Math 3000 Section 003 Intro to Abstract Math Homework 2


 Dana Quinn
 1 years ago
 Views:
Transcription
1 Math 3000 Section 003 Intro to Abstract Math Homework 2 Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Solutions (February 13, 2012) Please note that these solutions are only suggestions; different answers or proofs are also possible. Section 2.1: Statements 1. Give one sentence each about abstract mathematics (or the super bowl) that is (a) declarative and a statement; (b) declarative and open; (c) imperative; (d) interrogative; (e) exclamatory. Solution: Answers may vary, but here are some examples: (a) The super bowl is the annual championship game of the National Basketball Association. (False, but a declarative statement.) (b) This day is the biggest day for U.S. food consumption. (Note that this day is not specified but left open  however, you may guess that it s Thanksgiving Day, but be surprised that according to Wikipedia Super Bowl Sunday beats Christmas and takes second place.) (c) Wiggle wiggle wiggle wiggle wiggle! (d) Who wants chicken wings? (e) Touchdown! Section 2.2: The Negation of a Statement 2. Exercise 2.8: State the negation of each of the following statements. (Avoid the awkwardness of using double negation.) (a) 2 is a rational number. (b) 0 is not a negative integer. (c) 111 is a prime number. Solution: (a) 2 is not a rational (better: an irrational) number. (b) 0 is a negative integer. (Careful here: positive is not the opposite of negative because 0 is neither positive nor negative; to include zero with the positive numbers you have to say nonnegative; and similarly, you have to say nonpositive to include 0 with the negative numbers.) (c) 111 is not a prime number. (Careful again: the opposite of prime is not composite, and vice versa, the opposite of composite is not prime; for example, both 0 and 1 are neither composite nor prime.) Section 2.3: The Disjunction and Conjunction of Statements 3. Exercise 2.10: Let P : 15 is odd and Q: 21 is prime. State each of the following in words, and determine whether they are true or false. (a) P Q (b) P Q (c) ( P ) Q (d) P ( Q). Solution: (a) 15 is odd or 21 is prime (True). (b) 15 is odd and 21 is prime (False). (c) 15 is even or 21 is prime (False). (d) 15 is odd and 21 is not prime (True). Section 2.4: The Implication
2 Math Intro to Abstract Math Homework 2, UC Denver, Spring 2012 (Solutions) 2 4. Formulate four conditional sentences about abstract mathematics (or the super bowl) that correspond to the four possible truth assignments (T T, T F, F T, F F ) for protasis (condition) and apodasis (consequence). Explain why only one of your implications is logically false. Solution: Consider the following mathematical statement that be composed of two open sentences over the domain of positive integers: If P (x): x is an even prime number, then Q(x): x has exactly two divisors, one and itself. You will likely agree that this statement is generally true: although condition and consequence are both satisfied only for x = 2 (T T ), the general statement remains true also when the condition is violated for all odd primes (F T ) or for all even (or odd) composite numbers (F F ): the fact that the condition is false for x = 3 does not mean the implication is false; similarly, if x = 4 were a prime, then by definition it would have exactly two divisors, 1 and itself. Now imagine that your friend believes that 0 is a prime number, so that the condition P (0) would be true, but agrees with you that 0 has infinitely many divisors (all positive integers are divisors of 0), so that the consequence Q(0) is false (T F ). In this case, the implication is clearly false, and you should have no difficulties convincing your friend that 0 cannot be a prime number. You may have noticed that a logical explanation using nonmathematical sentences is quite difficult: for example, sentences like If Justin Bieber sang during the halftime show of Super Bowl XLVI, then the New England Patriots won or If LMFAO wiggled, then Tom Brady and Eli Manning were sexy and they knew it (if this sentences seems weird, just ignore it!) simply do not make much sense: they are not related, act on different domains, and include a linguistic shade that mathematical logic does not know: the distinction of conditional sentences as factual (in past or