Practice Test III, Math 314, Spring 2016
|
|
- Bruno Barnett
- 5 years ago
- Views:
Transcription
1 Practice Test III, Math 314, Spring 2016 Dr. Holmes April 26, 2016 This is the 2014 test reorganized to be more readable. I like it as a review test. The students who took this test had to do four sections of their choice (doing both problems in each section, with the usual grading scheme). Good performance on a section repeated from earlier exams could improve the grade on the earlier exam as well. They could do more sections if they wished, with best work counting. Your test will probably be shorter, but again, with more than four sections, some being review, and you having the choice of which four to do. 1
2 1 Logic Questions A set of logic strategies and rules is attached. 1. Give a proof of ((P Q) (Q R)) (P R) in the style taught in class. 2
3 2. State and prove one of de Morgan s laws in the style taught in class. 3
4 2 Formal arithmetic questions Some axioms and theorems of formal arithmetic are attached. 1. Do one of the following relatively easy formal arithmetic proofs. If you do both your best work will count. (a) Prove S(x) + y = S(x + y) using the axioms alone. (b) Prove 1x = x using the axioms alone (well, you may also use the definition of 1 and the theorem S(x) = x + 1). 4
5 2. Prove the right distributive law of multiplication using the axioms alone. You may if you wish choose one of the other main proofs in the formal arithmetic homework assignment (whose text is attached). If you choose to do one of the others, you are allowed to use the theorems appearing before it in the list in your proof. 5
6 3 General algebra questions 1. Do one of the following three (I hope relatively easy) problems. If you do more than one your best work will count. A list of axioms from Spivak is attached. (a) Prove the lemma If x+y = 0 then y = x from Spivak s axioms (P1)-(P9) (which will be given). Then prove a( b) = (ab). Your proofs should show full justifications from P1-9. You may use the theorem x0 = 0 without proving it. [This is not the same as this part of the original Test I, but similar in difficulty and very similar in approach]. (b) Prove that for any real numbers x, y if xy = 1 then y = x 1. Your proof should show full justifications from P1-9. Remember that you can only write x 1 if you know that x 0. You can use the theorem x0 = 0 without proving it. (c) Prove using either Spivak s order axioms (P10)-(P12) and algebra skills, or his alternative set of axioms and algebra skills, that if 0 < a < b it follows that a 2 < b 2 (yes, you know the definition of a 2 as aa). You should show justifications mentioning the order axioms (from whichever of the two sets you use); you do not need to show justifications from P1-9 (you can use algebra without comment). 6
7 2. Prove using familiar properties of even and odd numbers and your algebra skills that the square root of 2 is not a rational number. There is no need to mention any Spivak axioms by name in this proof. 7
8 4 Limit proofs Limit proofs are to include scratch work. Where you put it is up to you, as long as I can tell what is scratch work and what is proof. 1. Prove that lim x 2 x 2 = 4 in complete detail (yes, this is harder. You should be up to it). This question should demonstrate your acquaintance with the exact meaning of limit notation: in this or any limit proof, start by expanding out the statement using the definition. 8
9 2. State and prove a limit theorem, your choice from the constant multiple property of limits (computing lim x a cf(x)), the addition property of limits or the subtraction property of limits. 9
10 5 Chapter 5 Limit Questions 1. Explain why lim f(x) lim g(x) = lim f(x) g(x) x a x a x a is not a good way to state the multiplication property of limits. State it in a better way. Write an example with specific functions and numbers where this equation has one side defined and one side undefined. 10
11 2. Define D(x) = 0 if x is irrational and 1 if x is rational. Prove that lim x a D(x) does not exist for any number a. 11
12 3. Prove that lim x 0 1 x does not exist. 12
13 6 Continuity questions 1. Use the Intermediate Value Theorem (along with general algebra and simple calculus knowledge) to prove that there is a cube root of 5. Hint: you want to choose a suitable closed interval and a suitable function. Your proof should make it clear that you know the full statement of some version of the IVT. 13
14 2. Prove that if f is continuous at a and f(a) > 0, then f is bounded above and bounded below on some open interval (a δ, a + δ) You cannot use any Big Theorems: this is just an ɛ δ proof from the definitions of continuity and limit. You should also make it clear that you know what bounded above and bounded below mean. 14
15 3. A set D is dense iff every open interval contains an element of D. Suppose that a function f is continuous at every number a and further that f(x) = 0 for each x D, where D is a dense set. Prove that f(x) = 0 for every x. 15
