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<div style="font-size: 120%;font-family: 'times new roman';text-align: justify;text-indent: 0.5in"><p style="line-height: 2">Discrete mathematics describes processes that consist of a sequence of individual steps.Thiscontrasts with calculus, which describes processes that change in a continuous fashion Whereas the ideas of calculus were undamental to the science and technology of the industrial revolution, the ideas of discrete mathematics underlie the science and technology of the computer age. The main themes of a first course in discrete mathematics are logic and proof, induction and recursion, discrete structures, combinatorics and discrete probability, algorithms and their analysis, and applications and modeling...</p></div> | <div style="font-size: 120%;font-family: 'times new roman';text-align: justify;text-indent: 0.5in"><p style="line-height: 2">Discrete mathematics describes processes that consist of a sequence of individual steps.Thiscontrasts with calculus, which describes processes that change in a continuous fashion Whereas the ideas of calculus were undamental to the science and technology of the industrial revolution, the ideas of discrete mathematics underlie the science and technology of the computer age. The main themes of a first course in discrete mathematics are logic and proof, induction and recursion, discrete structures, combinatorics and discrete probability, algorithms and their analysis, and applications and modeling...</p></div> | ||
| + | =='''Logic and Proof '''== | ||
| + | <div style="font-size: 120%;font-family: 'times new roman';text-align: justify;text-indent: 0.5in"><p style="line-height: 2">Probably the most important goal of a first course in discrete mathematics is to help students develop the ability to think abstractly. This means learning to use logically valid forms of argument and avoid common logical errors, appreciating what it means to reason from definitions, knowing how to use both direct and indirect. </p></div> | ||
| + | =='''Induction and Recursion '''== | ||
| + | <div style="font-size: 120%;font-family: 'times new roman';text-align: justify;text-indent: 0.5in"><p style="line-height: 2">An exciting development of recent years has been the increased appreciation for the power and beauty of “recursive thinking.” To think recursively means to address a problem by assuming that similar problems of a smaller nature have already been solved and figuring out how to put those solutions together to solve the larger problem. Such thinking is widely used in the analysis of algorithms,where recurrence relations that result from recursive thinking often give rise to formulas that are verified by mathematical induction. </p></div> | ||
| + | =='''Discrete Structures'''== | ||
| + | <div style="font-size: 120%;font-family: 'times new roman';text-align: justify;text-indent: 0.5in"><p style="line-height: 2">Discrete mathematical structures are the abstract structures that describe, categorize, and reveal the underlying relationships among discrete mathematical objects Those studied in this book are the sets of integers and rational numbers, general sets, Boolean algebras, functions, relations, graphs and trees, formal languages and regular expressions, and finite-state automata.. </p></div> | ||
| + | =='''Combinatorics and Discrete Probability'''== | ||
| + | <div style="font-size: 120%;font-family: 'times new roman';text-align: justify;text-indent: 0.5in"><p style="line-height: 2">Combinatorics is the mathematics of counting and arranging objects, and probability is the study of laws concerning the measurement of random or chance events. Discrete probability focuses on situations involving discrete sets of objects, such as finding the likelihood of obtaining a certain number of heads when an unbiased coin is tossed a certain number of times. Skill in using combinatorics and probability is needed in almost every discipline where mathematics is applied, from economics to biology, to computer science, to chemistry and physics, to business management. </p></div> | ||
| + | =='''Algorithms and Their Analysis'''== | ||
| + | <div style="font-size: 120%;font-family: 'times new roman';text-align: justify;text-indent: 0.5in"><p style="line-height: 2">The word algorithm was largely unknown in the middle of the twentieth century, yet now it is one of the first words encountered in the study of computer science. To solve a problem on a computer, it is necessary to find an algorithm or step-by-step sequence of instructions for the computer to follow. Designing an algorithm requires an understanding of the mathematics underlying the problem to be solved Determining whether or not an algorithm is correct requires a sophisticated use of mathematical induction. Calculating the amount of time or memory space the algorithm will need in order to compare it to other algorithms that produce the same output requires knowledge of combinatorics, recurrence relations, functions, and O-, _-, and _-notations. </p></div> | ||
| + | =='''Applications and Modeling s'''== | ||
