(4) Using results you have studied, show that if x, y are real numbers,
|
|
- Sylvia Lee
- 6 years ago
- Views:
Transcription
1 Solutions to Homework 4, Math 310 (1) Give a direct proof to show that if a, b are integers which are squares of integers, then ab is the square of an integer. Proof. We show that if a, b are integers which are squares of integers, then ab is the square of an integer. We may write a = m 2, b = n 2 for some integers m, n by hypothesis. Then ab = m 2 n 2 = (mn) 2 and since mn is an integer, by closure, we see that ab is the square of the integer mn. This proves the result. (2) Write the converse to the statement appearing in the first problem. Is the converse true or false? If true, give a proof and if false, give a counterexample. The converse to the statement above is, if a, b are integers and ab is the square of an integer then so are a and b. The converse is false. For example, you may take a = b = 3, then ab = 9 is a square but neither a nor b is a square. (3) Give a constructive proof to show that the equation, x 5 x 4 + x 3 x 2 + x 1 = 0 has a solution in integers. Proof. We show that the equation x 5 x 4 + x 3 x 2 + x 1 = 0 has a solution in integers. One easy way to do this is to notice that 1 is a solution of the equation. Substituting 1 for x in the equation, we have, = = 0. Another way could be to write the expression as follows: f(x) = x 5 x 4 + x 3 x 2 + x 1 = (x 5 x 4 ) + (x 3 x 2 ) + (x 1) = x 4 (x 1) + x 2 (x 1) + (x 1) distributivity = (x 4 + x 2 + 1)(x 1) distributivity Thus we see that f(1) = ( )(1 1) = 3 0 = 0. So, 1 is a solution of the given equation. (4) Using results you have studied, show that if x, y are real numbers, then xy x 2 + y 2. Proof. Any two real numbers can be considered as the coordinates of a point in the plane and thus from trigonometry, we may assume that they are of the form x = r cos θ, y = r sin θ 1
2 2 for some real number r 0 and an angle θ. Thus, xy = r cos θ r sin θ = r 2 cos θ sin θ = r 2 cos θ sin θ (1) r 2 since cos θ 1 and sin θ 1 and so their product is atmost one. On the other hand, we have, x 2 + y 2 = r 2 cos 2 θ + r 2 sin 2 θ = r 2 cos 2 θ + sin 2 θ (2) = r 2 since cos 2 θ + sin 2 θ = 1. Now, using equations (1) and (2), we get that xy x 2 + y 2. (5) Use intermediate value theorem (and thus a non-constructive proof) to show the following: A hiker walks up a hill starting at 6am and reaching the top at 6pm. He walks back from the top to the bottom starting at 6am and reaching the bottom at 6pm along the same path. Show that there exists some point in his path where he was at the same time, going up or down. Proof. Let us denote by the variable t the time and by the variable x the distance along the path starting from the bottom of the hill. Then we have two continuous functions, f, g : I = [6am, 6pm] [0, A], where A is the distance to the top of the hill, defined as follows. f(t) for t I is the distance from the bottom the hiker has travelled on his way up. Similarly, g(t) is the distance from the bottom the traveller reached on his way down at time t. Then h(t) = f(t) g(t) is continuous. We also have, h(6am) = f(6am) g(6am) = 0 A = A h(6pm) = f(6pm) g(6pm) = A 0 = A By intermediate value theorem, therefore there exists a t I so that h(t) = 0. This means that at this time t, f(t) = g(t) or in other words that the hiker was at the same distance from the bottom on his way up and down at time t. This establishes the result we set out to show.
