**Universitas Islam IndonesiaJurusan Teknik InformatikaMatematika Diskret Yudi PrayudiSlides for a Course Based on the Text Discrete Mathematics & Its Applications (5th Edition) by Kenneth H. Rosen

**Module #0:Course OverviewPengantar Kuliah Awal Matematika Diskret, 18 slides, lecture

**Apakah Sesungguhnya Matematika Bukan hanya sekedar angka dan bilangan ! Matematika lebih dari sekedar pengertian tersebut,

These concepts can be about numbers, symbols, objects, images, sounds, anything!Mathematics is, most generally, the study of any and all absolutely certain truths about any and all perfectly well-defined concepts.

**Physics from MathematicsStarting from simple structures of logic & set theory,Mathematics builds up structures that include all the complexity of our physical universeExcept for a few loose ends.One theory of philosophy:Perhaps our universe is nothing other than just a complex mathematical structure!Its just one that happens to include us!From Max Tegmark, 98

**Matematika Diskret ?Definisi dari matematika diskret :The part of math devoted to the study of discrete objects and the discrete structures used to represent them.discrete distinct, unconnected, countableContoh :Bilangan integer adalah diskret .Bilangan real lebih cenderung dikatakan sebagai kontinyu dibandingkan dengan diskret.

**Matematika Diskret ? Apa pengertian struktur diskret / discrete structures ?Discrete ( discreet!) - Composed of distinct, separable parts. (Opposite of continuous.) discrete:continuous :: digital:analogStructures - Objects built up from simpler objects according to some definite pattern.Discrete Mathematics - The study of discrete, mathematical objects and structures.

*Apakah Struktur Diskrit itu?Struktur diskrit: cabang matematika yang mengkaji objek-objek diskrit. Apa yang dimaksud dengan kata diskrit (discrete)? Benda disebut diskrit jika:terdiri dari sejumlah berhingga elemen yang berbeda, atau elemen-elemennya tidak bersambungan (unconnected). Contoh: himpunan bilangan bulat (integer)

*Lawan kata diskrit: kontinyu atau menerus (continuous). Contoh: himpunan bilangan riil (real)

Komputer digital bekerja secara diskrit. Informasi yang disimpan dan dimanipulasi oleh komputer adalah dalam bentuk diskrit. Matematika diskrit merupakan ilmu dasar dalam pendidikan informatika atau ilmu komputer.

*Struktur diskrit memberikan landasan matematis untuk kuliah-kuliah lain di informatika. algoritma, struktur data, basis data, otomata dan teori bahasa formal, jaringan komputer, keamanan komputer, sistem operasi, teknik kompilasi, dsb. Struktur diskrit adalah matematika yang khas informatika Matematika-nya orang Informatika.

**Bentuk Struktur Diskret Yang Akan Dipelajari PropositionsPredicatesProofsSetsFunctionsOrders of GrowthAlgorithmsIntegersSummationsSequencesStringsPermutations CombinationsRelationsGraphsTreesLogic CircuitsAutomata

**Relasi antar bentuk struktur diskret : Can be defined in terms ofSetsSequencesn-tuplesMatricesNatural numbersIntegersRelationsFunctionsGraphsReal numbersComplex numbersStringsPropositionsProofsTreesOperatorsProgramsInfinite ordinalsVectorsGroupsBitsNot all possibilities are shown here.

**Notasi yang banyak digunakan

**Computer programming adalah sebuah studi tentang bagaimana membuat sebuah good software.Computer science adalah study dari computer programming plus beberapa fundamental tentang sintaks dan semantik dari bahasa komputer, sistem operasi, data basis serta analisa pemrograman. Termasuk didalamnya adalah proofs of the correctness of programs, proofs of the time (and space) complexity of programs.

**Pentingnya Matematika Diskret Dasar dari semua pemrosesan informasi digital adalah : manipulasi diskret dari struktur diskret yang direpresentasikan di memori . Merupakan fondasi utama dari hampir semua bidang dalam computer science.Konsep konsep pada Matematika Diskret juga digunakan secara luas pada bidang-bidang yang lain, seperti : math, science, engineering, economics, biology, etc., Matematika Diskret adalah tools yang sangat berguna untuk memahami cara berfikir yang rasional.

**Bentuk DiskretInformasi disimpan dan dimanipulasi oleh komputer dalam bentuk diskret . memorycell

1011011

**Matematika Diskret Pada Ilmu Komputer Advanced algorithms & data structuresProgramming language compilers & interpreters.Computer networksOperating systemsComputer architectureDatabase management systemsCryptographyError correction codesGraphics & animation algorithms, game engines, etc.I.e., the whole field!

