discrete math for computer science students solutions

Bài tập 8.57: Không có giải pháp nào thỏa mãn điều kiện đưa ra, tức là không có tập hợp nào để A/B > 21p3.

Ngày tải lên: 14/08/2014, 04:21

... David M Keil Framingham State University CSCI 317 Discrete Structures 2/14 Study questions on Discrete Structures for Computer Science The intention in providing these questions is to ... table contains (a) variables; (b) formulas; (c) values of formulas under one interpretation; (d) values of formulas under all interpretations; (e) operations If formulas  and  have the same truth ... (d) variables; (e) quantifiers A formula in logic is valid if (a) it is true for some interpretation; (b) it is true for all interpretations; (c) it is true for no interpretation; (d) it is an

Ngày tải lên: 17/11/2016, 22:23

... steam-curing at 60oC for 24 hours In order to optimize the usage of formwork, the products were cast and steam-cured initially for about hours The steam-curing was then stopped for some time to allow ... used in the case of reinforced Portland cement concrete columns and beams are applicable for reinforced geopolymer concrete columns Mid-span deflection at service load of reinforced geopolymer concrete ... “Bond performance of Reinforcing Bars in Inorganic Polymer Concrete (IPC)”, Journal of Materials Science, Sumajouw, M D J and Rangan, B.V., “Low-Calcium Fly Ash-Based Geopolymer Concrete: Reinforced

Ngày tải lên: 09/12/2016, 07:56

... Functions Nguyen An Khuong, Huynh Tuong Nguyen Chapter Sets and Functions Contents Sets Discrete Structures for Computer Science (CO1007) on Ngày tháng 10 năm 2016 Set Operation Functions One-to-one and ... fundamental discrete structure on which all discrete structures are built • Sets are used to group objects, which often have the same properties Contents Sets Set Operation Example • Set of all the students ... Operation Example • Set of all the students who are currently taking Discrete Mathematics course • Set of all the subjects that K2011 students have to take in Functions One-to-one and Onto Functions

Ngày tải lên: 29/03/2017, 18:30

... An Khuong, Huynh Tuong Nguyen Chapter Relations Contents Properties of Relations Discrete Structures for Computer Science (CO1007) on Ngày tháng 11 năm 2016 Combining Relations Representing Relations ... Closures of Relations Types of Relations Homeworks Nguyen An Khuong, Huynh Tuong Nguyen Faculty of Computer Science and Engineering University of Technology, VNU-HCM 4.1 Contents Relations Nguyen An ... equivalence classes of an equivalence relation R on a set S form a partition of S Homework Every partition of a set can be used to form an equivalence relation 4.31 Relations Example Nguyen An

Ngày tải lên: 29/03/2017, 18:30

... Counting Nguyen An Khuong, Huynh Tuong Nguyen Chapter Counting Contents Introduction Discrete Structures for Computer Science (CO1007) on Ngày 17 tháng 11 năm 2016 Counting Techniques Pigeonhole Principle ... project from one of three fields: Information system (32 projects), Software Engineering (12 projects) and Computer Science (15 projects) How many ways are there for a student to choose? Solution: ... • Probability Pigeonhole Principle • Statistics Permutations & Combinations • Computer science • Game theory • Information theory • 5.4 Problems Counting Nguyen An Khuong, Huynh Tuong Nguyen

Ngày tải lên: 29/03/2017, 18:30

... English as a foreign language, especially English for computer science in Vietnam. The main goal of ESP course, especially English for Computer Science is aimed at providing the students with ... questionnaires for the research. The ideas for these questionnaires were based on the literature, the challenges in reading English for Computer Science and the author’s observation of the students ... examine the areas of challenges in reading comprehension of English for Computer Science for second-year students in the department of information technology at Nghe An JTTC. To be more specific, the

Ngày tải lên: 18/12/2013, 15:37

... importance of reading English for Computer Science for IT students It is obvious that both teachers and students often encounter difficulties when teaching and learning English for CS The aims of this ... English for Specific Purposes CS: Computer Science IT: Information Technology L1: The first language L2: The second language v LIST OF TABLES AND FIGURES Figure: Students? ?? background Table 1: Students? ?? ... of English for Computer Science perceived by the second year students at VTTTU and present some useful techniques for improving the quality of teaching and learning to read English for CS In order

Ngày tải lên: 28/03/2015, 10:36

... difference/anti-difference pairs useful for summation. page 55. Get it? Despite all the parallels between continuous and discrete math, some continuous notions have no discrete analog. For example, the chain ... is the discrete analog of the continuous lnx. (We will define H, for noninteger x in Chapter 6, but integer values are good enough for present purposes. We’ll also see in Chapter 9 that, for large ... This formula indicates why harmonic numbers tend to pop up in the solutions to discrete problems like the analysis of quicksort, just as so-called natural logarithms arise naturally in the solutions

Ngày tải lên: 14/08/2014, 04:21

... notation for it; but alas, number theorists haven’t come up with a very good one yet. Therefore we cry: HEAR us, 0 MATHEMATICIANS OF THE WORLD! LETUS NOTWAITANYLONGER! WE CAN MAKEMANYFORMULAS ... ma-nb = 1. (4.32) (Actually we said m’m + n’n = gcd( m, n), but we can write 1 for gcd( m, n), a for m’, and b for -n’.) The Farey series gives us another proof of (4.32), because we can let ... entire equations, for which a slightly different notation is more convenient: a s b (mod m) amodm = bmodm. (4.35) For example, 9 = -16 (mod 5), because 9 mod 5 = 4 = (-16) mod 5. The formula ‘a =