3 I make a remark about problem solving. In the above, we have been told that we are supposed to use the Intermediate value theorem. We know that this theorem is a theorem about continuous functions. So, from the data given, we should somehow create a continuous function to make use of the theorem. Any function must have an input and an output. In our problem, there are not too many choices. More or less the only function is the distance travelled in time t. Once you reach this conclusion, the rest is straight forward. (6) Show that if a, b are postive real numbers and ab (a + b) 2 /4, then a b. Proof. We will prove the contrapositive of the above statement. Since the negation of a b is a = b and ab (a + b) 2 /4 is ab = (a + b) 2 /4, the contrapositive is: If a = b then ab = (a + b) 2 /4. So, assume that a = b. Then ab = a 2 and (a+b) 2 /4 = (2a) 2 /4 = a 2. Thus, we get ab = (a + b) 2 /4, establishing the result. (7) (a) If y is an irrational number and x 0 is a rational number, show that xy is irrational. Notice that this statement can be written as, ((y, irrational) (0 x Q)) (xy is irrational). This can also be written as ( y, irrational)( x Q, x 0)(xy irrational). Why is the above not same as (( y, irrational) ( x Q, x 0)) (xy irrational)? Proof. We will prove this statement by contradiction. The negation of the statement is, there exists an irrational number y and a rational number x 0 such that xy is rational. So, we assume this and reach a contradiction. If x is rational, we can write x = p/q where p, q are integers and q 0 by definition of a rational number. Since x 0, we see that p 0. But, then x 1 = q/p is also a rational number, by definition of a rational number. Then, we can write y = x 1 (xy) and if xy is rational, since rational numbers are closed under multiplication, we get that y is also rational. This is a contradiction which proves our result. (b) If x is a real number and y is an irrational number, show that either x + y or x + y is irrational. 3
4 4 Proof. We will show this by contradiction. So, assume the above statement is false. Then writing the negation, we see that we have a real number x, an irrational number y with both x + y and x + y rational. Since the set of rational numbers is closed under addition, we see by adding, (x + y) + ( x + y) = 2y is rational. Since the product of two rational numbers is also rational, we see that y = 2y 1 is also rational. This contradicts our 2 assumption that y is irrational. This proves our result. (c) Give an example of x, y as before, so that both x + y and x + y are irrational. Taking any irrational number y and letting x = 2y, we see that x + y = 3y is irrational, using the first part of the problem. Similarly, x + y = y is also irrational. (d) Give an example of x, y as before so that one of x+y, x+y is irrational and the other is rational. For, this we may take y to be any irrational number and let x = y. Then x + y = 2y is irrational, but x + y = 0 is rational. (8) If a, b, c are integers with a 2 + b 2 = c 2, then show that either a or b is even. Proof. We will prove this by contradiction. If the statement is false, we have odd integers a, b with a 2 + b 2 = c 2 for an integer c. Writing a = 2m + 1, b = 2n + 1 for some integers m, n, by definition of odd numbers, we get, a 2 + b 2 = (2m + 1) 2 + (2n + 1) 2 = 4m 2 + 4m n 2 + 4n + 1 = 2(2m 2 + 2n 2 + 2m + 2n + 1) = 2(2(m 2 + n 2 + m + n) + 1) From this we see that c 2 = a 2 + b 2 is even. We know that if c 2 is even, then so is c. So, writing c = 2d for an integer d, we see from the above, c 2 = 4d 2 = 2(2(m 2 + n 2 + m + n) + 1). Cancelling a 2, we get, 2d 2 = 2(m 2 + n 2 + m + n) + 1.
5 But the left hand side above is even and the right hand side is odd. This is the contradiction we desire, proving the result. 5
The following techniques for methods of proofs are discussed in our text: - Vacuous proof - Trivial proof
Ch. 1.6 Introduction to Proofs The following techniques for methods of proofs are discussed in our text - Vacuous proof - Trivial proof - Direct proof - Indirect proof (our book calls this by contraposition)
More informationMATH CSE20 Homework 5 Due Monday November 4
MATH CSE20 Homework 5 Due Monday November 4 Assigned reading: NT Section 1 (1) Prove the statement if true, otherwise find a counterexample. (a) For all natural numbers x and y, x + y is odd if one of
More informationProof by Contradiction
Proof by Contradiction MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Proof by Contradiction Fall 2014 1 / 12 Outline 1 Proving Statements with Contradiction 2 Proving
More informationMATH 271 Summer 2016 Practice problem solutions Week 1
Part I MATH 271 Summer 2016 Practice problem solutions Week 1 For each of the following statements, determine whether the statement is true or false. Prove the true statements. For the false statement,
More information(3,1) Methods of Proof
King Saud University College of Sciences Department of Mathematics 151 Math Exercises (3,1) Methods of Proof 1-Direct Proof 2- Proof by Contraposition 3- Proof by Contradiction 4- Proof by Cases By: Malek
More informationComputer Science Section 1.6
Computer Science 180 Solutions for Recommended Exercises Section 1.6. Let m and n be any two even integers (possibly the same). Then, there exist integers k and l such that m = k and n = l. Consequently,
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 informationMathematics 220 Midterm Practice problems from old exams Page 1 of 8
Mathematics 220 Midterm Practice problems from old exams Page 1 of 8 1. (a) Write the converse, contrapositive and negation of the following statement: For every integer n, if n is divisible by 3 then
More informationSome Review Problems for Exam 1: Solutions
Math 3355 Fall 2018 Some Review Problems for Exam 1: Solutions Here is my quick review of proof techniques. I will focus exclusively on propositions of the form p q, or more properly, x P (x) Q(x) or x
More informationMathematics 220 Homework 4 - Solutions. Solution: We must prove the two statements: (1) if A = B, then A B = A B, and (2) if A B = A B, then A = B.
