mathematics for computer science eric lehman

... 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 ... 1) for all n ∈ N. By the principle of induction, P (n) is true for all n ∈ N, which proves the claim. This proof would look quite mysterious to anyone not privy to the scratchwork we did beforehand. ... 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

... — 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 ... Meyer Department of Electrical Engineering and Computer Science and the Computer Science and AI Laboratory, Massachussetts Institute of Technology Creative Commons 2011, Eric Lehman, F Tom Leighton, Albert R ... 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...

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 ... 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 ... a, 6, and y and trying to find a closed form for the more general recurrence f(1) = cc; f(2n) = 2f(n) + fi, for n 3 1; (1.11) f(2n+1)=2f(n)+y, for n 3 1. (Our original recurrence had...

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 ... 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 ... to understand the general formula for the sum of a geometric progression: t a<k<b for c # 1. Every time we encounter a function f that might be useful as a closed form, we can compute its...

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 ... reduced). It is defined recursively by starting with For N > 1, we form ?$,+I by inserting two symbols just before the kNth symbol of ?N, for all k > 0. The two inserted symbols are k-l - ... -1; p(pk) = 0 for k > 1. Therefore by (4.52), we have the general formula ifm=pjpz p,; if m is divisible by some p2. (4.57) That’s F. If we regard (4.54) as a recurrence for the function...

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: 14/08/2014, 04:21

Ngày tải lên: 14/08/2014, 04:21

Ngày tải lên: 18/12/2013, 15:37

... 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

... 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

... proof theory and procedures for constructing formal proofs of for- mulae algorithmically. This book is designed primarily for computer scientists, and more gen- erally, for mathematically inclined ... Theorem for Formulae in NNF, 355 7.6.1 Skolem-Herbrand-Găodels Theorem in Unsatisability Form, 355 7.6.2 Skolem Normal Form, 357 7.6.3 Compound Instances, 359 7.6.4 Half of a Herbrand-like Theorem for ... 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 ơP 1 ...

Ngày tải lên: 29/03/2014, 09:20

... Li Department of Mathematics and Physics, Air Force Engineering University, China jianq_li@263.net Wanbiao Ma Department of Mathematics and Mechanics, School of Applied Science, University of Science ... Sports, Science and Technology, The Japanese Society for Mathematical Biology, The Society of Population Ecology, Mathematical Society of Japan, Japan Society for Industrial and Applied Mathematics, ... stochastic models and direct computer simulations. References 1. Anderson, R. M. and R. M. May (1991), Infectious diseases of humans. Oxford University Press, Oxford UK. 2. Kermack, W. O. and...

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