Premium Essay

Proofs of Discreet Mathematics

In: Other Topics

Submitted By kentderla22
Words 2416
Pages 10
LOGICAL INFERENCE & PROOFs
Debdeep Mukhopadhyay Dept of CSE, IIT Madras

Defn
• A theorem is a mathematical assertion which can be shown to be true. A proof is an argument which establishes the truth of a theorem.

Nature & Importance of Proofs
• In mathematics, a proof is:
– a correct (well-reasoned, logically valid) and complete (clear, detailed) argument that rigorously & undeniably establishes the truth of a mathematical statement.

• Why must the argument be correct & complete?
– Correctness prevents us from fooling ourselves. – Completeness allows anyone to verify the result.

• In this course (& throughout mathematics), a very high standard for correctness and completeness of proofs is demanded!!

Overview
• Methods of mathematical argument (i.e., proof methods) can be formalized in terms of rules of logical inference. • Mathematical proofs can themselves be represented formally as discrete structures. • We will review both correct & fallacious inference rules, & several proof methods.

Applications of Proofs
• An exercise in clear communication of logical arguments in any area of study. • The fundamental activity of mathematics is the discovery and elucidation, through proofs, of interesting new theorems. • Theorem-proving has applications in program verification, computer security, automated reasoning systems, etc. • Proving a theorem allows us to rely upon on its correctness even in the most critical scenarios.

Proof Terminology
• Theorem
– A statement that has been proven to be true.

• Axioms, postulates, hypotheses, premises
– Assumptions (often unproven) defining the structures about which we are reasoning.

• Rules of inference
– Patterns of logically valid deductions from hypotheses to conclusions.

More Proof Terminology
• Lemma - A minor theorem used as a steppingstone to proving a major theorem. • Corollary -...

Similar Documents

Premium Essay

Paper

...Module 7, Course 2, Session 18 Growth Strategies - External March 9, 2013 Discussion Topics Section Introductions and Segment Objectives The Acquisition Process – Target Research and Process Execution Case Studies Concluding Thoughts – Keys to Success Background Information on John Dickinson Page 2 3 7 15 17 CONFIDENTIAL 1 Introductions and Segment Objectives Introductions: • • John Dickinson – Session Leader Participants – Goals, objectives, questions, and backgrounds CONFIDENTIAL Session Objectives: • Why pursue external strategies? • • • Limited organic growth versus strategic goals New products / services, end markets, technologies, and customers, as well as sourcing synergies What are some of the strategies? • • • Joint ventures – private and public / governmental Acquisitions Divestitures and re-investing in the core business The Acquisition Process: • • Strategic goals and target identification / research Deal execution – the team, valuation / structure, and culture / integration 2 CONFIDENTIAL The Acquisition Process – Target Research and Deal Execution 3 Developing the Optimal Acquisition Process CONFIDENTIAL Optimizing the process begins with analyzing the unique dynamics of the Buyer’s situation, and then developing an appropriate and flexible strategy for achieving the Buyer’s goals. Process Considerations Current End Market Dynamics & Outlook Parent Company Objectives & Resource Commitment Management Team......

Words: 2731 - Pages: 11

Premium Essay

Progamming

...A FIRST COURSE IN PROBABILITY This page intentionally left blank A FIRST COURSE IN PROBABILITY Eighth Edition Sheldon Ross University of Southern California Upper Saddle River, New Jersey 07458 Library of Congress Cataloging-in-Publication Data Ross, Sheldon M. A first course in probability / Sheldon Ross. — 8th ed. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-13-603313-4 ISBN-10: 0-13-603313-X 1. Probabilities—Textbooks. I. Title. QA273.R83 2010 519.2—dc22 2008033720 Editor in Chief, Mathematics and Statistics: Deirdre Lynch Senior Project Editor: Rachel S. Reeve Assistant Editor: Christina Lepre Editorial Assistant: Dana Jones Project Manager: Robert S. Merenoff Associate Managing Editor: Bayani Mendoza de Leon Senior Managing Editor: Linda Mihatov Behrens Senior Operations Supervisor: Diane Peirano Marketing Assistant: Kathleen DeChavez Creative Director: Jayne Conte Art Director/Designer: Bruce Kenselaar AV Project Manager: Thomas Benfatti Compositor: Integra Software Services Pvt. Ltd, Pondicherry, India Cover Image Credit: Getty Images, Inc. © 2010, 2006, 2002, 1998, 1994, 1988, 1984, 1976 by Pearson Education, Inc., Pearson Prentice Hall Pearson Education, Inc. Upper Saddle River, NJ 07458 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Pearson Prentice Hall™ is a trademark of Pearson Education, Inc...