1. (4.46) Let A and B be sets. Prove that A B = A B if and only if A = B. Solution: We must prove the two statements: (1) if A = B, then A B = A B, and (2) if A B = A B, then A = B. Proof of (1): Suppose
More informationCarmen s Core Concepts (Math 135)
Carmen s Core Concepts (Math 135) Carmen Bruni University of Waterloo Week 3 1 Translating From Mathematics to English 2 Contrapositive 3 Example of Contrapositive 4 Types of Implications 5 Contradiction
More informationRED. Name: Instructor: Pace Nielsen Math 290 Section 1: Winter 2014 Final Exam
RED Name: Instructor: Pace Nielsen Math 290 Section 1: Winter 2014 Final Exam Note that the first 10 questions are true-false. Mark A for true, B for false. Questions 11 through 20 are multiple choice
More informationa + b = b + a and a b = b a. (a + b) + c = a + (b + c) and (a b) c = a (b c). a (b + c) = a b + a c and (a + b) c = a c + b c.
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
More informationProperties of the Integers
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
More informationShow Your Work! Point values are in square brackets. There are 35 points possible. Tables of tautologies and contradictions are on the last page.
Formal Methods Midterm 1, Spring, 2007 Name Show Your Work! Point values are in square brackets. There are 35 points possible. Tables of tautologies and contradictions are on the last page. 1. Use truth
More informationMore examples of mathematical. Lecture 4 ICOM 4075
More examples of mathematical proofs Lecture 4 ICOM 4075 Proofs by construction A proof by construction is one in which anobjectthat proves the truth value of an statement is built, or found There are
More informationMidterm Preparation Problems
Midterm Preparation Problems The following are practice problems for the Math 1200 Midterm Exam. Some of these may appear on the exam version for your section. To use them well, solve the problems, then
More information9/5/17. Fermat s last theorem. CS 220: Discrete Structures and their Applications. Proofs sections in zybooks. Proofs.
Fermat s last theorem CS 220: Discrete Structures and their Applications Theorem: For every integer n > 2 there is no solution to the equation a n + b n = c n where a,b, and c are positive integers Proofs
More informationFoundations of Discrete Mathematics
Foundations of Discrete Mathematics Chapter 0 By Dr. Dalia M. Gil, Ph.D. Statement Statement is an ordinary English statement of fact. It has a subject, a verb, and a predicate. It can be assigned a true
More informationIntroduction to proofs. Niloufar Shafiei
Introduction to proofs Niloufar Shafiei proofs Proofs are essential in mathematics and computer science. Some applications of proof methods Proving mathematical theorems Designing algorithms and proving
More informationSample Problems for the Second Midterm Exam
Math 3220 1. Treibergs σιι Sample Problems for the Second Midterm Exam Name: Problems With Solutions September 28. 2007 Questions 1 10 appeared in my Fall 2000 and Fall 2001 Math 3220 exams. (1) Let E
More informationMidterm Exam Solution
Midterm Exam Solution Name PID Honor Code Pledge: I certify that I am aware of the Honor Code in effect in this course and observed the Honor Code in the completion of this exam. Signature Notes: 1. This
More informationHomework 3: Solutions
Homework 3: Solutions ECS 20 (Fall 2014) Patrice Koehl koehl@cs.ucdavis.edu October 16, 2014 Exercise 1 Show that this implication is a tautology, by using a table of truth: [(p q) (p r) (q r)] r. p q
More informationSolutions to Homework Set 1
Solutions to Homework Set 1 1. Prove that not-q not-p implies P Q. In class we proved that A B implies not-b not-a Replacing the statement A by the statement not-q and the statement B by the statement
More informationContradiction MATH Contradiction. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Contradiction Benjamin V.C. Collins James A. Swenson Contrapositive The contrapositive of the statement If A, then B is the statement If not B, then not A. A statement and its contrapositive
More informationYour quiz in recitation on Tuesday will cover 3.1: Arguments and inference. Your also have an online quiz, covering 3.1, due by 11:59 p.m., Tuesday.
