Discrete Mathematics 3

Discrete Mathematics 3

Mathematics from high school to university

[None of our courses are produced using AI; they are all real-human products.]

S1. Introduction to the course
You will learn: about this course: its content and the optimal way of studying it together with the book.

S2. Some preliminaries for the sections devoted to sequences
You will learn: some basic algebraic techniques that are needed for the sections about sequences: solving systems of linear equations (2-by-2 and 3-by-3 determinants, Sarrus' rule, Cramer's rule), solving quadratic equations (with help of the discriminant, or by qualified guesses), some basics about polynomials, finding integer zeros of polynomials with integer coefficients, polynomial division, partial fraction decomposition, subspaces of vector spaces and their generators, span, some formulas for derivatives. [My advice is that you don't watch this section before you know why you need this stuff; I will refer to the appropriate videos when I'm about to use some of the concepts and techniques above for working with sequences.]

S3. A general introduction to sequences
You will learn: various ways of defining sequences (by a closed/explicit formula, by recursion, a verbal description, with help of a picture, by giving a number of elements that suggest a rule), arithmetic operations on sequences ([element-wise] addition, subtraction, multiplication, scaling), sequence of partial sums, sequence of differences, the vector space of all sequences of real numbers (a preparation for S7).

S4. Arithmetic progressions and arithmetic sums
You will learn: the definition and properties of arithmetic progressions; partial sums of an arithmetic progression; monotonicity of sequences (generally, and specifically of arithmetic progressions).

S5. Geometric progressions and geometric sums
You will learn: the definition and properties of geometric progressions; partial sums of a geometric progression; monotonicity of geometric progressions.

S6. Polynomial sequences
You will learn: how to deal with sequences that have constant sequence of (first, second, third, and so on) differences; you will learn that all such sequences are polynomial sequences and you will learn how to find their explicit formulas.

S7. Solving linear recurrence relations
You will learn: solving linear recurrence relations of order k with constant coefficients (mainly homogenous, but you will also see some simple examples of non-homogenous ones); some examples of counting problems that are modelled by linear recurrences; solution sets of linear recurrences with constant coefficients as subspaces of vector spaces of all sequences numbered from index 0; linearly independent solutions (optional).

S8. Generating functions
You will learn: the concept of a generating function for a sequence, with some arithmetic rules for working with it, some examples of generating functions of various sequences, and two examples of application.

S9. Some fun problems about sequences
You will learn: OK, this section is just for fun: it shows you some really cool problems about sequences.

S10. A brief introduction to Graph Theory
You will learn: you get a very brief and elementary introduction to Graph Theory; for this one, I recommend reading Chapter 2 from the DM Book, as the concepts are quite easy to grasp (even though the theory itself is surprisingly complicated!) and I will concentrate on some more difficult parts in the video lectures and, as always, I'll deliver plenty of illustrations; this should give you a decent preparation to a future serious course devoted entirely to Graph Theory (I have no plans to create such a course, though, so you will have to look somewhere else for it...).

S11. Some applications of Discrete Mathematics
You will learn: some applications of DM to CS; it will not be much, just some (very modest) words about algorithm complexity, Chinese Remainder Theorem, and RSA encryption; you get a bunch of practice problems (with solutions), both in the videos and in other resources; you will also get some advice for your further studies of DM, and all the students are welcome to leave references to their favorite resources on the QA under the last lecture in this section.

Note: This is the third (and last) part of our trilogy in Discrete Mathematics.

Make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.

A detailed description of the content of the course, with all the 181 videos and their titles, and with the texts of all the 320 problems solved during this course, is presented in the resource file

“001 List_of_all_Videos_and_Problems_Discrete_Mathematics_3.pdf”

under Video 1 ("Introduction to the course"). This content is also presented in Video 1.