Words: 121193 - Pages: 485

Premium Essay

Police

...BELHAVEN UNIVERSITY Jackson, Mississippi A CHRISTIAN UNIVERSITY OF LIBERAL ARTS AND SCIENCES FOUNDED IN 1883 CATALOGUE 2014-2015 EFFECTIVE JUNE 1, 2014 Directory of Communication Mailing Address: Belhaven University 1500 Peachtree St. Jackson, MS 39202 Belhaven University 535 Chestnut St. Suite 100 Chattanooga, TN 37402 Belhaven University 7111 South Crest Parkway Southaven, MS 38671 Belhaven University – LeFleur 4780 I-55 North Suite 125 Jackson, MS 39211 Belhaven University 15115 Park Row Suite 175 Houston, TX 77084 Belhaven University Online 1500 Peachtree St. Box 279 Jackson, MS 39202 Belhaven University 1790 Kirby Parkway Suite 100 Memphis, TN 38138 Belhaven University 4151 Ashford Dunwoody Rd. Suite 130 Atlanta, GA 30319 Belhaven University 5200 Vineland Rd. Suite 100 Orlando, FL 32811 Traditional Admission Adult and Graduate Studies Admission – Jackson Atlanta Chattanooga Desoto Houston Memphis Orlando Alumni Relations/Development Belhaven Fax Business Office Campus Operations Integrated Marketing Registrar Student Life Security Student Financial Planning Student Development Online Admission Online Student Services (601) 968-5940 or (800) 960-5940 (601) 968-5988 or Fax (601) 352-7640 (404) 425-5590 or Fax (404) 425-5869 (423) 265-7784 or Fax (423) 265-2703 (622) 469-5387 (281) 579-9977 or Fax (281) 579-0275 (901) 896-0184 or Fax (901) 888-0771 (407) 804-1424 or Fax (407) 367-3333 (601) 968-5980 (601) 968-9998 (601) 968-5901 (601) 968-5904 (601) 968-5930 (601)......

Words: 151104 - Pages: 605

Premium Essay

Tpote

...01-Ellis-45685:01-Ellis-45685 7/16/2008 7:00 PM Page 1 Chapter 1 The Study of Personality Introduction Chapter Goals • Provide an overview of the controversies in the field of personality • Explain the purpose and utility of studying personality to mental health professionals • Review the various definitions of human personality • Offer insights into the history of personality theories • Introduce some of the methods used to measure or evaluate personality • Present some of the major personality theorists who have developed the concepts we will be studying S ubdisciplines of psychology such as social psychology, cognitive psychology, and indus- trial psychology endeavor to find common principles that will explain everyone’s behavior. These subfields have achieved considerable success in doing so, since we are all similar in many ways. Despite our similarities, however, there is little doubt that each human being is unique—different from every other individual on the planet. Seeking to understand human commonalities and seeking to account for individual differences are complementary, insofar as we cannot fully apprehend differences if we cannot identify our common characteristics. Personality psychology looks for answers to numerous questions. In what ways do human beings differ? In what situations and along what dimensions do they differ? Why do they differ? How much do they differ? How consistent are human differences? Can they......

