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X bring you back to the exercise). Readers are encouraged to try these exercisesbeforelookingatthehelp. Discrete Mathematics i About the Tutorial Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool. Discrete mathematics deals with objects that come in discrete bundles, e.g., 1 or 2 babies. In contrast, continuous mathematics deals with objects that vary continuously, e.g., 3.42 inches from a wall. Think of digital watches versus analog watches (ones where the second hand loops around continuously without stopping).
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There are two ways: 1. Learn the material and practice until you master it. This may involve studying with other students and/or looking at other books and videos that cover the same material. Then, you actually do the exercises and double check your answers using alternative methods. When you are done, you will know a very important part of mathematics and you will feel justifiably good about your achievement. 2. You can find the answers with research and copy them without understanding the material. When you are done, you will know next to nothing and you might temporarily feel pretty smug that you passed a course without learning much. In the long run, you will be less competent and you’ll imagine the way to succeed in life is to take short cuts and cheat. This will encourage you to take more cheating short cuts and eventually you will almost certainly be caught. There is a small chance that you will get away with cheating your whole life in which case you will have lived a kind of “shadow” of a life.
Discrete Mathematical Structures.pdf - Free download Ebook, Handbook, Textbook, User Guide PDF files on the internet quickly and easily. C) The proposition is ∃s∃c∃m P (s, c, m) ∧ (c = junior) ∧ (m = mathematics). This is true, since there is a sophomore majoring in computer science. D) The proposition is ∀s ∃cP (s, c, computer science) ∨ ∃mP (s, sophomore, m). This is false, since there is a freshman mathematics major. Discrete Mathematics i About the Tutorial Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities.
Now, perhaps you really want to follow path one and learn the material but you are struggling with it and you feel getting the answers would help you learn it. It would be better to discuss the concepts with other people and look for other material on line. Maybe you think math comes “easy” to some and is impossible for others, like yourself. You can learn it, understand it and do it. Math comes “easy” to very very few people. The concepts of probability, for instance, are very counter-intuitive. I have speculated about why this might be in a couple articles. In nature, where we evolved, things are rarely if ever scattered in a truly random fashion. They are typically “clustered” and some of our mistaken intuitions about sampling and so on are based on that. You may have heard of the “Monte Hall” problem? Even Ph.D.’s in math often get this wrong.
Van morrison cds. You may discover that you are more of a “visual” or “motor” thinker. In that case, you might find on-line, or have to develop for yourself, alternative ways to represent some of the concepts in the book. You might also generate additional exercises that use the same principles but are based on your own central interests. For instance, if you are passionate about tennis, you should generate for *yourself* additional problems and answers about tennis players. Are break points more likely to be won by particular servers over the course of their careers than typical service points? If you like basketball, are “hot hands” a real phenomenon or are “streaks” just what you would expect from chance alone?
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$begingroup$As a programmer, mathematics is important basic knowledge to study some topics, especially Algorithms. Many websites, and my fellows suggest me to study Discrete Mathematics before going to Algorithms, so I want to know which Discrete Mathematics book is suitable for my needs?
11 Answers
$begingroup$Concrete Mathematics: A Foundation for Computer Science, By Donald Knuth himself!
Discrete Math knowledge is needed to become adept in proving the correctness and deriving the complexity of algorithms and data structures. You will be taught those in Algo/DS books, but you can only get the mathematical proficiency by practicing just discrete math.
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Knuth book is very good for that. But IMHO, you will only need it if you for doing advanced proofs in DS/Algorithms.
For a beginner, it would be great to go over 'Grimaldi' http://www.amazon.com/Discrete-Combinatorial-Mathematics-Applied-Introduction/dp/0201199122and then quickly move to Algorithms.
Otherwise, you will continue going deep in Discrete Math and never get to Algorithms/DS.
Remember, Discrete Math does not teach you how to design algorithms or Data structures. That will come only by practicing Algorithm problems @ topcoder, acm icpc , spoj etc and reading books on Algos/DS or courses on those.
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My 2 cents.
A very good textbook for discrete mathematics at an undergraduate level is the Kenneth Rosen book titled Discrete Mathematics and Its Applications.
The book provides solutions to half of the problems. You can also buy the Student's Solutions Guide. I don't own it, but I would suspect that it either provides the answers to the other half of the questions or provides a step-by-step guide to solving the problems (the book only provides final answers with minimal explanations of those answers).
It's used for the two-quarter sequence in Discrete Mathematics that is taken by computer science and software engineering majors, as well as a number of mathematics programs at my university. I kept this book around even after I took the course, and I'm currently using it to brush up on my discrete math skills for my Certified Software Development Associate exam.