**Garis Besar Kuliah 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)

*Materi-materi dalam Struktur Diskrit:Logika (logic) Teori Himpunan (set) Matriks (matrice) Relasi dan Fungsi (relation and function) Induksi Matematik (mathematical induction) Algoritma (algorithms)Teori Bilangan Bulat (integers) Barisan dan Deret (sequences and series)Teori Grup dan Ring (group and ring)Aljabar Boolean (Boolean algebra)Kombinatorial (combinatorics) Teori Peluang Diskrit (discrete probability)Fungsi Pembangkit dan Analisis RekurensTeori Graf (graph included tree) Kompleksitas Algoritma (algorithm complexity) Otomata & Teori Bahasa Formal (automata and formal language theory)

*Contoh-contoh persoalan di dalam Struktur Diskrit:Berapa banyak kemungkinan jumlah password yang dapat dibuat dari 8 karakter?Bagaimana nomor ISBN sebuah buku divalidasi?Berapa banyak string biner yang panjangnya 8 bit yang mempunyai bit 1 sejumlah ganjil?Bagaimana menentukan lintasan terpendek dari satu kota a ke kota b?Buktikan bahwa perangko senilai n (n 8) rupiah dapat menggunakan hanya perangko 3 rupiah dan 5 rupiah sajaDiberikan dua buah algoritma untuk menyelesaian sebuah persoalan, algoritma mana yang terbaik?

*Bagaimana rangkaian logika untuk membuat peraga digital yang disusun oleh 7 buah batang (bar)?

Dapatkah kita melalui semua jalan di sebuah kompleks perubahan tepat hanya sekali dan kembali lagi ke tempat semula?

Makanan murah tidak enak, makanan enak tidak murah. Apakah kedua pernyataan tersebut menyatakan hal yang sama?

*Buku PeganganKenneth H. Rosen, Discrete Mathematics and Application to Computer Science 5th Edition, Mc Graw-Hill, 2003.Rinaldi Munir, Matematika Diskrit (Edisi Keempat), Penerbit Informatika Bandung.. Jek Siang, Matematika Diskret, Penerbit Andi Offset Yogyakarta

**Tujuan dari Mata Kuliah Setelah menyelesaikan matakuliah ini maka mahasiswa diharapkan memiliki kemampuan dalam hal :Check validity of simple logical arguments (proofs).Check the correctness of simple algorithms.Creatively 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.

**Penilaian Komponen :Nilai UTS, UASTugas Kuliah dan QuizTatap Muka / Absensi / KeaktifanPenilaianSkor = (Total Absensi+keaktifan * 15%) + (Tugas Kuliah * 25 %) + (UTS * 30%) + (UAS*30%)

**Penggunaan KlasiberKlasiber.uii.ac.idMateri Tambahan bahanTugasQuiz

**KonsensusKuliah Sopan : Tidak bersandal dan berkaosBusana muslimah yang pantasToleransi terlambat masuk kelas : 15 Menit

Discrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*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.(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*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, thats 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 arent 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, well 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, its 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!(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*The title of this course (at UF) is Applications of Discrete Structures. Actually that title is misleading. Although discrete math has many applications, which well 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 dont get upset that were 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 dont mean something thats kept secret thats 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 well 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, well 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.(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*Here are just a few of the kinds of discrete structures and related concepts that well be studying this semester.(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*As I mentioned earlier, mathematical structures can often be built up or defined in terms of simpler structures. This diagram (which itself is an example of the type of discrete structure known as a graph) outlines some ways in which a variety of discrete (and continuous) structures can ultimately be built up from a very simple type of discrete structure known as a set, which is one of the first structures we will learn. This is just to give you a a bit of a feel for the inter-relatedness of mathematical objects. This diagram is vastly simplified; some of the dependencies are left out, many other kinds of structures and ways of defining structures in terms of other structures are also possible and are left out for simplicity. For example, sets themselves can be defined in terms of functions, or relations. No one type of structure is truly fundamental, since almost any structure can be defined in terms of almost any other. Sets are only one possible starting point; but they are a popular one because their definition is so simple. We will revisit this diagram in more detail throughout this course, and show how the various links work.(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*1st row: NOT, OR, XOR, IMPLIES, logical equvalence, FORALL2nd row: EXISTS, sets / index notation / ellipses, integers/naturals/reals, therefore, set builder notation, not a member of3rd row: empty set, subset, cardinality, union, set complement, intersection over a sequence of sets4th row: function signature, inverse, composition, floor, summation over a set, product over a sequence5th row: at most order/at least order/exactly order, minimum/maximum, not divisible by, greatest common denominator/least common multiple, modulo, modular congruence6th row: base-b number representation, matrix indexing, transpose, boolean product, matrix exponentiation, choose7th row: generalized combination, conditional probability, transitive closure, , , in-degree(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*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.(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*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.(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*(c)2001-2003, Michael P. FrankDiscrete Math - Module #0 - OverviewDiscrete Math - Module #0 - Overview*(c)2001-2003, Michael P. Frank*(c)2001-2003, Michael P. Frank