16 7 Least upper bound questions 1. Determine the greatest lower bound and least upper bound of each of the following sets, if any. Explain in each case whether each of the bounds is actually an element of the set. Give brief explanations of your answers. (a) The set of fractions n 1 for n a positive integer. This set is not n unbounded in either direction! (b) The set of negative even integers (c) The set { 1 n + ( 1)n : n N}. This is an example from Spivak; I think the best approach is to list some values and see how they act; I definitely expect an explanation of each of the two answers. Zero is not a natural number. 16
17 2. Prove that for every positive real ɛ > 0 there is a natural number n such that 1 < ɛ. You can prove this by first proving Spivak s theorem n that for every real number x there is a natural number n greater than x (using P13) or you could consider the greatest lower bound of the set { 1 : n N}: first show that it exists (you may assume and use the n theorem about greatest lower bounds, but you do need to show that you know what it says) and then show that it cannot be greater than zero. 17
18 3. Suppose that A and B are nonempty sets and every element of A is less than every element of B. First, show that the least upper bound of A, sup A is less than or equal to any y B. Then show that the least upper bound of A (sup(a)) is less than or equal to the greatest lower bound of B (sup(b)). This is problem 12 from the chapter 8 homework, which we went over in class. It is really an exercise in understanding the definitions of least upper bound and greatest lower bound. 18
19 8 Sequence questions 1. Prove the addition property for limits of sequences (hint: the letter δ should not appear in your work at all!) 19
20 2. Prove the Monotone Convergence Theorem: a sequence which is nondecreasing and bounded above must converge. Your presentation should include definitions of what it means for a sequence to be nondecreasing and what it means for a sequence to be bounded above. It will use the Least Upper Bound axiom. 20
21 3. A Cauchy sequence is a sequence {a n } such that for each ɛ > 0 there is N such that for any m, n if m, n > N then a m a n < ɛ. Prove that any sequence with a limit is Cauchy. 21
22 9 Propositional Logic Style Sheet Not all of these are relevant to the assigned proofs; you need to recognize which rules and strategies are appropriate. Conjunction (and): To prove A B, prove A (part 1), then prove B (part 2). The parts are not cases, and you will lose credit if you call something a proof by cases which isn t one. From A B, deduce A. From A B, deduce B. You need to explicitly break apart assumptions or lemmas which are and statements to use their components; you will lose credit if you don t. You are permitted to break apart statements by numbering the component statements directly without copying them over. Disjunction (or): To prove A B, assume A and adopt the new goal B [or assume B and adopt the new goal A; you do not need to do both]. From A, deduce A B. From B, deduce A B (rule of weakening). From A B and A, deduce B. From A B and B, deduce A. This is disjunctive syllogism. To deduce a conclusion C from an assumption or lemma A B, use proof by cases: in the first part (case 1) assume A and prove C; in the second part (case 2) assume B and prove C. Implication (if): To prove A B, assume A and adopt the new goal B. alternative strategy: to prove A B, assume B and adopt the new goal A. Given A and A B, deduce B (modus ponens). Given B and A B, deduce A (modus tollens). If you have an assumption A B you may want to try proving A (so that you can further conclude B). Negation (not): To prove A, assume A and try to prove (contradiction). From A deduce A (double negation). To prove any statement A, assume A and try to prove. This is proof by contradiction. 22
23 From A and A, deduce. If you have a negative assumption A, the commonest way to use it is to wait until your goal is a contradiction, then try to prove A to get the contradiction. From deduce anything! 23
24 10 Axioms and Theorems of Formal Arithmetic Here are the axioms of formal arithmetic is a natural number (in symbols, 0 N). 2. If x and y are natural numbers, so are S(x), x + y, and x y. ( xy N.S(x) N x + y N x y N) is not a successor. ( x.s(x) 0). Here we understand that x y abbreviates x = y. Here and in the following axioms we write our quantifiers unrestricted: we could write ( x N.S(x) 0) instead, but in this context we are only talking about natural numbers, so we can leave the restriction on our quantifiers implicit. 4. Numbers with the same successor are the same. ( xy.s(x) = S(y) x = y). 5. Let P(x) be any sentence about a natural number variable x. We assert P (0) ( y.p (y) P (S(y))) ( x.p (x)). This is a symbolic presentation of the familiar principle of mathematical induction. From an extremely technical standpoint, this is an infinite collection of axioms, one for each sentence P (x). If we are also willing to talk about sets of natural numbers, we can state it as a single axiom: ( A P(N).0 A ( y N.y A S(y) A) A = N). We will not use the set formulation now but we might use it later. P(N) is a notation for the collection of all sets of natural numbers. 6. ( x.x + 0 = x) 7. ( xy.x + S(y) = S(x + y)) 8. ( x.x 0 = 0) 9. ( x.x S(y) = x y + x) Here we assume the usual order of operations. Here are some theorems and a definition. Not all of these are necessarily of any use. definition of 1: 1 is defined as S(0). 24