| + | <div style="font-size: 120%;font-family: 'times new roman';text-align: justify;text-indent: 0.5in"><p style="line-height: 2">Mathematical topics are best understood when they are seen in a variety of contexts and used to solve problems in a broad range of applied situations One of the profound lessons of mathematics is that the same mathematical model can be used to solve problems in situations that appear superficially to be totally dissimilar. A goal of this book is to show students the extraordinary practical utility of some very abstract mathematical ideas. </p></div> | ||
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<div style="font-size: 120%;font-family: 'times new roman';text-align: justify;text-indent: 0.5in"><p style="line-height: 2">Pengertian Model : Suatu abstraksi atau penyederhanaan dari suatu realitas sistem yang kompleks dimana hanya komponen-komponen yang relevan atau faktor-faktor yang dominan dari masalah-masalah yang dianalisis diikutsertakan.Model menunjukkan hubungan (langsung atau tidak langsung) dari aksi dan reaksidalam pengertian sebab dan akibat. Model akan tampak kurang kompleks dibandingkenyataannya hal ini disebabkan karena penyederhanaan tadi. Dalam pembentukan modelperlu diperhatikan : Variabel, Hubungan, dan Teknik-teknik kuantitatif sepertiStatistik, Simulasi, dll. Model dapat diklasifikasikandalam banyak cara, misalnya berdasakan jenis, dimensi, fungsi, tujuan, subjek,atau derajad abstarksinya. Yang palingsederhana adalah berdasarkan jenis: Iconic (physical), analogue(diagramatic),dan symbolic (mathematical).</p></div> | <div style="font-size: 120%;font-family: 'times new roman';text-align: justify;text-indent: 0.5in"><p style="line-height: 2">Pengertian Model : Suatu abstraksi atau penyederhanaan dari suatu realitas sistem yang kompleks dimana hanya komponen-komponen yang relevan atau faktor-faktor yang dominan dari masalah-masalah yang dianalisis diikutsertakan.Model menunjukkan hubungan (langsung atau tidak langsung) dari aksi dan reaksidalam pengertian sebab dan akibat. Model akan tampak kurang kompleks dibandingkenyataannya hal ini disebabkan karena penyederhanaan tadi. Dalam pembentukan modelperlu diperhatikan : Variabel, Hubungan, dan Teknik-teknik kuantitatif sepertiStatistik, Simulasi, dll. Model dapat diklasifikasikandalam banyak cara, misalnya berdasakan jenis, dimensi, fungsi, tujuan, subjek,atau derajad abstarksinya. Yang palingsederhana adalah berdasarkan jenis: Iconic (physical), analogue(diagramatic),dan symbolic (mathematical).</p></div> | ||
<div style="font-size: 120%;font-family: 'times new roman';text-align: justify;text-indent: 0.5in"><p style="line-height: 2">Iconic : suatu penyajian fisik yang tampak seperti aslinyadari suatu sistem nyata dengan skala berbeda bisa scale up atau scale down.Contoh: mainan anak-anak, peta, potret, histogram,maket, struktur sel. | <div style="font-size: 120%;font-family: 'times new roman';text-align: justify;text-indent: 0.5in"><p style="line-height: 2">Iconic : suatu penyajian fisik yang tampak seperti aslinyadari suatu sistem nyata dengan skala berbeda bisa scale up atau scale down.Contoh: mainan anak-anak, peta, potret, histogram,maket, struktur sel. | ||
Revisi per 17 Oktober 2014 03.45
Discrete mathematics describes processes that consist of a sequence of individual steps.Thiscontrasts with calculus, which describes processes that change in a continuous fashion Whereas the ideas of calculus were undamental to the science and technology of the industrial revolution, the ideas of discrete mathematics underlie the science and technology of the computer age. The main themes of a first course in discrete mathematics are logic and proof, induction and recursion, discrete structures, combinatorics and discrete probability, algorithms and their analysis, and applications and modeling...
Daftar isi
Logic and Proof
Probably the most important goal of a first course in discrete mathematics is to help students develop the ability to think abstractly. This means learning to use logically valid forms of argument and avoid common logical errors, appreciating what it means to reason from definitions, knowing how to use both direct and indirect.
Induction and Recursion
An exciting development of recent years has been the increased appreciation for the power and beauty of “recursive thinking.” To think recursively means to address a problem by assuming that similar problems of a smaller nature have already been solved and figuring out how to put those solutions together to solve the larger problem. Such thinking is widely used in the analysis of algorithms,where recurrence relations that result from recursive thinking often give rise to formulas that are verified by mathematical induction.
Discrete Structures
Discrete mathematical structures are the abstract structures that describe, categorize, and reveal the underlying relationships among discrete mathematical objects Those studied in this book are the sets of integers and rational numbers, general sets, Boolean algebras, functions, relations, graphs and trees, formal languages and regular expressions, and finite-state automata..