present), predictive (in future), and speculative (in past, present, of future expressed in subjunctive mood using a modal verb could, might, would,... ). To avoid this confusion, we sometimes distinguish the logical conditional P Q (read if P, then Q or P implies Q ) from the material conditional P Q (read not P or Q ): Justin Bieber did not sing during the halftime show of Super Bowl XLVI, or the New England Patriots won (true and sensible) or LMFAO did not wiggle, or Tom and Eli were sexy and they knew it (not sure about that one). Although logical and material conditionals are equivalent in mathematical logic, their meaning and interpretation may seem different based on our past experience, intuition, and prepossession with natural language. Section 2.5: More on Implications 5. Exercise 2.20: In each of the following, two open sentences P (x) and Q(x) over a domain S are given. Determine all x S for which P (x) Q(x) is a true statement. (Hint: Use the logical equivalence between the two statements P (x) Q(x) ( P (x)) Q(x).) (a) P (x) : x 3 = 4; Q(x) : x 8; S = R (b) P (x) : x 2 1; Q(x) : x 1; S = R (c) P (x) : x 2 1; Q(x) : x 1; S = N (d) P (x) : x [ 1, 2]; Q(x) : x 2 2; S = [ 1, 1] Solution: This exercise (quite impressively) shows how the equivalent material conditional P (x) Q(x) ( P (x)) Q(x) can facilitate our understanding of a logical implication P (x) Q(x): (a) The equivalent disjunction x 3 4 or x 8 is true for all x 7. Therefore, over the domain of real numbers, the implication P (x) Q(x) is true for all real numbers but 7. (b) The equivalent disjunction x 2 < 1 or x 1 is true for all x > 1. Therefore, over the domain of real numbers, the implication P (x) Q(x) is true for all real numbers greater than 1. (c) An immediate consequence from (b), over the domain of positive integers, the implication P (x) Q(x) is always true. (d) In this case, it is easier to use the
3 Math Intro to Abstract Math Homework 2, UC Denver, Spring 2012 (Solutions) 3 logical conditional directly and observe that x for all x S = [ 1, 1]. Therefore, over the domain S = [ 1, 1], the implication P (x) Q(x) is always true. Alternatively, the same conclusion follows from the equivalent disjunction x / [ 1, 2] or x 2 2, which is true for all real numbers x / [ 2, 2] and thus for all real numbers in the domain S = [ 1, 1]. Section 2.6: The Biconditional 6. Exercise 2.22: Let P : 18 is odd and Q: 25 is even. State P Q in words. Is P Q true or false? Solution: 18 is odd if and only if 25 is even. This biconditional is logically true. If you are not convinced, state the two implications 18 is odd if 25 is even and 18 is odd only if 25 is even separately (rewrite as if..., then... if you prefer) and formulate them as material conditionals: 18 is odd or 25 is odd and 18 is even or 25 is even both of which are correct. Section 2.7: Tautologies and Contradictions 7. Exercise 2.32: For statements P and Q, show that (P (P Q)) Q is a tautology. Then state (P (P Q)) Q in words. (This is an important logical argument form, called modus ponens.) Solution: If P is true, and if P implies Q, then Q is true. (Using natural language, this means that a correct inference from a correct condition always yields a correct consequence). Section 2.8: Logical Equivalence P Q P Q P (P Q) (P (P Q)) Q T T T T T T F F F T F T T F T F F T F T 8. Exercise 2.34: For statements P and Q, the implication ( P ) ( Q) is called the inverse of the implication P Q. (a) Use a truth table to show that these statement are not (!) logically equivalent. (b) Find another implication that is logically equivalent to ( P ) ( Q) and verify your answer. Solution: (a) The truth table below shows that the truth values of P Q and its inverse ( P ) ( Q) are different when exactly one of the two statements P and Q is true and the other one is false. (b) A logically equivalent implication to ( P ) ( Q) is its contrapositive Q P which is itself equivalent to the material conditional ( Q) P. P Q P Q P Q ( P ) ( Q) Q P T T F F T T T T F F T F T F F T T F T F T F F T T T T T Section 2.9: Some Fundamental Properties of Logical Equivalence 9. Verify (mathematically) or explain (logically) correctness of the laws in Theorem 2.18 on page 49 in your text book. (These laws are very important and we will use them a lot, so please make sure that you understand their meaning.)