Ngày tải lên: 14/08/2014, 04:21

... h(n, k) are there for exactly k fans to get their own hats back? For example, if n = 4 and if the hats and fans are... integer x 3 0 (5 .44 ) This formula is valid for any function f(x) ... have the sum we did before; but now the m occurs in a different place. This problem is a bit harder yet than Problem 7, but (fortunately) we’re getting better at finding solutions. We can begin ... integers j is zero, by (5.24). Hence -S, is the sum for j < 0. To evaluate -S, for j < 0, let’s replace j by -k - 1 and sum for k 3 0: m! n! sm = (m+n+l)! k>O ~(-l)frn,+“k’l)

Ngày tải lên: 14/08/2014, 04:21

... numbers. We can therefore perform “absorption” in terms like n[z] and k{ T}, when we do proofs by mathematical induction. Every permutation is equivalent to a set of cycles. For example, consider ... nearly as often. Therefore it’s best for us just to list the simplest ones, for future reference when a tough Stirling nut needs to be cracked. Tables 250 and 251 contain the formulas that are most ... XpJ,(X)2Q TX>0 (6 .53 ) (iln& - = x&o,(x+n)zn; / Therefore we can obtain general convolution formulas for Stirling numbers, as we did for binomial coefficients in Table 202; the results appear

Ngày tải lên: 14/08/2014, 04:21

... F,+I , for n > 0. (We consider this to be a closed form for Tnr because the Fibonacci numbers are important enough to be considered “known!’ Also, F, itself has a closed form (6.123) ... before. tilings, for all n 3 0, and call it T: T =~+o+rn+~+m~+m+a+ (7.1) (The first term ‘I’ on the right stands for the null tiling of a 2 x 0 rectangle.) This sum T represents lots of information. ... and get a closed form for the coeffi- cients, but it’s bett,er to save that for later in the chapter after we’ve gotten more experience. So let’s divest ourselves of dominoes for the moment and

Ngày tải lên: 14/08/2014, 04:21

... Introduction “Algorithmic analysis” is a branch of computer science that derives quantitative information about the efficiency of computer methods “Probabilistic analysis of an algorithm” ... going... Let F(z) and G(z) be the pgf’s for X and Y, and let H(z) be the pgf for X + Y Then H(z) = F(z)G(z), and our formulas (8.28) through must have (8.31) for mean and variance tell us that ... as before. And if j < k we have E(F,,j F,,k) = Pr(rr has both j and k as fixed points) = (n - 2)!/n! = l/n(n - 1). Therefore E(FfJ = ; + n ;! = 0 2 n(n-1) 2, for n

Ngày tải lên: 14/08/2014, 04:21

... stands for “‘real part.” valid for !.Xz > 9%~. The asymptotic formula for Bernoulli numbers B, in Table 438 illustrates this principle. On the other hand, the asymptotic formulas for H,, ... “common” base-10 logarithm that students learn in high school-the “common” logarithm that turns out to be very uncommon in mathematics and computer science? Yes; and many mathematicians confuse the ... a formula. Problem 2: Perturbation of Stirling’s formula. Stirling’s approximation for n! is undoubtedly the most famous asymp- totic formula of all. We will prove it later in this chapter; for

Ngày tải lên: 14/08/2014, 04:21

... So there are ,at most four solutions, of which two (x = 0 and x = 1) don’t qualify for the title “n-digit number” unless n = 1. The other two solutions have the forms x and 1 On + 1 x, and ... + j, a3 , , a, ,,; b2, , b,; z) for 0 6 j 6 d, thereby eliminating an upper parameter and a lower parameter Thus, for example, we get closed forms for F( a, b; a - 1; z), F( a, ... applications to computer graphics because it represents all rational directions in the plane.) 4.5 Lk = (A :) and Rk = (Ly) ; this holds even when k < 0. (We will find a general formula for any

Ngày tải lên: 14/08/2014, 04:21

... original formula for q. Recall that our proof of the formula we had in Exercise 1.4-5 did not explain why the product of three factorials appeared in the denominator, it simply proved the formula ... distinct elements. There are n choices for the first number in the list. For each way of choosing the first element, there are n −1choices for the second. For each choice of the first two elements, ... the product in the denominator of the formula in Exercise 1.4-5 for the number of labellings with three labels is what it is, and could generalize this formula to four or more labels. Equivalence...

Ngày tải lên: 21/02/2014, 09:20

... fundamental one in discrete mathematics. The principle of mathematical induction is usually described in two forms. The one we have talked about so far is called the “weak form.” The Weak Principle of Mathematical ... −r = r n b + a 1 −r n 1 −r . Therefore by the principle of mathematical induction, our formula holds for all integers n greater than 0. Corollary 4.2.2 The formula for the sum of a geometric series ... equation p(x)=0ofdegree m there are no real numbers for solutions. 11. Let p(x) stand for “x is a prime,” q(x) for “x is even,” and r(x, y) stand for “x = y.” Write down the statement “There is one...

Ngày tải lên: 31/03/2014, 15:58

... elegant way to model a computer in mathematical terms. This was a breakthrough, because it allowed the tools of mathemat- ics to be brought to bear on questions of computation. For example, with his ... truth of mathematics. 2.3 Using Induction Induction is by far the most important proof technique in computer science. Generally, induction is used to prove that some statement holds for all natural ... divisibility hold. 1. If a | b, then a | bc for all c. 2. If a | b and b | c, then a | c. 3. If a | b and a | c, then a | sb + tc for all s and t. 4. For all c = 0, a | b if and only if ca | cb. Proof....

Ngày tải lên: 05/03/2014, 23:20