25 Theorem 1: ( x.x + 1 = S(x)) Theorem 2: ( x.x = 0 ( y.s(y) = x)) Instructions: You may if you wish write any of the following proofs to meet requirements for the second problem in section 2. The commutative laws are not recommended. You may write a second one of these if you wish, to replace any other question on the test but the last one (you cannot escape the major limit theorem). You are only allowed to use one of these theorems without proving it if you are proving one that appears later on this list. commutativity of addition: ( xy.x + y = y + x) (not recommended for you to write this proof) right distributivity of multiplication over addition: ( xyz.x(y + z) = xy + xz) associativity of addition: ( xyz.(x + y) + z = x + (y + z)) left distributivity of multiplication over addition: ( xyz.(x + y)z = xz + yz) associativity of multiplication: ( xyz.(xy)z = x(yz)) commutativity of multiplication: ( xy.xy = yx) (I do not recommend that you write this proof). 25
26 11 Axioms from Spivak algebra axioms (P1): For any numbers x, y, z, x+(y +z) = (x+y)+z. [associative property of addition] (P2): For any number x, x + 0 = 0 + x = x [identity property of addition] (P3): For any number x, x + ( x) = ( x) + x = 0 [inverse property of addition] (P4): For any numbers x, y, x+y = y+x [commutative property of addition] (P5): For any numbers x, y, z, x(yz) = (xy)z. [associative property of multiplication] (P6): For any number x, x1 = 1x = x. 1 0 [identity property of multiplication] (P7): For any number x 0, xx 1 = x 1 x = 1 [inverse property of multiplication] (P8): For any numbers x, y, xy = yx [commutative property of multiplication] (P9): For any numbers x, y, z, x(y + z) = xy + xz [distributive property of multiplication over addition] order axioms P10 For each number x, exactly one of the following is true: x P ; x = 0; x P. P11: For any numbers x,y, if x P and y P, then x + y P. P11: For any numbers x,y, if x P and y P, then xy P. alternative order axioms P10 For any numbers x, y, exactly one of the following is true: x < y;x = y;x > y 26
27 P11 For any numbers x, y, z, if x < y and y < z, then x < z. P12 For any numbers x, y, z, if x < y then x + z < y + z. P13 For any numbers x, y, z, if x < y and z > 0, then xz < yz. 27
Logic Overview, I. and T T T T F F F T F F F F
Logic Overview, I DEFINITIONS A statement (proposition) is a declarative sentence that can be assigned a truth value T or F, but not both. Statements are denoted by letters p, q, r, s,... The 5 basic logical
More informationMath 3336: Discrete Mathematics Practice Problems for Exam I
Math 3336: Discrete Mathematics Practice Problems for Exam I The upcoming exam on Tuesday, February 26, will cover the material in Chapter 1 and Chapter 2*. You will be provided with a sheet containing
More informationFoundations of Mathematics MATH 220 FALL 2017 Lecture Notes
Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements
More informationManual of Logical Style
Manual of Logical Style Dr. Holmes January 9, 2015 Contents 1 Introduction 2 2 Conjunction 3 2.1 Proving a conjunction...................... 3 2.2 Using a conjunction........................ 3 3 Implication
More informationManual of Logical Style (fresh version 2018)
Manual of Logical Style (fresh version 2018) Randall Holmes 9/5/2018 1 Introduction This is a fresh version of a document I have been working on with my classes at various levels for years. The idea that
More informationChapter 1 Elementary Logic
2017-2018 Chapter 1 Elementary Logic The study of logic is the study of the principles and methods used in distinguishing valid arguments from those that are not valid. The aim of this chapter is to help
More informationPacket #2: Set Theory & Predicate Calculus. Applied Discrete Mathematics
CSC 224/226 Notes Packet #2: Set Theory & Predicate Calculus Barnes Packet #2: Set Theory & Predicate Calculus Applied Discrete Mathematics Table of Contents Full Adder Information Page 1 Predicate Calculus
More informationProof Worksheet 2, Math 187 Fall 2017 (with solutions)
Proof Worksheet 2, Math 187 Fall 2017 (with solutions) Dr. Holmes October 17, 2017 The instructions are the same as on the first worksheet, except you can use all the rules in the strategies handout. We
More information1. For each statement, either state that it is True or else Give a Counterexample: (a) If a < b and c < d then a c < b d.