Theres many different areas to discrete math, and many good books.
theres Graph Theory by Diestel, which has a free pdf version available at
theres generatingfunctionology by wilf, free pdf version at
Other books that are good include Enumerative combinatorics 1 and 2 by Richard P Stanley (a book which is sufficiently dense that having at least 1 analysis and algebra course each will help).
that being said, for more introductory expositions in terms of expected mathematical maturity, I'd suggest googling around and looking at various lecture notes of the 'intro to combinatorics' or 'mathematics for computer scientists' sorts. I found that MIT OCW's 'mathematics for Computer Scientists' notes were quite nice when I looked at them several years ago.
has a link to the lecture notes. There are some really funny asides in it. One of my favorites '.. anyone who says that is wrong, and you should make fun of them until they cry'.
Also, If you want to dig even deeper into discrete math/ combinatorics, the value of building up a wee bit of mathematical basics in other areas of math. Complex Analysis, real analysis (at the level of at least baby rudin, and perhaps even up to functional analysis), maybe some probability up to its measure theory formulation level, and at least a smidge of abstract algebra. Then you can do stuff like look at the combinatorics of random processes (great for analyzing randomized algorithms) and look at cool problems like percolation.
theres probably other things I should suggest, but the point is discrete math is accessible without that much of a background, but is also rewards you for enriching that mathematics background with some amazingly beautiful stuff thats 1) awesome and fun 2) useful.
I very much like Norman Biggs' Discrete Mathematics. I would not recommend the second edition. Rather, get the first edition (the 'revised' first edition if you can). The text claims to be self-contained (seems so to me).
Since there is not much info on this edition of the text online (the preview on Amazon is the second edition), here is an outline:
Part 1: Numbers and Counting
- Integers (ordering, recursion, induction, divisibility, gcd, factorization)
- Function and counting (surjections, injections, bijections, pigeonhole principle, finite vs infinite)
- Principles of counting (Euler's function, addition principle, words, permutations)
- Subsets and designs (binomial theorem, sieve principle, designs, $t$-designs)
- Partition (equivalence relations,distributions, multinomial numbers, classification of permutations)
- Modular arithmetic (congruences, $mathbb{Z}_m$, cyclic constructions, Latin squares)
Part 2: Graphs and Algorithms:
- Algorithms and efficiency (proving correctness, $O$ notation, comparison, sorting)
- Graphs (isomorphism of graphs, valency, paths, cycles, trees, coloring, greedy algorithm)
- Trees, sorting, searching (counting leaves, sorting algorithms, spanning trees, MST problem, depth-first, breadth-first, shortest path problem)
- Bipartite graphs (relations, edge colorings, matchings, maximum matchings, transversals)
- Digraphs, networks, flows (critical paths, flows and cuts, max-flow min-cut theorem, labelling algorithm)
- Recursive techniques (linear recursion,recursive bisection, recursive optimization, dynamic programming)
Part 3: Algebraic Methods:
- Groups (axioms, isomorphisms,cyclic groups, subgroups, cosets)
- Groups of permutations (definitions, orbits, stabilizers, size/number of orbits, representation of groups by permutations)
- Rings, fields, polynomials (division algorithm, Euclidean algorithm, factorization)
- Finite fields (order, construction, primitive element theorem, finite geometry, projective planes, existence)
- Error correction (words, codes, errors, linear codes, cyclic codes)
- Generating functions (power series, partial fractions, binomial theorem, linear recursion)
- Partitions of a positive integer (conjugate partitions, generating functions, mysterious identity)
- Symmetry and counting (cyclic and dihedral symmetry,3D symmetry, inequivalent colorings, colorings and generating functions, Polya's theorem)
I really like Discrete Mathematics by Ross and Wright:
Mathematical Thinking: Problem-Solving and Proofs.
John P. D'Angelo, Douglas B. West.
Available at Amazon.
This is supposed to be an introduction to mathematical proofs. As such it is not restricted to discrete mathematics. But it does a very good job for discrete mathematics. You would also see some proof in real analysis; but you can focus only on the discrete part ignoring this.
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I found the book Elements of Discrete Mathematics by C. L. Liuextremely helpful.
It is at a very basic level and is great if you are looking for an introduction into discrete mathematics.
The best book as far as i know are these two:
or
The best book to study Discrete mathematics is ' Discrete mathematics and Structures' by Satinder Bal Gupta'. It is published by University Science Press. The language of the book is very simple. It contained hundreds of solved and unsolved problems with hints.
Fundamentals of Discrete mathematical structures, 3rd Edition. It is written as per ACM-Curriculum, comprises lot of GATE level questions, and written by a Computer science Professor.