mathematics for computer science mit book

... important proof technique in computer science. Generally, induction is used to prove that some statement holds for all natural values of a variable. For example, here is a classic formula: 46 Number Theory ... 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 ... 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 ... b for some s 2 Sg: For example, if we let Œr; s denote set of numbers in the interval from r to s on the real line, then f 1 .Œ1; 2/ D Œ1=4; 1. For another example, let’s take the “search for...

Ngày tải lên: 23/03/2014, 23:20

Ngày tải lên: 18/12/2013, 15:37

... of any of the books already there. (Notice that if book 2 and book 1 are on shelf 7 in that order, putting book 3 to the immediate right of book 2 means putting it between book2 and book 1.) Thus ... may we place k distinct books on n shelves of a bookcase (all books pushed to the left as far as possible) if there must be at least one book on each shelf? 8. The formula for the number of multisets ... put the books on the shelves as follows: put all the books before the first piece of wood on shelf 1, all the books between the first and second on shelf 2, and so on until you put all the books...

Ngày tải lên: 21/02/2014, 09:20

... 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

... Edition) This book is intended as an introduction to mathematical logic, with an em- phasis on proof theory and procedures for constructing formal proofs of for- mulae algorithmically. This book is ... primarily for computer scientists, and more gen- erally, for mathematically inclined readers interested in the formalization of proofs, and the foundations of automatic theorem-proving. The book is ... 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 ∨...

Ngày tải lên: 29/03/2014, 09:20

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Ngày tải lên: 05/05/2014, 15:02

... same is true for the examples that follow. Example 1.15 Perform the multiplication 53u 555++ 15== 53u 53u 35u= 39 25u 39 25u +39+25u +975975=== 53 18–u Mathematics for Business, Science, ... Operations 1-12 Mathematics for Business, Science, and Technology, Second Edition Orchard Publications tomary to place the minus sign in front of the bar that separates them. For instance, We must not forget ... enter A-4, and so forth. Chapter 1 Numbers and Arithmetic Operations 1-20 Mathematics for Business, Science, and Technology, Second Edition Orchard Publications Solution: For each of (a), (b)...

Ngày tải lên: 19/02/2014, 17:05

... telephone)?  Which informations we may consider for the future (e.g. email, birthday, bankaccount, webpage, ip, image, holographic picture, etc )?  By means of which information should we sort ... course Historical development of databases Databases Introduction Michael Emmerich Leiden Institute for Advanced Computer Science, Leiden University January 17, 2012 Michael T. M. Emmerich Databases Preliminaries Introductory ... stone-plates, books, libraries, microfiche, etc.  not to be ingnored, many concepts/principles developed then - look at town library’s organization!  often hierarchical sorting of information (efficient...

Ngày tải lên: 16/03/2014, 16:20

Ngày tải lên: 29/05/2014, 17:21

... In this problem, we will be reading in formatted data and generating a report. One of the common formats for interchange of formatted data is ’tab delimited’ where each line corresponds to ... } 6 MIT OpenCourseWare http://ocw .mit. edu 6.087 Practical Programming in C January (IAP) 2010 For information about citing these materials or our Terms of Use, visit: http://ocw .mit. edu/terms . ... characters. 4 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.087: Practical Programming in C IAP 2010 Problem Set 3 Control flow. Functions....

Ngày tải lên: 25/04/2013, 08:07

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Ngày tải lên: 09/09/2013, 12:24

... LET S LEARN ENGLISH BOOK 2’ – Hai An Primary school Test 35 minutes Full name:………………………………………………. Subject: English Class: 4 Date: / 12 / 2010 Mark Teacher’s remarks TEST FOR THE FIRST TERM I.Circle ... TERM I.Circle the odd one out( 2 marks) 1. A. August B. Thursday C. Sunday D. Saturday 2. A. Pen B. Book C. Notebook D. Candy 3. A. English B. Maths C. Ruler D. Art 4. A. May B. Monday C. June D. July 5. ... Draw B. Ten C. Read D. Write II. Choose the right words to fill in the blank(2 marks) 1.……… are my books.( This/ These) 2. Those…… my pencils.( is/ are) 3. The maps are………the wall.( on/ in) 4. During...

Ngày tải lên: 28/10/2013, 13:11

... consistent case format for each kind of identifier makes your identifiers easier to under- stand. For example, we use title case (uppercase for the first letter of each word) for class names. ... Fig. 1.12. To get an intuitive feel for how much information these devices can store, con- sider that all the information in this book, including pictures and formatting, requires about 6 MB of ... Java 2 Platform is organized into three major groups: ◗ Java 2 Platform, Standard Edition (J2SE) ◗ Java 2 Platform, Enterprise Edition (J2EE) ◗ Java 2 Platform, Micro Edition (J2ME) This book focuses...

Ngày tải lên: 12/12/2013, 21:15

... degree-of-freedom pose space for a 3D object and also does not account for any non- rigid deformations. Therefore, we use broad bin sizes of 30 degrees for orientation, a factor of 2 for scale, and 0.25 ... operations are performed on image data that has been transformed relative to the assigned orientation, scale, and location for each feature, thereby providing invariance to these transformations. 4. ... while allowing for shifts in their position results in much better classification under 3D rotation. For example, recognition accuracy for 3D objects rotated in depth by 20 degrees increased from 35% for...

Ngày tải lên: 13/12/2013, 16:15

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