Name: Instructions. Show all work in the space provided. Indicate clearly if you continue on the back side, and write your name at the top of the scratch sheet if you will turn it in for grading. No books
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics. Introductory Notes in Discrete Mathematics Solution Guide
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics Introductory Notes in Discrete Mathematics Solution Guide Marcel B. Finan c All Rights Reserved 2015 Edition Contents
More informationHandout on Logic, Axiomatic Methods, and Proofs MATH Spring David C. Royster UNC Charlotte
Handout on Logic, Axiomatic Methods, and Proofs MATH 3181 001 Spring 1999 David C. Royster UNC Charlotte January 18, 1999 Chapter 1 Logic and the Axiomatic Method 1.1 Introduction Mathematicians use a
More informationTHE REAL NUMBERS Chapter #4
FOUNDATIONS OF ANALYSIS FALL 2008 TRUE/FALSE QUESTIONS THE REAL NUMBERS Chapter #4 (1) Every element in a field has a multiplicative inverse. (2) In a field the additive inverse of 1 is 0. (3) In a field
More informationMath 10850, fall 2017, University of Notre Dame
Math 10850, fall 2017, University of Notre Dame Notes on first exam September 22, 2017 The key facts The first midterm will be on Thursday, September 28, 6.15pm-7.45pm in Hayes-Healy 127. What you need
More informationProofs. Chapter 2 P P Q Q
Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,
More informationSequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.
Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence
More informationCOMP 182 Algorithmic Thinking. Proofs. Luay Nakhleh Computer Science Rice University
COMP 182 Algorithmic Thinking Proofs Luay Nakhleh Computer Science Rice University 1 Reading Material Chapter 1, Section 3, 6, 7, 8 Propositional Equivalences The compound propositions p and q are called
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More informationSupplementary Logic Notes CSE 321 Winter 2009
1 Propositional Logic Supplementary Logic Notes CSE 321 Winter 2009 1.1 More efficient truth table methods The method of using truth tables to prove facts about propositional formulas can be a very tedious
More informationProofs. Chapter 2 P P Q Q
Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,
More informationPacket #1: Logic & Proofs. Applied Discrete Mathematics
Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 3-13 Course Objectives At the conclusion of this course, you should
More informationAdvanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010
Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010 1. Real Number System 1.1. Introduction. Numbers are at the heart of mathematics. By now you must be fairly familiar with
More information0.Axioms for the Integers 1
0.Axioms for the Integers 1 Number theory is the study of the arithmetical properties of the integers. You have been doing arithmetic with integers since you were a young child, but these mathematical
More informationLogic. Quantifiers. (real numbers understood). x [x is rotten in Denmark]. x<x+x 2 +1
Logic One reason for studying logic is that we need a better notation than ordinary English for expressing relationships among various assertions or hypothetical states of affairs. A solid grounding in
More informationmeans is a subset of. So we say A B for sets A and B if x A we have x B holds. BY CONTRAST, a S means that a is a member of S.
1 Notation For those unfamiliar, we have := means equal by definition, N := {0, 1,... } or {1, 2,... } depending on context. (i.e. N is the set or collection of counting numbers.) In addition, means for
More informationProofs: A General How To II. Rules of Inference. Rules of Inference Modus Ponens. Rules of Inference Addition. Rules of Inference Conjunction
Introduction I Proofs Computer Science & Engineering 235 Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu A proof is a proof. What kind of a proof? It s a proof. A proof is a proof. And when
More informationConjunction: p q is true if both p, q are true, and false if at least one of p, q is false. The truth table for conjunction is as follows.
