...Q1. a. Torencia Products assembles two models, busterStop and Speed Luster, using the electronic components A and B. * The decision variables: unit quantities of each model to manufacture * Objective function maximize profits = $24BusterStop + $40SpeedLuster LaserStop Unit number to assemble LaserStop models. SpeedBuster Unit number to assemble SpeedBuster models . $24 the profit per unit of LaserStop model. $40 the profit per unit of SpeedBuster model. * Constraints: 1. 18LaserStop + 12SpeedBuster ≤ 4000 One unit of LaserStop model requires 18 A components and one unit of SpeedBuster model requires 12 A components. The total number of component A is limited to 4000. 2. 6LaserStop + 10SpeedBuster ≤ 3500 One unit of LaserStop model requires 6 B components and one unit of SpeedBuster model requires 10 B components. The total number of component B is limited to 3500. 3. LaserStop ≥ 0 and LaserStop = int LaserStop is non-negative and should be an integer. 4. SpeedBuster≥ 0 and SpeedBuster = int SpeedBuster is non-negative and should be an integer. b. A linear optimization model Maximize profits = 24LaserStop + 40SpeedBuster 18LaserStop + 12SpeedBuster ≤ 4000 6LaserStop + 10SpeedBuster ≤ 3500 LaserStop ≥ 0 LaserStop = int SpeedBuster≥ 0, SpeedBuster = int ...
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...developing a program including the program description, analysis, test plan, design, and implementation with C code. Program Description: This program will sum two integer numbers to yield a third integer number. It will also divide two float numbers to yield a third float number. Once the calculations are made the results of all the numbers will be printed to the output screen. Analysis: We will use sequential programming statements. We will define 3 integer numbers: a,b,c. c will store the sum of a and b. We will define 3 float numbers: f,g,h. h will store the quotient of f and g. Test Plan: To understand this program the following input numbers could be used for testing: a = 10 b = 20 c = a + b = 10 + 20 = 30 f = 75.0 g = 3.0 h = f/g = 75.0/3.0 = 25.0 Design using Pseudocode: // This program will sum two integer numbers to yield a third integer number. // It will also divide two float numbers to yield a third float number. // Declare variables Declare a,b,c as Integer Declare f as Float // Set values of Integers Set a=10 Set b=20 Set c=a+b // Print c Print a,b,c Set f=75.0 Set g=3.0 Set h=f+g // Print h print f,g,h C Code The following is the C Code that will compile in execute in the online compilers. // C code // This program will sum two integer numbers to yield a third integer number. // It will also divide two float numbers to yield a third float number. // Developer: Faculty CMIS102 // Date: Jan 31, 2014 #include <stdio.h> ...
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...The Concept of Prime Numbers and Zero MTH/110 March 14, 2011 The Concept of Prime Numbers and Zero Have you ever wondered about the origins of prime numbers or the numeral zero? The ancient philosophers and mathematicians from such early civilizations in Egypt, Greece, Babylon, and India did. Their efforts have provided the basic fundamentals for mathematics that are used today. Prime Numbers A prime number is “any integer other than a 0 or + 1 that is not divisible without a remainder by any other integers except + 1 and + the integer itself (Merriam-Webster, 1996). These numbers were first studied in-depth by ancient Greek mathematicians who looked to numbers for their mystical and numerological properties, seeking perfect and amicable numbers. (O’Connor & Robertson, 2009) In 300 BC, Greek mathematician, Euclid of Alexandria proved and documented in his Book IX of the Elements that prime numbers were infinite. He started with what he believed to be a comprehensive list of prime numbers, created a new number, N, by multiplying all of the prime numbers together and adding 1. This resulted in a number not on his list and not divisible by any of his prime numbers. N therefore had to be either prime itself or be a composite number that was a product of at least two other prime numbers not on his list. In 1747, a mathematician named of Euler demonstrated that all even numbers were perfect numbers. However, one hundred years later in 200 BC, Eratosthenes of...