Friday, February 15 Today we will begin Course Notes 3.2: Methods of Proof. Your quiz in recitation on Tuesday will cover 3.1: Arguments and inference. Your also have an online quiz, covering 3.1, due
More informationHomework 3 Solutions, Math 55
Homework 3 Solutions, Math 55 1.8.4. There are three cases: that a is minimal, that b is minimal, and that c is minimal. If a is minimal, then a b and a c, so a min{b, c}, so then Also a b, so min{a, b}
More informationMidterm 2 Sample Introduction to Higher Math Fall 2018 Instructor: Linh Truong
Midterm Sample Introduction to Higher Math Fall 018 Instructor: Linh Truong Name: Instructions: Print your name in the space above. Show your reasoning. Write complete proofs. You have 75 minutes. No notes,
More informationCSE 20 DISCRETE MATH WINTER
CSE 20 DISCRETE MATH WINTER 2016 http://cseweb.ucsd.edu/classes/wi16/cse20-ab/ Today's learning goals Evaluate which proof technique(s) is appropriate for a given proposition Direct proof Proofs by contraposition
More informationLimit Laws- Theorem 2.2.1
Limit Laws- Theorem 2.2.1 If L, M, c, and k are real numbers and lim x c f (x) =L and lim g(x) =M, then x c 1 Sum Rule: lim x c (f (x)+g(x)) = L + M 2 Difference Rule: lim x c (f (x) g(x)) = L M 3 Constant
More informationCSE 20 DISCRETE MATH SPRING
CSE 20 DISCRETE MATH SPRING 2016 http://cseweb.ucsd.edu/classes/sp16/cse20-ac/ Today's learning goals Evaluate which proof technique(s) is appropriate for a given proposition Direct proof Proofs by contraposition
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 informationMATH 2200 Final LC Review
MATH 2200 Final LC Review Thomas Goller April 25, 2013 1 Final LC Format The final learning celebration will consist of 12-15 claims to be proven or disproven. It will take place on Wednesday, May 1, from
More informationUNIVERSITY OF VICTORIA DECEMBER EXAMINATIONS MATH 122: Logic and Foundations
UNIVERSITY OF VICTORIA DECEMBER EXAMINATIONS 2013 MATH 122: Logic and Foundations Instructor and section (check one): K. Mynhardt [A01] CRN 12132 G. MacGillivray [A02] CRN 12133 NAME: V00#: Duration: 3
More informationUniversity of Toronto Faculty of Arts and Science Solutions to Final Examination, April 2017 MAT246H1S - Concepts in Abstract Mathematics
University of Toronto Faculty of Arts and Science Solutions to Final Examination, April 2017 MAT246H1S - Concepts in Abstract Mathematics Examiners: D. Burbulla, P. Glynn-Adey, S. Homayouni Time: 7-10
More informationADVICE. Since we will need to negate sentences, it is important to review seccons 7 and 11 before proceeding.
20.CONTRADICTION AND CONTRAPOSITIVE We address, once again, the ques2on of how to prove a mathema2cal statement of the form if A, then B. We have used so far the template for a direct proof: unravel A,
More informationCSE 20 DISCRETE MATH. Winter
CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Today's learning goals Evaluate which proof technique(s) is appropriate for a given proposition Direct proof Proofs by contraposition
More informationIn 1854, Karl Weierstrauss gave an example of a continuous function which was nowhere di erentiable: cos(3 n x) 2 n. sin(3 n x), 2
Why non-pictured analysis? CHAPTER 1 Preliminaries f is continuous at x if lim f(x + h) = f(x) h!0 and f(x + h) f(x) f is di erentiable at x if lim h!0 h Then but Di erentiability =) continuity, continuity
More informationMath.3336: Discrete Mathematics. Proof Methods and Strategy
Math.3336: Discrete Mathematics Proof Methods and Strategy Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu Fall
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 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 informationIntro to Algebra Today. We will learn names for the properties of real numbers. Homework Next Week. Due Tuesday 45-47/ 15-20, 32-35, 40-41, *28,29,38
Intro to Algebra Today We will learn names for the properties of real numbers. Homework Next Week Due Tuesday 45-47/ 15-20, 32-35, 40-41, *28,29,38 Due Thursday Pages 51-53/ 19-24, 29-36, *48-50, 60-65
More informationDiscrete Math I Exam II (2/9/12) Page 1
Discrete Math I Exam II (/9/1) Page 1 Name: Instructions: Provide all steps necessary to solve the problem. Simplify your answer as much as possible. Additionally, clearly indicate the value or expression
More informationCSC Discrete Math I, Spring Propositional Logic
CSC 125 - Discrete Math I, Spring 2017 Propositional Logic Propositions A proposition is a declarative sentence that is either true or false Propositional Variables A propositional variable (p, q, r, s,...)