Chapter 1 Logic 1.1 Introduction and Definitions Definitions. A sentence (statement, proposition) is an utterance (that is, a string of characters) which is either true (T) or false (F). A predicate is
More informationLECTURE NOTES DISCRETE MATHEMATICS. Eusebius Doedel
LECTURE NOTES on DISCRETE MATHEMATICS Eusebius Doedel 1 LOGIC Introduction. First we introduce some basic concepts needed in our discussion of logic. These will be covered in more detail later. A set is
More informationFirst order Logic ( Predicate Logic) and Methods of Proof
First order Logic ( Predicate Logic) and Methods of Proof 1 Outline Introduction Terminology: Propositional functions; arguments; arity; universe of discourse Quantifiers Definition; using, mixing, negating
More informationTools for reasoning: Logic. Ch. 1: Introduction to Propositional Logic Truth values, truth tables Boolean logic: Implications:
Tools for reasoning: Logic Ch. 1: Introduction to Propositional Logic Truth values, truth tables Boolean logic: Implications: 1 Why study propositional logic? A formal mathematical language for precise
More informationAxiomatic systems. Revisiting the rules of inference. Example: A theorem and its proof in an abstract axiomatic system:
Axiomatic systems Revisiting the rules of inference Material for this section references College Geometry: A Discovery Approach, 2/e, David C. Kay, Addison Wesley, 2001. In particular, see section 2.1,
More informationUnderstanding Decimal Addition
2 Understanding Decimal Addition 2.1 Experience Versus Understanding This book is about understanding system architecture in a quick and clean way: no black art, nothing you can only get a feeling for
More informationProofs. Joe Patten August 10, 2018
Proofs Joe Patten August 10, 2018 1 Statements and Open Sentences 1.1 Statements A statement is a declarative sentence or assertion that is either true or false. They are often labelled with a capital
More informationSeminaar Abstrakte Wiskunde Seminar in Abstract Mathematics Lecture notes in progress (27 March 2010)
http://math.sun.ac.za/amsc/sam Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics 2009-2010 Lecture notes in progress (27 March 2010) Contents 2009 Semester I: Elements 5 1. Cartesian product
More informationMATH 341, Section 001 FALL 2014 Introduction to the Language and Practice of Mathematics
MATH 341, Section 001 FALL 2014 Introduction to the Language and Practice of Mathematics Class Meetings: MW 9:30-10:45 am in EMS E424A, September 3 to December 10 [Thanksgiving break November 26 30; final
More information4.1 Real-valued functions of a real variable
Chapter 4 Functions When introducing relations from a set A to a set B we drew an analogy with co-ordinates in the x-y plane. Instead of coming from R, the first component of an ordered pair comes from
More informationMTH 299 In Class and Recitation Problems SUMMER 2016
MTH 299 In Class and Recitation Problems SUMMER 2016 Last updated on: May 13, 2016 MTH299 - Examples CONTENTS Contents 1 Week 1 3 1.1 In Class Problems.......................................... 3 1.2 Recitation
More informationReadings: Conjecture. Theorem. Rosen Section 1.5
Readings: Conjecture Theorem Lemma Lemma Step 1 Step 2 Step 3 : Step n-1 Step n a rule of inference an axiom a rule of inference Rosen Section 1.5 Provide justification of the steps used to show that a
More informationIntroduction to Metalogic
Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)
More informationSection 1.2: Propositional Logic
Section 1.2: Propositional Logic January 17, 2017 Abstract Now we re going to use the tools of formal logic to reach logical conclusions ( prove theorems ) based on wffs formed by some given statements.
More informationProof. Theorems. Theorems. Example. Example. Example. Part 4. The Big Bang Theory
Proof Theorems Part 4 The Big Bang Theory Theorems A theorem is a statement we intend to prove using existing known facts (called axioms or lemmas) Used extensively in all mathematical proofs which should
More informationLecture 4: Constructing the Integers, Rationals and Reals
Math/CS 20: Intro. to Math Professor: Padraic Bartlett Lecture 4: Constructing the Integers, Rationals and Reals Week 5 UCSB 204 The Integers Normally, using the natural numbers, you can easily define
More informationMathematics 114L Spring 2018 D.A. Martin. Mathematical Logic
Mathematics 114L Spring 2018 D.A. Martin Mathematical Logic 1 First-Order Languages. Symbols. All first-order languages we consider will have the following symbols: (i) variables v 1, v 2, v 3,... ; (ii)
More informationMat 243 Exam 1 Review
OBJECTIVES (Review problems: on next page) 1.1 Distinguish between propositions and non-propositions. Know the truth tables (i.e., the definitions) of the logical operators,,,, and Write truth tables for
More informationLECTURE NOTES DISCRETE MATHEMATICS. Eusebius Doedel
LECTURE NOTES on DISCRETE MATHEMATICS Eusebius Doedel 1 LOGIC Introduction. First we introduce some basic concepts needed in our discussion of logic. These will be covered in more detail later. A set is
More informationLecture Notes on DISCRETE MATHEMATICS. Eusebius Doedel
Lecture Notes on DISCRETE MATHEMATICS Eusebius Doedel c Eusebius J. Doedel, 009 Contents Logic. Introduction............................................................................... Basic logical
More informationA Guide to Proof-Writing
A Guide to Proof-Writing 437 A Guide to Proof-Writing by Ron Morash, University of Michigan Dearborn Toward the end of Section 1.5, the text states that there is no algorithm for proving theorems.... Such
More informationMultivariable Calculus and Matrix Algebra-Summer 2017
Multivariable Calculus and Matrix Algebra-Summer 017 Homework 4 Solutions Note that the solutions below are for the latest version of the problems posted. For those of you who worked on an earlier version
More informationOutline. We will now investigate the structure of this important set.
