Discrete Math - Module #0 - Overview 8/20/2019 University of Aberdeen, Computing Science CS3511 Discrete Methods Kees van Deemter Slides adapted from Michael P. Frank’s Course Based on the Text Discrete Mathematics & Its Applications (5th Edition) by Kenneth H. Rosen A word about organization: Since different courses have different lengths of lecture periods, and different instructors go at different paces, rather than dividing the material up into fixed-length lectures, we will divide it up into “modules” which correspond to major topic areas and will generally take 1-3 lectures to cover. Within modules, we have smaller “topics”. Within topics are individual slides. The instructor can bring several modules to each lecture with him, to make sure he has enough material to fill the lecture, or in case he wants to preview or review slides from upcoming or recent past lectures. 8/20/2019 Michael P. Frank; Kees van Deemter (c)2001-2003, Michael P. Frank

Michael P. Frank; Kees van Deemter Practicalities Lectures Mon 2PM (MT6) Tue 11AM (Old Senate Room) Practicals Tue 1-3PM (MT011) Tue 3-5PM (MT011) Don’t go by this info alone, since things may change. Consult web page at least twice per week, and watch email to level3-students 8/20/2019 Michael P. Frank; Kees van Deemter

Michael P. Frank; Kees van Deemter Key resources Lectures based on Michael Frank’s slides (Univ. of Florida) We’re not using all his lectures Many changes in those that we do use Lectures will appear on the web Book: Ken Rosen, Discrete Mathematics and its Applications. 5th or 6th edition. Then e.g. (5th ed.): http://www.mhhe.com/math/advmath/rosen/r5 8/20/2019 Michael P. Frank; Kees van Deemter

Other practical issues Assessment:Standard arrangments 75% exam 25% cont. assessment Arguably, this course comes too late in your curriculum You will be somewhat familiar with much in this course “A little knowledge is a dangerous thing” 8/20/2019 Michael P. Frank; Kees van Deemter

Module #0: Course Overview Discrete Math - Module #0 - Overview 8/20/2019 Module #0: Course Overview A few general slides about the subject matter of this course. 14 slides, ½ lecture 8/20/2019 Michael P. Frank; Kees van Deemter (c)2001-2003, Michael P. Frank

What is Mathematics, really? Discrete Math - Module #0 - Overview 8/20/2019 What is Mathematics, really? It’s not just about numbers! Mathematics is much more than that: These concepts can be about numbers, symbols, ideas, images, sounds, anything! Mathematics is, most generally, the study of any and all certain truths (about any kind of concepts) As you may have already concluded from the title of the textbook, this is a course about mathematics. But, it is a type of mathematics that may be unfamiliar to you. You may have concluded from your exposure to math so far that mathematics is primarily about numbers. But really, that’s not true at all. The essence of mathematics is just the study of formal systems in general. A formal system is any kind of entity that is defined perfectly precisely, so that there can be no misunderstanding about what is meant. Once such a system has been set forth, and a set of rules for reasoning about it has been laid out, then, anything you can deduce about that system while following those rules is an absolute certainty, within the context of that system and those rules. Number systems aren’t the only kinds of system that we can reason about, just one kind that is widely useful. For example, we also reason about geometrical figures in geometry, surfaces and other spaces in topology, images in image algebra, and so forth. Any time you precisely define what you are talking about, set forth a set of definite axioms and rules for some system, then it is by definition a mathematical system, and any time you discover some unarguable consequence of those rules, you are doing mathematics. In this course, we’ll encounter many kinds of perfectly precisely defined (and thus, by definition, mathematical) concepts other than just numbers. You will learn something of the richness and variety of mathematics. And, we will also teach you the language and methods of logic, which allow us to describe mathematical proofs, which are linguistic presentations that establish conclusively the absolute truth of certain statements (theorems) about these concepts. In a sense, it’s easy to create your own new branch of mathematics, never before discovered: just write down a set of axioms and inference rules, make sure their meanings are perfectly clear, and then work out their consequences. Whether anyone else finds your new area of mathematics particularly interesting is another matter. The areas of math that have been thoroughly explored by large numbers of people are generally ones that have a large number of applications in helping to understand some other subject, whether it be another area of mathematics, or physics, or economics, biology, sociology, or whatever other field. See http://mathworld.wolfram.com/ for a survey of some other areas of mathematics, to show you just how much there is to mathematics, besides just numbers! 8/20/2019 Michael P. Frank; Kees van Deemter (c)2001-2003, Michael P. Frank

