discrete mathematics for computer science with student solutions manual cdrom

... numbers than most of us can imagine. However in discrete mathematics we often work with functions from a finite set S with s elements to a finite set T with t elements. Then there are only a finite ... 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 ... the formula was correct. With this idea in hand, we could now explain why the product in the denominator of the formula in Exercise 1.4-5 for the number of labellings with three labels is what...

Ngày tải lên: 21/02/2014, 09:20

... n: • If student 0 gets candy, then student 1 also gets candy. • If student 1 gets candy, then student 2 also gets candy. • If student 2 gets candy, then student 3 also gets candy, and so forth. Now ... are student 17. By these rules, are you entitled to a miniature candy bar? Well, student 0 gets candy by the first rule. Therefore, by the second rule, student 1 also gets candy, which means student ... proof to anyone who disagrees with you. Intimidation. Truth is asserted by someone with whom disagrement seems unwise. Mathematics its own notion of “proof”. In mathematics, a proof is a verification...

Ngày tải lên: 05/03/2014, 23:20

... — page i — #1 Mathematics for Computer Science revised Thursday 10 th January, 2013, 00:28 Eric Lehman Google Inc. F Thomson Leighton Department of Mathematics and the Computer Science and AI ... proposition for each possible set of truth values for the variables. For example, the truth table for the proposition “P AND Q” has four lines, since there are four settings of truth values for the ... or not a T ever appears, but as with testing validity, this approach quickly bogs down for formulas with many variables because truth tables grow exponentially with the number of variables. Is...

Ngày tải lên: 23/03/2014, 23:20

... defined for SO = ao; all n 3 0 .) S, = S-1 + a,, for n > 0. (2.6) Therefore we can evaluate sums in closed form by using the methods we learned in Chapter 1 to solve recurrences in closed form. For ... 3 These two inequalities, together with the trivial solution for n = 0, yield To =O; T, = 2T+1 +l , for n > 0. (1.1) (Notice that these formulas are consistent with the known values TI = ... much happier. That is, we’d like a nice, neat, “closed form” for T,, that lets us compute it quickly, even for large n. With a closed form, we can understand what T,, really is. So how do...

Ngày tải lên: 14/08/2014, 04:21

... exponents for rising factorial powers, analogous to (2.52)? Use this to define XC”. 10 The text derives the following formula for the difference of a product: A(uv) = uAv + EvAu. How can this formula ... different form, n = [n/21 + [n/2]. If we had wanted the parts to be in nondecreasing order, with the small groups coming before the larger ones, we could have proceeded in the same way but with [n/mJ ... circle. 34 Let f(n) = Et=, [lgkl. Find a closed form for f(n) , when n 3 1. L Provethatf(n)=n-l+f([n/2~)+f(~n/Z])foralln~l. 35 Simplify the formula \(n + 1 )‘n! e] mod n. Simplify it, but 36...

Ngày tải lên: 14/08/2014, 04:21

... (mod 4). QED for the case m = 12. QED: Quite Easily So far we’ve proved our congruence for prime m, for m = 4, and for m = Done. 12. Now let’s try to prove it for prime powers. For concreteness ... fractions m/n with m > 0 and n 6 N (including fractions that are not reduced). It is defined recursively by starting with For N > 1, we form ?$,+I by inserting two symbols just before the kNth symbol ... with 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...

Ngày tải lên: 14/08/2014, 04:21

... harder to start with f(k) and to figure out its indefinite sum x f(k) 6k = g(k) + C; this function g might not have a simple form. For example, there is apparently no simple form for x (E) ... every closed form for hypergeometrics leads to additional closed forms and to additional entries in the database. For example, the identities in exercises 25 and 26 tell us how to transform one hypergeometric ... > 2m. Therefore this limit gives us exactly the sum (5.20) we began with. 5.6 HYPERGEOMETRIC TRANSFORMATIONS It should be clear by now that a database of known hypergeometric closed forms is a...

Ngày tải lên: 14/08/2014, 04:21

... haven’t come up with a closed formula for them. We haven’t found closed forms for Stirling numbers, Eulerian numbers, or Bernoulli numbers either; but we were able to discover the closed form H, ... NUMBERS Before we stop to marvel at our derivation, we should check its accuracy. For n = 0 the formula correctly gives Fo = 0; for n = 1, it gives F1 = (+ - 9)/v%, which is indeed 1. For higher ... (-l).", for n > 0. (6.103) When n = 6, for example, Cassini’s identity correctly claims that 1 3.5-tS2 = 1. A polynomial formula that involves Fibonacci numbers of the form F,,+k for small...

Ngày tải lên: 14/08/2014, 04:21

Ngày tải lên: 14/08/2014, 04:21

Ngày tải lên: 14/08/2014, 04:21

Ngày tải lên: 14/08/2014, 04:21

Ngày tải lên: 14/08/2014, 04:21

Ngày tải lên: 14/08/2014, 04:21

Ngày tải lên: 18/12/2013, 15:37

... consider the subfields within an area of study than it is to define the area of study. So it is with computer science. 1.1 What Is Computer Science? In some respects, computer science is a new discipline; ... of computer science is computer programming. Some people prefer the term used in many European languages, informatics, over what is called computer science in the United States. Computer science ... departments of computer science. But computer science has benefited from work done in such older disciplines as mathematics, psychology, electricalengineering, physics, andlinguistics. Computer science...

Ngày tải lên: 19/03/2014, 14:12

... 170 Introduction to Programming 3 Computers have a fixed set of instructions that they can perform for us. The specific instruction set depends upon the make and model of a computer. However, these instructions ... that the computer always attempts to do precisely what you tell it to do. Say, for example, you tell the computer to divide ten by zero, it tries to do so and fails at once. If you tell the computer ... instructions that tell the computer every step to take in the proper sequence in order to solve a problem for a user. A programmer is one who writes the computer program. When the computer produces a...

Ngày tải lên: 19/03/2014, 14:14

... to mathematical logic, with an em- phasis on proof theory and procedures for constructing formal proofs of for- mulae algorithmically. This book is designed primarily for computer scientists, and ... proposition is a Horn formula iff it is a conjunction of basic Horn formulae. (a) Show that every Horn formula A is equivalent to a conjunction of distinct formulae of the form, P i , or ¬P 1 ∨ ... Sets (Languages Without Equality), 194 5.4.7 Completeness: Special Case of Languages Without Func- tion Symbols and Without Equality, 197 PROBLEMS, 205 5.5 Completeness for Languages with Function...

Ngày tải lên: 29/03/2014, 09:20

... such that n>0 and for all integers m>n, for every polynomial 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,” ... −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 with r =1is n−1  i=0 r i = 1 ... dreary) proof of this formula by plugging in our earlier formula for binomial coefficients into all three terms and verifying that we get an equality. A guiding principle of discrete mathematics is that when...

Ngày tải lên: 31/03/2014, 15:58

... of undergraduate computer science curricula and the mathematics which underpins it. Indeed, the whole relationship between mathematics and computer science has changed so that mathematics is now ... rigorous way the core mathematics requirement for undergraduate computer science students at British universities and polytechnics. Selections from the material could also form a one- or two- semester ... Therefore Jack is not a reasonable man. 4. All gamblers are bound for ruin. No one bound for ruin is happy. Therefore no gamblers are happy. 5. All computer scientists are clever or wealthy. No computer...

Ngày tải lên: 31/03/2014, 15:58