The Reals Outline As we have seen, the set of real numbers, R, has cardinality c. This doesn't tell us very much about the reals, since there are many sets with this cardinality and cardinality doesn't
More information3 The language of proof
3 The language of proof After working through this section, you should be able to: (a) understand what is asserted by various types of mathematical statements, in particular implications and equivalences;
More informationRecitation 7: Existence Proofs and Mathematical Induction
Math 299 Recitation 7: Existence Proofs and Mathematical Induction Existence proofs: To prove a statement of the form x S, P (x), we give either a constructive or a non-contructive proof. In a constructive
More informationMAT 243 Test 1 SOLUTIONS, FORM A
t MAT 243 Test 1 SOLUTIONS, FORM A 1. [10 points] Rewrite the statement below in positive form (i.e., so that all negation symbols immediately precede a predicate). ( x IR)( y IR)((T (x, y) Q(x, y)) R(x,
More informationWhy write proofs? Why not just test and repeat enough examples to confirm a theory?
P R E F A C E T O T H E S T U D E N T Welcome to the study of mathematical reasoning. The authors know that many students approach this material with some apprehension and uncertainty. Some students feel
More informationResearch Methods in Mathematics Extended assignment 1
Research Methods in Mathematics Extended assignment 1 T. PERUTZ Due: at the beginning of class, October 5. Choose one of the following assignments: (1) The circle problem (2) Complete ordered fields 2
More information1.1 Language and Logic
c Oksana Shatalov, Spring 2018 1 1.1 Language and Logic Mathematical Statements DEFINITION 1. A proposition is any declarative sentence (i.e. it has both a subject and a verb) that is either true or false,
More informationMATH 2001 MIDTERM EXAM 1 SOLUTION
MATH 2001 MIDTERM EXAM 1 SOLUTION FALL 2015 - MOON Do not abbreviate your answer. Write everything in full sentences. Except calculators, any electronic devices including laptops and cell phones are not
More informationProofs. Introduction II. Notes. Notes. Notes. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Fall 2007
Proofs Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Fall 2007 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 1.5, 1.6, and 1.7 of Rosen cse235@cse.unl.edu
More informationLogic, Sets, and Proofs
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Operators. A logical statement is a mathematical statement that can be assigned a value either true or false.
More informationCITS2211 Discrete Structures Proofs
CITS2211 Discrete Structures Proofs Unit coordinator: Rachel Cardell-Oliver August 13, 2017 Highlights 1 Arguments vs Proofs. 2 Proof strategies 3 Famous proofs Reading Chapter 1: What is a proof? Mathematics
More informationIntro to Logic and Proofs
Intro to Logic and Proofs Propositions A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both. Examples: It is raining today. Washington
More informationMath 414, Fall 2016, Test I
Math 414, Fall 2016, Test I Dr. Holmes September 23, 2016 The test begins at 10:30 am and ends officially at 11:45 am: what will actually happen at 11:45 is that I will give a five minute warning. The
More informationDo not start until you are given the green signal
SOLUTIONS CSE 311 Winter 2011: Midterm Exam (closed book, closed notes except for 1-page summary) Total: 100 points, 5 questions. Time: 50 minutes Instructions: 1. Write your name and student ID on the
More informationThe least element is 0000, the greatest element is 1111.
