ciencias at Universidad Central de Venezuela. You can check the current course website (in Spanish).

## Linear Algebra and Applications

## M301 Linear Algebra and Applications

This is an undergraduate course offered by the Math Department at Indiana University. According to their webpage, in this course the students are exposed to: "Solving systems of linear equations, matrix algebra, determinants, vector spaces, eigenvalues and eigenvectors. Selection of advanced topics. Applications throughout. Computer used for theory and applications". I assisted Prof. Jack Hanson and the following material is of his creation. This is a very useful material as a complement to the textbook used in this course: Linear Algebra and Its Applications, 4th Edition by Lay.

## Notes on Lay

- Definitions
- Determinants
- Lay Section 1
- Lay Section 2
- Lay Section 3
- Lay Section 5
- Lay Section 6

## Exams

- Exam 1 (no solution provided)
- Final Exam (no solution provided)
- Practice Final - [Solution]

## Mathematical Foundations of Informatics

## I201 Mathematical Foundations of Informatics

This is a course offered by the School of Informatics and Computing at Indiana University as part of the Informatics major.

I was an Associate Instructor for this course several times: Summer 2013, Fall 2013 and Summer 2014. The two main books used for this course are: Discrete Mathematics with Applications by Epp and Discrete Mathematics and Its Applications by Rosen. These are really good books but unfortunately expensive. If you do buy them, you will have a resource for a long time not just for the duration of the course. Note that in previous years (Summer 2013) this course was done differently and we used the textbook Informatics I201 by Haghverdi, a must have resource for a solid foundation in logic to whoever wishes to learn more about Computer Science. That year we also used Tarski's World: Revised and Expanded (Center for the Study of Language and Information - Lecture Notes), an interactive and gentle introduction to first-order logic. The following are some useful resources for this course. All materials from Summer 2013 are from Dr. Esfandiar Haghverdi.

## Assignments

- Assignment 1 (Summer 2013)
- Assignment 2 (Summer 2013)
- Assignment 3 (Summer 2013)
- Assignment 4 (Summer 2013)
- Assignment 5 (Summer 2013)
- Assignment 6 (Summer 2013)
- Assignment 7 (Summer 2013)
- Assignment 8 (Summer 2013)

## Quizzes

- Quiz 1 (Summer 2013)
- Quiz 2 (Summer 2013)
- Quiz 3 (Summer 2013)
- Quiz 6 (Summer 2013)
- Quiz 7 (Summer 2013)
- Quiz 8 (Summer 2013)

## Exams

- Final Exam (Summer 2013)
- Sample Final Exam (Summer 2013)
- MidTerm 2 (Summer 2013)
- Sample MidTerm 2 (Summer 2013)
- Sample MidTerm 1 (Summer 2013)

## Mathematics for Elementary Teachers I

## T101 Mathematics for Elementary Teachers I

This is a course offered by the Math Department at Indiana University. The course's intended audience are pre-service teachers that will teach at the elementary school level. This is a -somewhat- rigorous course on the foundations of all mathematics: the whole numbers and their arithmetic. The course also covers integers and rational numbers with their respective operations. The idea is for students to be able to understand why the math works and explain them to others. The textbook used for this class is Elementary Mathematics for Teachers by Parker and Baldridge that follows the Singapore Math method. The book is a nice resource for elementary teachers to have. Next I will provide some resources I think are useful for this course and finally, some advice of my own -based on my experience teaching this course 3 times- before taking this class.

## Class Activities

- Activity Notes: all prepared by Serife Sevis (some may contain errors, be careful!)

## Exams

- Exam 1 (no solutions provided)
- Exam 2 (no solutions provided)
- Exam 3 - [Solutions]

## Resources

- Divisibility Criteria
- Russian Algorithm prepared by Justin Cyr
- Clock Congruence prepared by Sisi Tang
- Final Exam (from a different but similar class)

## Some advice before enrolling in this course

The goal of the course is that students build upon their previous arithmetic foundation to understand why the fundamental operations (addition, multiplication, subtraction and division) on numbers (whole numbers, integers and rational numbers) work, and be able to intuitively explain them to others. And here in lies the biggest obstacle for students in this class: a weak arithmetical foundation. Even though students know -in principle- how to add, subtract, multiply and divide whole numbers/integers/fractions, some do not have a practical knowledge, i.e., the kind of knowledge that would allow them to do these operations in a heartbeat.

Since students lack this practical knowledge, some of the more advance (and interesting!) material, such as elementary number theoretical proofs (for example, show that the sum of two even numbers is even) gets lost in the way. This is a clear example of Mathematics as a subject that builds upon itself where previous material is essential to grasp more advance material. If students cannot add properly, it is nearly impossible to fully grasp the beauty of elementary proofs or even be able to understand the algebra needed for said proofs.

My advice: make sure you have a clear understanding of the elementary operations before enrolling in this course. And by that I mean: throw away the calculator and start doing the math in your head. In particular, you should be able to add fractions (I found this to be the more challenging of the operations). The course is intended for you to understand why the operations work and be able to explain them to others. However, there is rarely time to go over the mechanics. You have to take care of that before class. This is a really fun class that sheds light on many things we do routinely in our daily lives. The only way to get the full out of the class is to come prepare for it!

Let me finish by giving you an example of what I mean by coming prepared to class. Suppose you want to get your driver's license. As you may know, you have to do both a theoretical and a practical exam in order to obtain a driver's license. In the practical exam you actually get to drive a car with an approved instructor sitting next to you, taking notes on how you perform. Ok, so while driving the car you would certainly not ask the instructor: "should I stop at this red light?". You are suppose to know what to do -the mechanics- when presented with various driving signals. Instead, you could ask -and maybe you wouldn't, but just for the sake of argument- why the light is red and not blue or any other color. There is a reason why the light is red and not any other color (click here if you are interested). Asking whether or not to stop in your driver's test would be the equivalent of asking why 1/2 + 1/3 = 5/6 in our math class. This you should know and be second nature (like stoping at a red light!). Instead, you could ask why is it that we need a common denominator to add fractions and we can't just add top and bottom like 1/2 + 1/3 = (1+1)/(2+3)?

In terms of the proofs we do in T101, a common question is "why 2**k** + 2**l** = 2(**k**+**l**)"? (here k and l are whole numbers). You should know the distributive property (even if you do not know the name "distributive" that is ok), and in a heartbeat know that **a**(**b**+**c**) = **ab**+**ac**. Immediately. The question "why 2**k** + 2**l** = 2(**k**+**l**)"? is another equivalent to "should I stop at this red light?" in your driving test. If you find yourself asking these type of questions, and by that I mean questions about the result of an arithmetical operation, then you are not prepared for class. Instead, an interesting question would be why the distributive property works the way it does and not in some other way. Why do we have **a**(**b**+**c**) = **ad** + **ac** and not **a**(**b**+**c**) = **ab** + **c**, for instance. (Hint: think of the area of a rectangle with base **a** and height **b**+**c**, now subdivide the rectangle into 4 smaller rectangles). If you find yourself thinking these questions then you are on the right track! The beauty of this course is that after it, you should know why elementary math works and even more, be able to explain it and draw pictures accordingly. Pretty cool stuff!

## Estructuras Discretas para el Análisis de Algoritmos

## Recursos

Prácticas y soluciones. Guías y otros materiales. Here you can download practice problems and solutions. Also, study guides and other materials.

## Guías de Estudio - Study Guides

- Good guide about the Inclusion-Exclusion Principle
- Big-O examples
- Relaciones de Recurrencia

For the moment the following resources are all in Spanish.