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...Discrete Mathematics Lecture Notes, Yale University, Spring 1999 L. Lov´sz and K. Vesztergombi a Parts of these lecture notes are based on ´ ´ L. Lovasz – J. Pelikan – K. Vesztergombi: Kombinatorika (Tank¨nyvkiad´, Budapest, 1972); o o Chapter 14 is based on a section in ´ L. Lovasz – M.D. Plummer: Matching theory (Elsevier, Amsterdam, 1979) 1 2 Contents 1 Introduction 2 Let 2.1 2.2 2.3 2.4 2.5 us count! A party . . . . . . . . Sets and the like . . . The number of subsets Sequences . . . . . . . Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 7 7 9 12 16 17 21 21 23 24 27 27 28 29 30 32 33 35 35 38 45 45 46 47 51 51 52 53 55 55 56 58 59 63 64 69 3 Induction 3.1 The sum of odd numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Subset counting revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Counting regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Counting subsets 4.1 The number of ordered subsets . . . . 4.2 The number of subsets of a given size 4.3 The Binomial Theorem . . . . . . . . 4.4 Distributing presents . . . . . . . . . . 4.5 Anagrams . . . . . . . . . . . . . . . . 4.6 Distributing money . . . . . . . . . . ...
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...Strategies For Top Scores produce GMAT and GMAC are registered trademarks of the Graduate Management Admission Council which neither sponsors nor endorses th"o :M.anhattanG MAT·Prep the new standard 1. DIVISIBIUTY & PRIMES In Action Problems Solutions 11 21 23 2. ODDS & EVENS In Action Problems Solutions 27. 33 35 3. POSITIVES & NEGATIVES In Action Problems Solutions 37 43 45 4. CONSECUTIVE INTEGERS InAction Problems Solutions 47 5S 57 5. EXPONENTS In Action Problems Solutions 61 71 73 PART I: GENERAL TABLE OF CONTENTS 6. ROOTS IrfActiort,;Problems So1utioQS 75 83 85 7. PEMDAS In Action Problems .Solutions 87 91 93 8. STRATEGIES FOR DATASUFFICIENCY Sample Data Sufficiency Rephrasing 95 103 9. OmCIAL GUIDE PROBLEMS: PART I Problem Solving List Data Sufficiency List 109 112 113 :M.anliattanG MAT'Prep the new standard 10. DMSIBIUTY & PRIMES: ADVANCED 115 133 135 In Action Problems Solutions II. ODDS & EVENS/POSITIVES & NEGATIVES/CONSEC. INTEGERS: ADVANCED In Action Problems Solutions 145 153 155 12. EXPONENTS & ROOTS: ADVANCED In Action Problems Solutions 161 167 169 13. OmCIAL GUIDE PROBLEMS: PART II 173 Problem Solving List Data Sufficiency List 176 177 PART II:...
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...Sczence and Technolog9 Unzverszty of Tokyo (Received November 11, 1994; Final August 25, 1995) A circular connected-(r>s)-out-of-(m,n):Flattice system has m n compoaen%s is consisted of n and and m circles. The system fails if and only if a connected (r,s)-matrix of components fail. Fixskly, this paper proposes a recursive algorithm for the reliability of $he system>which requires O(rnn2sm) computing time. In the statisticaIly independent identically distributed case7the c o m p u t i ~ ~ g is able to be reduced. time Secondly, the upper a ~ lower bounds for the system reliability are obtained for the circular connectedd (r,s)-out-of-(m,n):l? lattice system. FinalIy, it is proved that the reliability of the large system tmds to where p 1A and 7 are exp[-pArs] as n = , m Q - I 2 n -+ m if every component has failure probability ~ n - ' / / ~ , constant, ,LA, A > Ol 7 > r , ri rays Abstract I . INTRODUCTION The linear (circular) consecutive-k-out-of-n:F system has n llnewly (circularly) ordered components. Each compnent either functkms or fails. The systems fail if and only if k consecutive compnents fail (see Figure 1.1). Defining the rmdom variable Zi by if component i functions zi = if component i Fdils, for i = 1 2 , .*, n, the reliability of the linear consect~tive-k-out-of-n:F system can be expressed as PI-{ n{zi= 0, a s i s a + k - 1)') and the circular consecutive-k-out-of-n:F system as a=l n-k+l pro{^^ =Q a s i sa+k-I)'), a=l n whereZi z Z i...
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...Gravner for a number of corrections and comments. Abstract. These are some notes on introductory real analysis. They cover the properties of the real numbers, sequences and series of real numbers, limits of functions, continuity, differentiability, sequences and series of functions, and Riemann integration. They don’t include multi-variable calculus or contain any problem sets. Optional sections are starred. c John K. Hunter, 2014 Contents Chapter 1. Sets and Functions 1 1.1. Sets 1 1.2. Functions 5 1.3. Composition and inverses of functions 7 1.4. Indexed sets 8 1.5. Relations 11 1.6. Countable and uncountable sets 14 Chapter 2. Numbers 21 2.1. Integers 22 2.2. Rational numbers 23 2.3. Real numbers: algebraic properties 25 2.4. Real numbers: ordering properties 26 2.5. The supremum and infimum 27 2.6. Real numbers: completeness 29 2.7. Properties of the supremum and infimum 31 Chapter 3. Sequences 35 3.1. The absolute value 35 3.2. Sequences 36 3.3. Convergence and limits 39 3.4. Properties of limits 43 3.5. Monotone sequences 45 3.6. The lim sup and lim inf 48 3.7. Cauchy sequences 54 3.8. Subsequences 55 iii iv Contents 3.9. The Bolzano-Weierstrass theorem Chapter 4. Series 4.1. Convergence of series 4.2. The Cauchy condition 4.3...