Note: this worksheet has been modified to emphasize the Boolean algebra content. Some problems have been deleted.; this, for instance, is why the first problem is #5 rather than #1. 5. Let A be the set
More informationIn Exercises 1 12, list the all of the elements of the given set. 2. The set of all positive integers whose square roots are less than or equal to 3
APPENDIX A EXERCISES In Exercises 1 12, list the all of the elements of the given set. 1. The set of all prime numbers less than 20 2. The set of all positive integers whose square roots are less than
More informationMATH10040: Numbers and Functions Homework 1: Solutions
MATH10040: Numbers and Functions Homework 1: Solutions 1. Prove that a Z and if 3 divides into a then 3 divides a. Solution: The statement to be proved is equivalent to the statement: For any a N, if 3
More informationStructure of R. Chapter Algebraic and Order Properties of R
Chapter Structure of R We will re-assemble calculus by first making assumptions about the real numbers. All subsequent results will be rigorously derived from these assumptions. Most of the assumptions
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter The Real Numbers.. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {, 2, 3, }. In N we can do addition, but in order to do subtraction we need to extend
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter 1 The Real Numbers 1.1. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {1, 2, 3, }. In N we can do addition, but in order to do subtraction we need
More informationSome Applications of MATH 1090 Techniques
1 Some Applications of MATH 1090 Techniques 0.1 A simple problem about Nand gates in relation to And gates and inversion (Not) Gates. In this subsection the notation A means that A is proved from the axioms
More informationA lower bound for X is an element z F such that
Math 316, Intro to Analysis Completeness. Definition 1 (Upper bounds). Let F be an ordered field. For a subset X F an upper bound for X is an element y F such that A lower bound for X is an element z F
More informationCSE 1400 Applied Discrete Mathematics Proofs
CSE 1400 Applied Discrete Mathematics Proofs Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Axioms 1 Logical Axioms 2 Models 2 Number Theory 3 Graph Theory 4 Set Theory 4
More informationBefore you get started, make sure you ve read Chapter 1, which sets the tone for the work we will begin doing here.
Chapter 2 Mathematics and Logic Before you get started, make sure you ve read Chapter 1, which sets the tone for the work we will begin doing here. 2.1 A Taste of Number Theory In this section, we will
More informationWriting proofs for MATH 61CM, 61DM Week 1: basic logic, proof by contradiction, proof by induction
Writing proofs for MATH 61CM, 61DM Week 1: basic logic, proof by contradiction, proof by induction written by Sarah Peluse, revised by Evangelie Zachos and Lisa Sauermann September 27, 2016 1 Introduction
More informationBasic properties of the Integers
Basic properties of the Integers Branko Ćurgus May 2, 2017 1 Axioms for the Integers In the axioms below we use the standard logical operators: conjunction, disjunction, exclusive disjunction, implication,
More informationCMPSCI 250: Introduction to Computation. Lecture 11: Proof Techniques David Mix Barrington 5 March 2013
CMPSCI 250: Introduction to Computation Lecture 11: Proof Techniques David Mix Barrington 5 March 2013 Proof Techniques Review: The General Setting for Proofs Types of Proof: Direct, Contraposition, Contradiction
More informationFor all For every For each For any There exists at least one There exists There is Some
Section 1.3 Predicates and Quantifiers Assume universe of discourse is all the people who are participating in this course. Also let us assume that we know each person in the course. Consider the following
More information1. Is the set {f a,b (x) = ax + b a Q and b Q} of all linear functions with rational coefficients countable or uncountable?