More informationSection Summary. Proof by Cases Existence Proofs
Section 1.8 1 Section Summary Proof by Cases Existence Proofs Constructive Nonconstructive Disproof by Counterexample Uniqueness Proofs Proving Universally Quantified Assertions Proof Strategies sum up
More informationa. See the textbook for examples of proving logical equivalence using truth tables. b. There is a real number x for which f (x) < 0. (x 1) 2 > 0.
For some problems, several sample proofs are given here. Problem 1. a. See the textbook for examples of proving logical equivalence using truth tables. b. There is a real number x for which f (x) < 0.
More informationRational Exponents Connection: Relating Radicals and Rational Exponents. Understanding Real Numbers and Their Properties
Name Class 6- Date Rational Exponents Connection: Relating Radicals and Rational Exponents Essential question: What are rational and irrational numbers and how are radicals related to rational exponents?
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 informationMath 38: Graph Theory Spring 2004 Dartmouth College. On Writing Proofs. 1 Introduction. 2 Finding A Solution
Math 38: Graph Theory Spring 2004 Dartmouth College 1 Introduction On Writing Proofs What constitutes a well-written proof? A simple but rather vague answer is that a well-written proof is both clear and
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 informationMath 104: Homework 1 solutions
Math 10: Homework 1 solutions 1. The basis for induction, P 1, is true, since 1 3 = 1. Now consider the induction step, assuming P n is true and examining P n+1. By making use of the result (1 + +... +
More informationCHAPTER 8: EXPLORING R
CHAPTER 8: EXPLORING R LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN In the previous chapter we discussed the need for a complete ordered field. The field Q is not complete, so we constructed
More informationMath 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 8 Solutions Please write neatly, and in complete sentences when possible.
Math 320: Real Analysis MWF pm, Campion Hall 302 Homework 8 Solutions Please write neatly, and in complete sentences when possible. Do the following problems from the book: 4.3.5, 4.3.7, 4.3.8, 4.3.9,
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 informationProof by contrapositive, contradiction
Proof by contrapositive, contradiction Margaret M. Fleck 9 September 2009 This lecture covers proof by contradiction and proof by contrapositive (section 1.6 of Rosen). 1 Announcements The first quiz will
More informationAN INTRODUCTION TO MATHEMATICAL PROOFS NOTES FOR MATH Jimmy T. Arnold
AN INTRODUCTION TO MATHEMATICAL PROOFS NOTES FOR MATH 3034 Jimmy T. Arnold i TABLE OF CONTENTS CHAPTER 1: The Structure of Mathematical Statements.............................1 1.1. Statements..................................................................
More informationBasic Proof Examples
Basic Proof Examples Lisa Oberbroeckling Loyola University Maryland Fall 2015 Note. In this document, we use the symbol as the negation symbol. Thus p means not p. There are four basic proof techniques
More informationMATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017
MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017 Definition: A set A is finite if there exists a nonnegative integer c such that there exists a bijection from A
More information1. From Lewis Carroll: extract a pair of premises and finish the conclusion.
Math 2200 2. Treibergs σιι First Midterm Exam Name: Sample January 26, 2011 Sample First Midterm Questions. Sept. 17, 2008 and Sept. 16, 2009. Some questions from Math 3210 Midterms of 1. From Lewis Carroll:
More informationSummer HSSP Week 1 Homework. Lane Gunderman, Victor Lopez, James Rowan
Summer HSSP Week 1 Homework Lane Gunderman, Victor Lopez, James Rowan July 9, 2014 Questions 1 Chapter 1 Homework Questions These are the questions that should be turned in as homework. As mentioned in
More informationMathematics Review for Business PhD Students
Mathematics Review for Business PhD Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationnot to be republished NCERT REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results
REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results Euclid s Division Lemma : Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r < b. Euclid s Division
More informationMath 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 2 Solutions Please write neatly, and in complete sentences when possible.