Michael P. Frank; Kees van Deemter Uses of Discrete Math Starting from simple structures of logic and set theory, theories are constructed that capture aspects of reality: Physics Biology (DNA) Common-sense reasoning (logic) Natural Language (trees, sets, functions, ..) … Anything that we want to describe precisely 8/20/2019 Michael P. Frank; Kees van Deemter

So, what’s this class about? Discrete Math - Module #0 - Overview 8/20/2019 So, what’s this class about? What are “discrete structures” anyway? “Discrete” ( “discreet”!) - Composed of distinct, separable parts. (Opposite of continuous.) discrete:continuous :: digital:analog “Structures” - Objects built up from simpler objects. “Discrete Mathematics” - The mathematical study of discrete objects and structures. The title of this course (at UF) is “Applications of Discrete Structures.” Actually that title is misleading. Although discrete math has many applications, which we’ll survey in a moment, this class is not really about applications so much as it is about basic mathematical skills and concepts. My guess is that the only reason we call the course “Applications of Discrete Structures” rather than just “Discrete Mathematics” so that the people over in the Math department don’t get upset that we’re teaching our own basic math course over here, rather than leaving it to them. Anyway, that issue aside, you may be wondering, what is a “discrete structure” anyway? Well, by “discrete” we don’t mean something that’s kept secret – that’s “discreet” with a double-e, a homonym, a completely different English word with a completely different meaning. No, our word “discrete” just means something made of distinct, atomic parts, as opposed to something that is smooth and continuous, like the curves you studied in calculus. Later in the course we’ll learn how to formalize this distinction more precisely using the concepts of countable versus uncountable sets. But for now, this intuitive concept suffices. By “structures” we just mean objects that can be built up from simpler objects according to some definite, regular pattern, as opposed to unstructured, amorphous blobs. In this course, we’ll see how various types of discrete mathematical objects can be built up from other discrete objects in a well-defined way. The field of discrete mathematics can be summarized as the study of discrete, mathematical (that is, well-defined, conceptual) objects, both primitive objects and more complex structures. 8/20/2019 Michael P. Frank; Kees van Deemter (c)2001-2003, Michael P. Frank

Michael P. Frank; Kees van Deemter Discrete Mathematics When using numbers, we’re more likely to use N (natural numbers) and Z (whole numbers) than Q (fractions) and R (real numbers). Reason: Q and R are densely ordered This notion can be defined precisely. Some DM notation will help: (Notation should become clear later on.) 8/20/2019 Michael P. Frank; Kees van Deemter

Michael P. Frank; Kees van Deemter Q,<  is densely ordered because xQ yQ (x<y  z (x<z & z<y) ) (“if x<y then there exists at least one z in between”) “Opposite” of densely ordered: discretely ordered 8/20/2019 Michael P. Frank; Kees van Deemter

Michael P. Frank; Kees van Deemter Yet, Q and R can be defined in terms of discrete concepts (as we have seen) This means that Discrete Mathematics has no exact borders Different books and courses treat different topics 8/20/2019 Michael P. Frank; Kees van Deemter

Discrete Structures We’ll Study Discrete Math - Module #0 - Overview 8/20/2019 Discrete Structures We’ll Study Propositions Predicates Proofs Sets Functions (Orders of Growth) (Algorithms) Integers (Summations) (Sequences) Strings Permutations Combinations Relations Graphs Trees (Logic Circuits) (Automata) Here are just a few of the kinds of discrete structures and related concepts that we’ll be studying this semester. 8/20/2019 Michael P. Frank; Kees van Deemter (c)2001-2003, Michael P. Frank