Combinatorics and Discrete Probability
Combinatorics is the mathematics of counting and arranging objects, and probability is the study of laws concerning the measurement of random or chance events. Discrete probability focuses on situations involving discrete sets of objects, such as finding the likelihood of obtaining a certain number of heads when an unbiased coin is tossed a certain number of times. Skill in using combinatorics and probability is needed in almost every discipline where mathematics is applied, from economics to biology, to computer science, to chemistry and physics, to business management.
Algorithms and Their Analysis
The word algorithm was largely unknown in the middle of the twentieth century, yet now it is one of the first words encountered in the study of computer science. To solve a problem on a computer, it is necessary to find an algorithm or step-by-step sequence of instructions for the computer to follow. Designing an algorithm requires an understanding of the mathematics underlying the problem to be solved Determining whether or not an algorithm is correct requires a sophisticated use of mathematical induction. Calculating the amount of time or memory space the algorithm will need in order to compare it to other algorithms that produce the same output requires knowledge of combinatorics, recurrence relations, functions, and O-, _-, and _-notations.
Applications and Modeling s
Mathematical topics are best understood when they are seen in a variety of contexts and used to solve problems in a broad range of applied situations One of the profound lessons of mathematics is that the same mathematical model can be used to solve problems in situations that appear superficially to be totally dissimilar. A goal of this book is to show students the extraordinary practical utility of some very abstract mathematical ideas.
Pengertian Model : Suatu abstraksi atau penyederhanaan dari suatu realitas sistem yang kompleks dimana hanya komponen-komponen yang relevan atau faktor-faktor yang dominan dari masalah-masalah yang dianalisis diikutsertakan.Model menunjukkan hubungan (langsung atau tidak langsung) dari aksi dan reaksidalam pengertian sebab dan akibat. Model akan tampak kurang kompleks dibandingkenyataannya hal ini disebabkan karena penyederhanaan tadi. Dalam pembentukan modelperlu diperhatikan : Variabel, Hubungan, dan Teknik-teknik kuantitatif sepertiStatistik, Simulasi, dll. Model dapat diklasifikasikandalam banyak cara, misalnya berdasakan jenis, dimensi, fungsi, tujuan, subjek,atau derajad abstarksinya. Yang palingsederhana adalah berdasarkan jenis: Iconic (physical), analogue(diagramatic),dan symbolic (mathematical).
Iconic : suatu penyajian fisik yang tampak seperti aslinyadari suatu sistem nyata dengan skala berbeda bisa scale up atau scale down.Contoh: mainan anak-anak, peta, potret, histogram,maket, struktur sel.
Analogue : lebih abstrak dibanding iconic karena tidak kelihatan sama antara modeldengan sistem nyata. Contohnya jaringan pipa tempat air mengalir dapatdigunakan dengan pengertian yang sama dengan pendistribusian aliran listrik.Warna-warna pada peta. Mathematic : adalah model paling abstrak. Menggunakan seperangkat simbol matematik untukmenunjukkan komponen-komponen (dan hubungan antar mereka dari sistem nyata). Tidak semua sistem nyata selalu dapat diekspresikan dalam rumusanmatematik.
Model ini dibagi dua: Deterministik dan probabilistik.Ada beberapa cara penyederhanaan dalam membuat model
- Melinierkan hubungan - Mengurangi banyaknya variabel/kendala - Mengubah variable dari diskrit ke kontinu - Mengganti tujuan yang kompleks menjadi tunggal - Mengeluarkan unsur dinamik menjadi statik - Mengasumsikan variabel random menjadi tunggal
Pembentukan model adalah hal paling esensial dalam RO – Philips, Ravindran, dan Solberg (1976)memberikan sepuluh prinsip dalam membentuk model: - Jangan membuat model yang rumit jika yang sederhana cukup - Perumusan masalah hendaknnya disesuaikan dengan teknik penyelesaian - Dalam memecahkan model hendaknya jangan melakukan kesalahan matematik - Pastikan kecocokan model sebelum diputuskan untuk diterapkan - Model tidak boleh keliru dengan sistem nyata - Jangan membuat model yang tidak diharapkan - Hati-hati dengan model yang terlalu banyak - Pembentukan model hendaknya dapat memberikan beberapa keuntungan - GIGO – Garbage In Garbage Out – suatu model tidak lebih baik dari pada datanya
- Model tidak dapat menggantikan pengambil keputusan.