Math 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 2 Solutions Please write neatly, and in complete sentences when possible. Do the following problems from the book: 1.4.2, 1.4.4, 1.4.9, 1.4.11,
More information3.6. Disproving Quantified Statements Disproving Existential Statements
36 Dproving Quantified Statements 361 Dproving Extential Statements A statement of the form x D, P( if P ( false for all x D false if and only To dprove th kind of statement, we need to show the for all
More informationMATH 13 FINAL EXAM SOLUTIONS
MATH 13 FINAL EXAM SOLUTIONS WINTER 2014 Problem 1 (15 points). For each statement below, circle T or F according to whether the statement is true or false. You do NOT need to justify your answers. T F
More informationProperties of Rational and Irrational Numbers
Properties of Rational and Irrational Numbers September 8, 2016 Definition: The natural numbers are the set of numbers N = {1, 2, 3,...}, and the integers are the set of numbers Z = {..., 2, 1, 0, 1, 2,...}.
More informationLogic 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 informationGraphing Square Roots - Class Work Graph the following equations by hand. State the domain and range of each using interval notation.
Graphing Square Roots - Class Work Graph the following equations by hand. State the domain and range of each using interval notation. 1. y = x + 2 2. f(x) = x 1. y = x +. g(x) = 2 x 1. y = x + 2 + 6. h(x)
More informationMATH 2200 Final Review
MATH 00 Final Review Thomas Goller December 7, 01 1 Exam Format The final exam will consist of 8-10 proofs It will take place on Tuesday, December 11, from 10:30 AM - 1:30 PM, in the usual room Topics
More informationCOT 2104 Homework Assignment 1 (Answers)
1) Classify true or false COT 2104 Homework Assignment 1 (Answers) a) 4 2 + 2 and 7 < 50. False because one of the two statements is false. b) 4 = 2 + 2 7 < 50. True because both statements are true. c)
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 informationMATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions.
MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if
More information(e) Commutativity: a b = b a. (f) Distributivity of times over plus: a (b + c) = a b + a c and (b + c) a = b a + c a.
Math 299 Midterm 2 Review Nov 4, 2013 Midterm Exam 2: Thu Nov 7, in Recitation class 5:00 6:20pm, Wells A-201. Topics 1. Methods of proof (can be combined) (a) Direct proof (b) Proof by cases (c) Proof
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 informationProof Techniques 1. Math Camp n = (1 = n) + (2 + (n 1)) ( n 1. = (n + 1) + (n + 1) (n + 1) + n + 1.
Math Camp 1 Prove the following by direct proof. Proof Techniques 1 Math Camp 01 Take any n N, then n is either even or odd. Suppose n is even n m for some m n(n + 1) m(n + 1) n(n + 1) is even. Suppose
More informationMath Real Analysis
1 / 28 Math 370 - Real Analysis G.Pugh Sep 3 2013 Real Analysis 2 / 28 3 / 28 What is Real Analysis? Wikipedia: Real analysis... has its beginnings in the rigorous formulation of calculus. It is a branch
More informationMathematical Reasoning (Part I) 1
c Oksana Shatalov, Spring 2017 1 Mathematical Reasoning (art I) 1 Statements DEFINITION 1. A statement is any declarative sentence 2 that is either true or false, but not both. A statement cannot be neither
More informationHOMEWORK 1: SOLUTIONS - MATH 215 INSTRUCTOR: George Voutsadakis
HOMEWORK 1: SOLUTIONS - MATH 215 INSTRUCTOR: George Voutsadakis Problem 1 Make truth tables for the propositional forms (P Q) (P R) and (P Q) (R S). Solution: P Q R P Q P R (P Q) (P R) F F F F F F F F
More informationExample 1: Identifying the Parts of a Conditional Statement
"If p, then q" can also be written... If p, q q, if p p implies q p only if q Example 1: Identifying the Parts of a Conditional Statement Identify the hypothesis and conclusion of each conditional. A.