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...A Solution Manual for: A First Course In Probability: Seventh Edition by Sheldon M. Ross. John L. Weatherwax∗ September 4, 2007 Introduction Acknowledgements Special thanks to Vincent Frost and Andrew Jones for helping find and correct various typos in these solutions. Miscellaneous Problems The Crazy Passenger Problem The following is known as the “crazy passenger problem” and is stated as follows. A line of 100 airline passengers is waiting to board the plane. They each hold a ticket to one of the 100 seats on that flight. (For convenience, let’s say that the k-th passenger in line has a ticket for the seat number k.) Unfortunately, the first person in line is crazy, and will ignore the seat number on their ticket, picking a random seat to occupy. All the other passengers are quite normal, and will go to their proper seat unless it is already occupied. If it is occupied, they will then find a free seat to sit in, at random. What is the probability that the last (100th) person to board the plane will sit in their proper seat (#100)? If one tries to solve this problem with conditional probability it becomes very difficult. We begin by considering the following cases if the first passenger sits in seat number 1, then all ∗ wax@alum.mit.edu 1 the remaining passengers will be in their correct seats and certainly the #100’th will also. If he sits in the last seat #100, then certainly the last passenger cannot sit there (in fact he will end up in seat #1). If he sits in any of the 98...
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...COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher, except for reading and browsing via the World Wide Web. Users are not permitted to mount this file on any network servers. Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu Lecture Notes in Microeconomic Theory Ariel Rubinstein Updates to the Printed Version The file you are viewing contains the printed version of the book. In relevant places throughout the text you will find small icons indicating the existence of updates to the text: A red icon indicates there is a correction for a mistake on this line. A green icon indicates an addition to the text at this point. The corrected and added text can be obtained from the author's homepage at http://arielrubinstein.tau.ac.il/ . October 21, 2005 12:18 master Sheet number 1 Page number 1 October 21, 2005 12:18 master Sheet number 2 Page number 2 October 21, 2005 12:18 master Sheet number 3 Page number i Lecture Notes in Microeconomic Theory October 21, 2005 12:18 master Sheet number 4 Page number ii October 21...
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...Algebra for the Utterly Confused This page intentionally left blank. Algebra for the Utterly Confused Larry J. Stephens McGraw-Hill New York San Francisco Washington, D.C. Auckland Bogotá Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. 0-07-143095-4 The material in this eBook also appears in the print version of this title: 0-07-135514-6 All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. For more information, please contact George Hoare, Special Sales, at george_hoare@mcgraw-hill.com or (212) 904-4069. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc...
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... b. There is a person p in my class such that p is c. There is a person p in my class with the property that for every person q in my class, p is . Solution a. person in my class; every person in my class b. at least as old as every person in my class c. at least as old as q ■ Some of the most important mathematical concepts, such as the definition of limit of a sequence, can only be defined using phrases that are universal, existential, and conditional, and they require the use of all three phrases “for all,” “there is,” and “if-then.” For example, if a1 , a2 , a3 , . . . is a sequence of real numbers, saying that the limit of an as n approaches infinity is L means that for all positive real numbers ε, there is an integer N such that for all integers n, if n > N then −ε < an − L < ε. Test Yourself Answers to Test Yourself questions are located at the end of each section. 1. A universal statement asserts that a certain property is for . 2. A conditional statement asserts that if one thing . some other thing then 3. Given a property that may or may not be true, an existential for which the property is true. statement asserts that Exercise Set 1.1 Appendix B contains either full or partial solutions to all exercises with blue numbers. When the solution is not complete, the exercise number has an H next to it. A ✶ next to an exercise number signals that the exercise is more challenging than usual. Be careful not to get into the habit of turning to the solutions too...