Words: 11226 - Pages: 45

Free Essay

Torts Cases

...[G.R. No. 132266. December 21, 1999] CASTILEX INDUSTRIAL CORPORATION, petitioner, vs. VICENTE VASQUEZ, JR. and LUISA SO VASQUEZ, and CEBU DOCTORS HOSPITAL, INC., respondents. The pivotal issue in this petition is whether an employer may be held vicariously liable for the death resulting from the negligent operation by a managerial employee of a company-issued vehicle. The antecedents, as succinctly summarized by the Court of Appeals, are as follows: On 28 August 1988, at around 1:30 to 2:00 in the morning, Romeo So Vasquez, was driving a Honda motorcycle around Fuente Osmea Rotunda. He was traveling counter-clockwise, (the normal flow of traffic in a rotunda) but without any protective helmet or goggles. He was also only carrying a Students Permit to Drive at the time. Upon the other hand, Benjamin Abad [was a] manager of Appellant Castilex Industrial Corporation, registered owner [of] a Toyota Hi-Lux Pick-up with plate no. GBW-794. On the same date and time, Abad drove the said company car out of a parking lot but instead of going around the Osmea rotunda he made a short cut against [the] flow of the traffic in proceeding to his route to General Maxilom St. or to Belvic St. In the process, the motorcycle of Vasquez and the pick-up of Abad collided with each other causing severe injuries to the former. Abad stopped his vehicle and brought Vasquez to the Southern Islands Hospital and later to the Cebu Doctors Hospital. On September 5, 1988, Vasquez died at the Cebu......

Words: 82604 - Pages: 331

Premium Essay

Writing Papers

...First edition 2000 Second edition 2001 Third edition 2002 Fourth edition 2007 Published by EnglishforResearch.com The Whole World Company Press, Cambridge, CB7 5EQ, England © Stephen Howe and Kristina Henriksson 2000–2007 Printed by Biddles Limited, King’s Lynn, England The authors hereby assert their moral rights to be identified as the authors of the PhraseBook. You may not remove or alter the authors’ names, publisher’s name, copyright notice, disclaimers or, from the digital version, the End User Licence Agreement. All rights reserved worldwide Copyright is reserved in English and all other languages and countries of the world. PhraseBook for Writing, EnglishforResearch.com, EnglishforStudents.com and EnglishforSchool.com are worldwide trademarks and/or service marks of The Whole World Company Limited. Microsoft and Microsoft Word are trademarks or registered trademarks of Microsoft Corporation. All other trademarks and registered trademarks are the property of their respective owners and are hereby acknowledged. Do not make illegal, unauthorized copies of the PhraseBook. The PhraseBook and digital version are protected by copyright law and international treaties. The publisher and authors have striven to ensure the accuracy and correctness of the PhraseBook; however, they can accept no responsibility for any loss or inconvenience as a consequence of use, information or advice contained in the PhraseBook. PhraseBook versions ISBN 978-1-903384-02-2...

Words: 48522 - Pages: 195

Premium Essay

Human Resource Devlopment

...Managing Human Resources for Health in India A case study of Madhya Pradesh & Gujarat 2007 Central Bureau of Health Intelligence Directorate General of Health services Ministry of Health & Family Welfare Nirman Bhawan, New Delhi – 110011 Website: www.cbhidghs.nic.in, www.prodindia.nic.in Email: dircbhi.nb.nic In collaboration with WHO - India Country Office Managing Human Resources for Health in India A case study of Gujarat & Madhya Pradesh October 2007 Central Bureau of Health Intelligence, Dte.GHS, MoHFW, GOI In collaboration with World Health Organisation - India Country Office The Report on Managing Human Resources in India may be freely reviewed, abstracted, reproduced or translated, in part or whole, but is not for sale or for use in conjunction with commercial purposes. Any relevant enquiry may be addressed to the office of Director, Central Bureau of Health Intelligence, Directorate General of Health Services, 401, A-wing, Nirman Bhavan, New Delhi 110011(India). © Central Bureau of Health Intelligence, Dte. General of Health Services, Ministry of Health & Family Welfare, Govt. of India. 2007 Technical Support: Advent Healthcare Group 102, Tribhuvan Complex, Mathura Road, New Delhi 110065 (India) Tel: 91-11-65689631, 26312631 Email: info@adventhcg.com 2 Table of Contents Foreword ............................................................................................................. 5 Preface ........................................