More informationSolution. 1 Solution of Homework 7. Sangchul Lee. March 22, Problem 1.1
Solution Sangchul Lee March, 018 1 Solution of Homework 7 Problem 1.1 For a given k N, Consider two sequences (a n ) and (b n,k ) in R. Suppose that a n b n,k for all n,k N Show that limsup a n B k :=
More information8. Given a rational number r, prove that there exist coprime integers p and q, with q 0, so that r = p q. . For all n N, f n = an b n 2
MATH 135: Randomized Exam Practice Problems These are the warm-up exercises and recommended problems taken from all the extra practice sets presented in random order. The challenge problems have not been
More informationChapter 2: The Logic of Quantified Statements. January 22, 2010
Chapter 2: The Logic of Quantified Statements January 22, 2010 Outline 1 2.1- Introduction to Predicates and Quantified Statements I 2 2.2 - Introduction to Predicates and Quantified Statements II 3 2.3
More informationMATH 135 Fall 2006 Proofs, Part IV
MATH 135 Fall 006 s, Part IV We ve spent a couple of days looking at one particular technique of proof: induction. Let s look at a few more. Direct Here we start with what we re given and proceed in a
More informationWhat is a proof? Proofing as a social process, a communication art.
Proof Methods What is a proof? Proofing as a social process, a communication art. Theoretically, a proof of a mathematical statement is no different than a logically valid argument starting with some premises
More informationQuiz 1. Directions: Show all of your work and justify all of your answers.
Quiz 1 1. Let p and q be the following statements. p : Maxwell is a mathematics major. q : Maxwell is a chemistry major. (1) a. Write each of the following in symbolic form using logical connectives. i.
More informationMATH 201 Solutions: TEST 3-A (in class)
MATH 201 Solutions: TEST 3-A (in class) (revised) God created infinity, and man, unable to understand infinity, had to invent finite sets. - Gian Carlo Rota Part I [5 pts each] 1. Let X be a set. Define
More informationSTRATEGIES OF PROBLEM SOLVING
STRATEGIES OF PROBLEM SOLVING Second Edition Maria Nogin Department of Mathematics College of Science and Mathematics California State University, Fresno 2014 2 Chapter 1 Introduction Solving mathematical
More informationProving Things. 1. Suppose that all ravens are black. Which of the following statements are then true?
Proving Things 1 Logic 1. Suppose that all ravens are black. Which of the following statements are then true? (a) If X is a raven, then X is black. (b) If X is black, then X is a raven. (c) If X is not
More information1-2 Study Guide and Intervention
1- Study Guide and Intervention Real Numbers All real numbers can be classified as either rational or irrational. The set of rational numbers includes several subsets: natural numbers, whole numbers, and
More information5. Use a truth table to determine whether the two statements are equivalent. Let t be a tautology and c be a contradiction.
Statements Compounds and Truth Tables. Statements, Negations, Compounds, Conjunctions, Disjunctions, Truth Tables, Logical Equivalence, De Morgan s Law, Tautology, Contradictions, Proofs with Logical Equivalent
More informationChapter 0 Preliminaries
Chapter 0 Preliminaries MA1101 Mathematics 1A Semester I Year 2017/2018 FTMD & FTI International Class Odd NIM (K-46) Lecturer: Dr. Rinovia Simanjuntak 0.1 Real Numbers and Logic Real Numbers Repeating
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationMath Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. A prime number
More informationNormal Forms Note: all ppts about normal forms are skipped.
Normal Forms Note: all ppts about normal forms are skipped. Well formed formula (wff) also called formula, is a string consists of propositional variables, connectives, and parenthesis used in the proper
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 informationMath1a Set 1 Solutions
Math1a Set 1 Solutions October 15, 2018 Problem 1. (a) For all x, y, z Z we have (i) x x since x x = 0 is a multiple of 7. (ii) If x y then there is a k Z such that x y = 7k. So, y x = (x y) = 7k is also
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 informationConnectives Name Symbol OR Disjunction And Conjunction If then Implication/ conditional If and only if Bi-implication / biconditional
Class XI Mathematics Ch. 14 Mathematical Reasoning 1. Statement: A sentence which is either TRUE or FALSE but not both is known as a statement. eg. i) 2 + 2 = 4 ( it is a statement which is true) ii) 2
More informationHOMEWORK ASSIGNMENT 6
HOMEWORK ASSIGNMENT 6 DUE 15 MARCH, 2016 1) Suppose f, g : A R are uniformly continuous on A. Show that f + g is uniformly continuous on A. Solution First we note: In order to show that f + g is uniformly
More information1.1 Language and Logic
c Oksana Shatalov, Fall 2017 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 information