Some relevant notations Discrete Math - Module #0 - Overview 8/20/2019 Some relevant notations 1st row: NOT, OR, XOR, IMPLIES, logical equvalence, FORALL 2nd row: EXISTS, sets / index notation / ellipses, integers/naturals/reals, therefore, set builder notation, not a member of 3rd row: empty set, subset, cardinality, union, set complement, intersection over a sequence of sets 4th row: function signature, inverse, composition, floor, summation over a set, product over a sequence 5th row: at most order/at least order/exactly order, minimum/maximum, not divisible by, greatest common denominator/least common multiple, modulo, modular congruence 6th row: base-b number representation, matrix indexing, transpose, boolean product, matrix exponentiation, choose 7th row: generalized combination, conditional probability, transitive closure, …, … , in-degree 8/20/2019 Michael P. Frank; Kees van Deemter (c)2001-2003, Michael P. Frank

Discrete Math for computing The basis of all of computing is: manipulations of discrete structures represented in memory. DM is the conceptual foundation for all of computer science. 8/20/2019 Michael P. Frank; Kees van Deemter

Michael P. Frank; Kees van Deemter Some examples: Algorithms & data structures Compilers & interpreters. Formal specification & verification Databases Cryptography Digital circuits etc. DM is relevant for all aspects of computing! 8/20/2019 Michael P. Frank; Kees van Deemter

Plan of Course (as per Rosen 5th ed.) Discrete Math - Module #0 - Overview 8/20/2019 Plan of Course (as per Rosen 5th ed.) Logic (§1.1-4) Proof methods (§1.5) Set theory (§1.6-7) Functions (§1.8) (Algorithms (§2.1)) (Orders of Growth (§2.2)) (Complexity (§2.3)) Number theory (§2.4-5) (Number theory apps. (§2.6)) (Matrices (§2.7)) Proof strategy (§3.1) (Sequences (§3.2)) (Summations (§3.2)) (Countability (§3.2)) Inductive Proofs (§3.3) Recursion (§3.4-5) Program verification (§3.6) (Combinatorics (ch. 4)) (Probability (ch. 5)) (Recurrences (§6.1-3)) Relations (ch. 7) Graph Theory (chs. 8+9) (Boolean Algebra (ch. 10)) (Computing Theory (ch.11)) Here is the order of topics covered, and corresponding section numbers in the textbook (5th edition). We are basically following the same order of material as the textbook. However, we will skip over some sections in the later chapters. 8/20/2019 Michael P. Frank; Kees van Deemter (c)2001-2003, Michael P. Frank

Discrete Math - Module #0 - Overview 8/20/2019 Course Objectives Upon completion of this course, the student should be able to: Check validity of simple logical arguments (proofs). Check the correctness of simple algorithms. Recognise and construct simple instances of valid logical arguments and correct algorithms. Describe the definitions and properties of a variety of specific types of discrete structures. Correctly read, represent and analyze various types of discrete structures using standard notations. Preparation for CS4026, Formal Models This summarizes the course objectives. The syllabus goes into this in more detail. The most important objective (in my opinion) is that after taking this course (if not before), the student should be capable of performing logical thought and reasoning that is creative, yet precise and correct. Working with specific kinds of discrete structures is a good way to practice this skill, and the knowledge gained is useful due to the widespread use of these structures. 8/20/2019 Michael P. Frank; Kees van Deemter (c)2001-2003, Michael P. Frank

Michael P. Frank; Kees van Deemter Have Fun! Many people find Discrete Mathematics more enjoyable than, for example, Analysis: Applicable to just about anything Some nice puzzles Highly varied 8/20/2019 Michael P. Frank; Kees van Deemter