Words: 26665 - Pages: 107

Free Essay

Art of Thinking Clearly by Rolf Dobelli

...CONTE NTS Introduction 1 WHY YOU SHOULD VISIT CEMETERIES: Survivorship Bias 2 DOES HARVARD MAKE YOU SMARTER?: Swimmer’s Body Illusion 3 WHY YOU SEE SHAPES IN THE CLOUDS: Clustering Illusion 4 IF 50 MILLION PEOPLE SAY SOMETHING FOOLISH, IT IS STILL FOOLISH: Social Proof 5 WHY YOU SHOULD FORGET THE PAST: Sunk Cost Fallacy 6 DON’T ACCEPT FREE DRINKS: Reciprocity 7 BEWARE THE ‘SPECIAL CASE’: Confirmation Bias (Part 1) 8 MURDER YOUR DARLINGS: Confirmation Bias (Part 2) 9 DON’T BOW TO AUTHORITY: Authority Bias 10 LEAVE YOUR SUPERMODEL FRIENDS AT HOME: Contrast Effect 11 WHY WE PREFER A WRONG MAP TO NO MAP AT ALL: Availability Bias 12 WHY ‘NO PAIN, NO GAIN’ SHOULD SET ALARM BELLS RINGING: The It’llGet-Worse-Before-It-Gets-Better Fallacy 13 EVEN TRUE STORIES ARE FAIRYTALES: Story Bias 14 WHY YOU SHOULD KEEP A DIARY: Hindsight Bias 15 WHY YOU SYSTEMATICALLY OVERESTIMATE YOUR KNOWLEDGE AND ABILITIES: Overconfidence Effect 16 DON’T TAKE NEWS ANCHORS SERIOUSLY: Chauffeur Knowledge 17 YOU CONTROL LESS THAN YOU THINK: Illusion of Control 18 NEVER PAY YOUR LAWYER BY THE HOUR: Incentive Super-Response Tendency 19 THE DUBIOUS EFFICACY OF DOCTORS, CONSULTANTS AND PSYCHOTHERAPISTS: Regression to Mean 20 NEVER JUDGE A DECISION BY ITS OUTCOME: Outcome Bias 21 LESS IS MORE: The Paradox of Choice 22 YOU LIKE ME, YOU REALLY REALLY LIKE ME: Liking Bias 23 DON’T CLING TO THINGS: Endowment Effect 24 THE INEVITABILITY OF UNLIKELY Events: Coincidence 25 THE CALAMITY OF CONFORMITY: Groupthink 26......

Words: 75018 - Pages: 301

Free Essay

English Vinglish

...Copyediting & Proofreading FOR DUMmIES by Suzanne Gilad ‰ Copyediting & Proofreading For Dummies® Published by Wiley Publishing, Inc. 111 River St. Hoboken, NJ 07030-5774 www.wiley.com Copyright © 2007 by Wiley Publishing, Inc., Indianapolis, Indiana Published by Wiley Publishing, Inc., Indianapolis, Indiana Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600. Requests to the Publisher for permission should be addressed to the Legal Department, Wiley Publishing, Inc., 10475 Crosspoint Blvd., Indianapolis, IN 46256, 317-572-3447, fax 317-572-4355, or online at http://www.wiley.com/go/permissions. Trademarks: Wiley, the Wiley Publishing logo, For Dummies, the Dummies Man logo, A Reference for the Rest of Us!, The Dummies Way, Dummies Daily, The Fun and Easy Way, Dummies.com, and related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc., and/or its affiliates in the United States and other countries, and may not be used without written permission. All...