Name: Instructions. Show all work in the space provided. Indicate clearly if you continue on the back side, and write your name at the top of the scratch sheet if you will turn it in for grading. No books
More informationProof strategies, or, a manual of logical style
Proof strategies, or, a manual of logical style Dr Holmes September 27, 2017 This is yet another version of the manual of logical style I have been working on for many years This semester, instead of posting
More informationAxioms for the Real Number System
Axioms for the Real Number System Math 361 Fall 2003 Page 1 of 9 The Real Number System The real number system consists of four parts: 1. A set (R). We will call the elements of this set real numbers,
More informationHerbrand Theorem, Equality, and Compactness
CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Herbrand Theorem, Equality, and Compactness The Herbrand Theorem We now consider a complete method for proving the unsatisfiability of sets of first-order
More informationBoolean Algebra and Proof. Notes. Proving Propositions. Propositional Equivalences. Notes. Notes. Notes. Notes. March 5, 2012
March 5, 2012 Webwork Homework. The handout on Logic is Chapter 4 from Mary Attenborough s book Mathematics for Electrical Engineering and Computing. Proving Propositions We combine basic propositions
More informationPROBLEM SET 3: PROOF TECHNIQUES
PROBLEM SET 3: PROOF TECHNIQUES CS 198-087: INTRODUCTION TO MATHEMATICAL THINKING UC BERKELEY EECS FALL 2018 This homework is due on Monday, September 24th, at 6:30PM, on Gradescope. As usual, this homework
More informationAdvanced Calculus: MATH 410 Real Numbers Professor David Levermore 1 November 2017
Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 1 November 2017 1. Real Number System 1.1. Introduction. Numbers are at the heart of mathematics. By now you must be fairly familiar with
More informationDiscrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 016 Seshia and Walrand Note 1 Proofs In science, evidence is accumulated through experiments to assert the validity of a statement. Mathematics, in
More informationSolutions for Homework Assignment 2
Solutions for Homework Assignment 2 Problem 1. If a,b R, then a+b a + b. This fact is called the Triangle Inequality. By using the Triangle Inequality, prove that a b a b for all a,b R. Solution. To prove
More information2/2/2018. CS 103 Discrete Structures. Chapter 1. Propositional Logic. Chapter 1.1. Propositional Logic
CS 103 Discrete Structures Chapter 1 Propositional Logic Chapter 1.1 Propositional Logic 1 1.1 Propositional Logic Definition: A proposition :is a declarative sentence (that is, a sentence that declares
More informationDue date: Monday, February 6, 2017.
Modern Analysis Homework 3 Solutions Due date: Monday, February 6, 2017. 1. If A R define A = {x R : x A}. Let A be a nonempty set of real numbers, assume A is bounded above. Prove that A is bounded below
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationWhy Proofs? Proof Techniques. Theorems. Other True Things. Proper Proof Technique. How To Construct A Proof. By Chuck Cusack
Proof Techniques By Chuck Cusack Why Proofs? Writing roofs is not most student s favorite activity. To make matters worse, most students do not understand why it is imortant to rove things. Here are just
More informationProving logical equivalencies (1.3)
EECS 203 Spring 2016 Lecture 2 Page 1 of 6 Proving logical equivalencies (1.3) One thing we d like to do is prove that two logical statements are the same, or prove that they aren t. Vocabulary time In
More information1. Introduction to commutative rings and fields
1. Introduction to commutative rings and fields Very informally speaking, a commutative ring is a set in which we can add, subtract and multiply elements so that the usual laws hold. A field is a commutative
More informationInference and Proofs (1.6 & 1.7)
EECS 203 Spring 2016 Lecture 4 Page 1 of 9 Introductory problem: Inference and Proofs (1.6 & 1.7) As is commonly the case in mathematics, it is often best to start with some definitions. An argument for
More informationSection 3.1: Direct Proof and Counterexample 1
Section 3.1: Direct Proof and Counterexample 1 In this chapter, we introduce the notion of proof in mathematics. A mathematical proof is valid logical argument in mathematics which shows that a given conclusion
More informationSample Problems for all sections of CMSC250, Midterm 1 Fall 2014
Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014 1. Translate each of the following English sentences into formal statements using the logical operators (,,,,, and ). You may also use mathematical
More informationBasic Logic and Proof Techniques
Chapter 3 Basic Logic and Proof Techniques Now that we have introduced a number of mathematical objects to study and have a few proof techniques at our disposal, we pause to look a little more closely
More informationMathematical Reasoning & Proofs
Mathematical Reasoning & Proofs MAT 1362 Fall 2018 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike 4.0
More informationPropositional natural deduction
Propositional natural deduction COMP2600 / COMP6260 Dirk Pattinson Australian National University Semester 2, 2016 Major proof techniques 1 / 25 Three major styles of proof in logic and mathematics Model
More informationDiscrete Mathematics. W. Ethan Duckworth. Fall 2017, Loyola University Maryland
Discrete Mathematics W. Ethan Duckworth Fall 2017, Loyola University Maryland Contents 1 Introduction 4 1.1 Statements......................................... 4 1.2 Constructing Direct Proofs................................
More informationECOM Discrete Mathematics
ECOM 2311- Discrete Mathematics Chapter # 1 : The Foundations: Logic and Proofs Fall, 2013/2014 ECOM 2311- Discrete Mathematics - Ch.1 Dr. Musbah Shaat 1 / 85 Outline 1 Propositional Logic 2 Propositional
More informationLecture Notes 1 Basic Concepts of Mathematics MATH 352
Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,
More informationContribution of Problems
Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions
More information