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...Lesson 1) Compare 27 and 35. < > Billions |Hundred millions |Ten millions |Millions |Hundred thousands |Ten thousands |Thousands |Hundreds |Tens |ones | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | We should compare numbers by starting . Billions |Hundred millions |Ten millions |Millions |Hundred thousands |Ten thousands |Thousands |Hundreds |Tens |ones | | | | | | | | | | | | | | | | | | | | | | | |Comparing and Explaining Examples 2) ___________ ____ _____________ because of the place Billions |Hundred millions |Ten millions |Millions |Hundred thousands |Ten thousands |Thousands |Hundreds |Tens |ones | | | | | | | | | | | | | | | | | | | | | | | | 3) ___________ __ _____________ because of the place Billions |Hundred millions |Ten millions |Millions |Hundred thousands |Ten thousands |Thousands |Hundreds |Tens |ones | | | | | | | | | | | | | | | | | | | | | | | | 4) 200,324 98,133 because of the place Billions |Hundred millions |Ten millions |Millions |Hundred thousands |Ten thousands |Thousands |Hundreds |Tens |ones | | | | | | | | | | | | | | | | | | | | | | | | 5) 44,454 43,999 because...
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...Exercise 4.1 (4,7 & 18) 4.A wheel of fortune has the integers from 1 to 25 placed on it in a random manner. Show that regardless of how the numbers are positioned on the wheel, there are three adjacent numbers whose sum is at least 39. Adding all 25 inequalities, we find that 3∑_(i=1)^25▒xi = 3∑_(i=1)^25▒i < 25(39) = 975. But 3∑_(i=1)^25▒i=(25)(26)/2=325 gives us the contradiction that 988 = 3(325) < 975 7. A lumberjack has 4n + 110 logs in a pile consisting of n layers. Each layer has two more logs than the layer directly above it. If the top layer has six logs, how many layers are there? . . 18. Consider the following four equations: 1) 1 =1 2) 2 + 3 + 4 = 1 + 8 3) 5 + 6 + 7 + 8 + 9 = 8 + 27 4) 10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64 1) 1=1 n = 1 2) 2 + 3 + 4 = 1 + 8 n = 2 3) 5 + 6 + 7 + 8 + 9 = 8 + 27 n = 3 4) 10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64 n = 4 Formula (n-1)²+1, or n²-2n+1+1or n²-2n+2. n²-2n+2 Exercise 4,3 10 & 15 10. If n ∈ Z+, and n is odd, prove that 8|(n2 − 1). 15. Write each of the following (base-10) integers in base 2 and base 16. a)22 b) 527 c) 1234 d) 6923 22) base 2 10110 base 16 = 16 527 base 2 = 1000001111 base 16 = 20F 1234 base 2 = 10011010010 base 16 = 4D2 6923 base 2 = 1101100001011 base 16 = 1B0B Exercise 4.4 (1&14) 1. For each of the following pairs a, b ∈ Z+, determine gcd(a, b) and express it as a linear combination...
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...Dustin has a set of 180 distinct blocks. Each of these blocks is made of either wood or plastic and comes in one of three sizes (small, medium, large), five colors (red, white, blue, yellow, green), and six shapes (triangular, square, rectangular, hexagonal, octagonal, circular). How many of the blocks in this set differ from a) the small red wooden square block in exactly one way? (For example, the small red plastic square block is one such block.) b) the large blue plastic hexagonal block in exactly two ways? (For example, the small red plastic hexagonal block is one such block.) Problem 15(a) a) How many of the 9000 four-digit integers 1000, 1001, 1002, . . . , 9998, 9999 have four distinct digits that are either increasing (as in 1347 and 6789) or decreasing (as in 6421 and 8653)? b) How many of the 9000 four-digit integers 1000, 1001, 1002, . . . , 9998, 9999 have four digits that are either nondecreasing...
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...GMAT MATH PRACTICE TEST 1 ® The numbers on this test are real numbers. You may assume that positions of points, lines, and angles are in the order shown. A figure accompanying a problem-solving question is intended to provide information useful in solving the problem. Figures are drawn as accurately as possible EXCEPT when it is stated in a specific problem that its figure is not drawn to scale. Straight lines may sometimes appear jagged. All figures lie in a plane unless otherwise indicated. ® ® In data-sufficiency problems that ask for the value of a quantity, the data given in the statements is sufficient only when it is possible to determine exactly one numerical value for the quantity. ® Example 1 What is the value of x + y? (1) 0.5 (x + y) = −1 1 1 (2) x + y = 2 5 5 A B C D • E From (1): x + y = −2, so (1) is sufficient. From (2): x + y = 10, so (2) is sufficient. Example 2 DIRECTIONS: Data-sufficiency problems consist of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements is sufficient for answering the question. Using the data given in the statements plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you must indicate whether A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone...
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