Words: 125743 - Pages: 503

Premium Essay

Montaigne

...1575 ESSAYS by Michel de Montaigne translated by Charles Cotton I. OF CUSTOM, AND THAT WE SHOULD NOT EASILY CHANGE A LAW RECEIVED. HE seems to have had a right and true apprehension of the power of custom, who first invented the story of a countrywoman who, having accustomed herself to play with and carry, a young calf in her arms, and daily continuing to do so as it grew up, obtained this by custom, that, when grown to be a great ox, she was still able to bear it. For, in truth, custom is a violent and treacherous schoolmistress. She, by little and little, slily and unperceived, slips in the foot of her authority, but having by this gentle and humble beginning, with the benefit of time, fixed and established it, she then unmasks a furious and tyrannic countenance, against which we have no more the courage or the power so much as to lift up our eyes. We see her, at every turn, forcing and violating the rules of nature: "Usus efficacissimus rerum omnium magister." I refer to her Plato's cave in his Republic, and the physicians, who so often submit the reasons of their art to her authority; as the story of that king, who by custom brought his stomach to that pass, as to live by poison, and the maid that Albertus reports to have lived upon spiders. In that new world of the Indies, there...

Words: 179434 - Pages: 718

Premium Essay

Doc, Docx, Pdf, Wps, Rtf, Odt

...Dan Brown Deception Point Deception Point by Dan Brown Acknowledgments With warm thanks to Jason Kaufman for his superb guidance and insightful editorial skills; Blythe Brown for her tireless research and creative input; my good friend Jake Elwell at Wieser & Wieser; the National Security Archive; the NASA Public Affairs Office; Stan Planton, who continues to be a source for information on all things; the National Security Agency; glaciologist Martin O. Jeffries; and the superb minds of Brett Trotter, Thomas D. Nadeau, and Jim Barrington. Thanks also to Connie and Dick Brown, the U.S. Intelligence Policy Documentation Project, Suzanne O’Neill, Margie Wachtel, Morey Stettner, Owen King, Alison McKinnell, Mary and Stephen Gorman, Dr. Karl Singer, Dr. Michael I. Latz of Scripps Institute of Oceanography, April at Micron Electronics, Esther Sung, the National Air and Space Museum, Dr. Gene Allmendinger, the incomparable Heide Lange at Sanford J. Greenburger Associates, and John Pike at the Federation of American Scientists. Author’s Note The Delta Force, the National Reconnaissance Office, and the Space Frontier Foundation are real organizations. All technologies described in this novel exist. “If this discovery is confirmed, it will surely be one of the most stunning insights into our universe that science has ever uncovered. Its implications are as far-reaching and awe-inspiring as can be imagined. Even as it promises answers to some of our oldest questions, it poses still...

Words: 144469 - Pages: 578

Premium Essay

The Deception Point

...Acknowledgments With warm thanks to Jason Kaufman for his superb guidance and insightful editorial skills; Blythe Brown for her tireless research and creative input; my good friend Jake Elwell at Wieser & Wieser; the National Security Archive; the NASA Public Affairs Office; Stan Planton, who continues to be a source for information on all things; the National Security Agency; glaciologist Martin O. Jeffries; and the superb minds of Brett Trotter, Thomas D. Nadeau, and Jim Barrington. Thanks also to Connie and Dick Brown, the U.S. Intelligence Policy Documentation Project, Suzanne O'Neill, Margie Wachtel, Morey Stettner, Owen King, Alison McKinnell, Mary and Stephen Gorman, Dr. Karl Singer, Dr. Michael I. Latz of Scripps Institute of Oceanography, April at Micron Electronics, Esther Sung, the National Air and Space Museum, Dr. Gene Allmendinger, the incomparable Heide Lange at Sanford J. Greenburger Associates, and John Pike at the Federation of American Scientists.   Author's Note The Delta Force, the National Reconnaissance Office, and the Space Frontier Foundation are real organizations. All technologies described in this novel exist.   If this discovery is confirmed, it will surely be one of the most stunning insights into our universe that science has ever uncovered. Its implications are as far-reaching and awe-inspiring as can be imagined. Even as it promises answers to some of our oldest questions, it poses still others even more......

Words: 144424 - Pages: 578

Premium Essay

English

...|Word |Synonym |Sentence | |Abase |to humiliate, degrade, debase, corrupt, |Don’t abase yourself by accepting this offer. | | |disgrace | | |Abash |to bewilder, confound, confuse, puzzle, amaze,|That poor man felt abashed in the company of rich man & women. He was not at | |[uh-bash] |stun, leave speechless, bamboozle. |all abashed by his open admiration (প্রশংসা). | |Abate |to remove, lessen |We resumed our journey when the rain abated. Rather than leaving immediately, | | | |they waited for the storm to abate. | |Abdicate |to forsake, give up, (পরিত্যাগ করা, অস্বীকার |Edwar VIII abdicated the British throne to marry the woman he loved. Saddam was| | |করা), resign, relinquish, hand over, step down|abdicated from his throne. | | |from, abandon | ......

Words: 27003 - Pages: 109

Premium Essay

Educationa Research

...1 EDUCATIONAL RESEARCH Unit Structure 1.0 Objectives 1.1 Introduction 1.2 Sources Acquiring Knowledge 1.3 Meaning, Steps and Scope of Educational Research 1.4 Scientific Method, aims and characteristics of research as a scientific activity 1.5 Ethical considerations in Educational Research 1.6 Paradigms of Educational research 1.7 Types of Research 1.7.a Fundamental 1.7.b Applied Research 1.7.c. Action Research 1.0 OBJECTIVES : After reading this unit, you will be able to: 1 To explain the concept of Educational Research 2 To describe the scope of Educational Research 3 To state the purpose of Educational Research 4 To explain what is scientific enquiry. 5 To explain importance of theory development. 6 To explain relationship among science, education and educational research. 7 To Identity fundamental research 8 To Identity applied research 9 To Identify action research 10 To Differentiate between fundamental, applied, and action research 11 To Identify different paradigms of research 2 1.1 INTRODUCTION : Research purifies human life. It improves its quality. It is search for knowledge. If shows how to Solve any problem scientifically. It is a careful enquiry through search for any kind of Knowledge. It is a journey from known to unknown. It is a systematic effort to gain new knowledge in any kind of discipline. When it Seeks a solution of any educational problem it leads to educational...

Words: 78659 - Pages: 315

Premium Essay

Saczxdsfadsfdsf

...A ∑ This eBook is provided by www.PlentyofeBooks.net E= mc 2 Plenty of eBooks is a blog with an aim of helping people, especially students, who cannot afford to buy some costly books from the market. For more Free eBooks and educational material visit www.PlentyofeBooks.net Uploaded By Bhavesh Pamecha (samsexy98) 1 Handwriting Analysis The CompZete Basic Book NEW PAGE BOOKS A division of The Career Press, Inc. Franklin Lakes, NJ Copyright 0 1980 by Karen Amend and Mary S . Ruiz All rights reserved under the Pan-American and International Copyright Conventions. This book may not be reproduced, in whole or in part, in any form or by any means electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system now known or hereafter invented, without written permission from the publisher, The Career Press. HANDWRITING ANALYSIS ISBN 0-87877-050-X Printed in the U.S.A. by Book-mart Press To order this title, please call toll-fiee I-800-CAREER-1 (NJ and Canada: 201-848-0310) to order using VISA or MasterCard, or for fbrther information on books fiom Career Press. The Career Press, Inc., 3 Tice Road, PO Box 687, Franklin Lakes, NJ 07417 The author of this book does not dispense medical advice nor prescribe the use of any technique as a form of treatment for medical problems without the advice of a physician, either directly or indirectly. The intent of the author is only to offer information.of a......

Words: 44082 - Pages: 177