4 Math Intro to Abstract Math Homework 2, UC Denver, Spring 2012 (Solutions) 4 Solution: Use the truth tables below to verify these laws, and your common sense to convince yourself of their correctness. Commutative and associative laws should be clear, but do think a little (and maybe formulate a few examples) about distributive and De Morgen s laws. (a) Commutative Laws (b) Associative Laws (c) Distributive Laws P Q P Q Q P T T T T T F T T F T T T F F F F P Q P Q Q P T T T T T F F F F T F F F F F F P Q R P Q Q R (P Q) R P (Q R) T T T T T T T T T F T T T T T F T T T T T T F F T F T T F T T T T T T F T F T T T T F F T F T T T F F F F F F F P Q R P Q Q R (P Q) R P (Q R) T T T T T T T T T F T F F F T F T F F F F T F F F F F F F T T F T F F F T F F F F F F F T F F F F F F F F F F F P Q R P Q P R Q R P (Q R) (P Q) (P R) T T T T T T T T T T F T T F T T T F T T T F T T T F F T T F T T F T T T T T T T F T F T F F F F F F T F T F F F F F F F F F F F
5 Math Intro to Abstract Math Homework 2, UC Denver, Spring 2012 (Solutions) 5 (d) De Morgan s Laws P Q R P Q P R Q R P (Q R) (P Q) (P R) T T T T T T T T T T F T F T T T T F T F T T T T T F F F F F F F F T T F F T F F F T F F F T F F F F T F F T F F F F F F F F F F Section 2.10: Quantified Statements P Q P Q P Q (P Q) ( P ) ( Q) T T F F T F F T F F T T F F F T T F T F F F F T T F T T P Q P Q P Q (P Q) ( P ) ( Q) T T F F T F F T F F T F T T F T T F F T T F F T T F T T 10. Exercise 2.48: Determine the truth value of each of the following statements. (a) x R : x 2 x = 0 (b) n N : n (c) x R : x 2 = x (d) x Q : 3x 2 27 = 0 (e) x R, y R : x + y + 3 = 8 (f) x, y R : x + y + 3 = 8 (g) x, y R : x 2 + y 2 = 9 (h) x R, y R : x 2 + y 2 = 9 (i) x R : y R : x = y (new!) (j) x R : y R : x = y (new!) Solution: (a) True (Examples: x {0, 1} R). (b) True (Proof: n 1 for all n N). (c) False (Counterexample: x = 1 R but ( 1) 2 = 1). (d) True (Examples: x { 3, 3} Q). (e) True (Example: (x, y) = (3, 2) R R). (f) False (Counterexample: (x, y) = (3, 1) R R). (g) True (Example: (x, y) = (3, 0) R R). (h) False (Counterexample: (x, y) = (3, 1) R R). (i) True (Proof: We need to show that given any real number x, there exists a real number y so that x = y. This is (almost) trivial: Let x be the real number that we are given, and choose y = x.). (j) False: This statement says that there exists a real number x such that x = y for all real numbers y, or worded slightly differently, that there exists a real number that is equal to all real numbers. This statement is clearly false. Section 2.11: Characterizations of Statements 11. Exercise 2.52: Give a definition of each of the following, and then state a characterization of each.
6 Math Intro to Abstract Math Homework 2, UC Denver, Spring 2012 (Solutions) 6 (a) two lines in the planes are perpendicular (b) a rational number Solution: (a) Possible definition: Two lines in the plane are said to be perpendicular if they form congruent adjacent angles (a Tshape). Possible characterizations: (i) Two lines in the plane are perpendicular if and only if they intersect at an angle of 90 degrees (you can say a right angle if you define (or assume the reader knows) that a right angle is an angle that measures 90 degrees, or π/2 radians). (ii) Two lines in the plane are perpendicular if and only if they have opposite reciprocal slopes (the product of their slopes is 1) or if one line is horizontal and the other line is vertical (because the slope of a vertical line is usually described as undefined or infinity, you need to treat vertical and horizontal lines as a special case). (iii) Two lines in the plane are said to be perpendicular if the dot product between the two direction vectors that describe these lines equals zero. (iv) Let a, b, p, q R be real numbers and L 1 : y = ax+p and L 2 : y = bx+q be two lines in the plane. Then L 1 and L 2 are perpendicular, denoted by L 1 L 2, if and only if ab = 1. (Note that this characterization does not say that vertical and horizontal lines are not perpendicular, because vertical lines can not be represented as shown here and thus do not fall into the domain of this result.) (iv) Let a, b, c, d, p, q R be real numbers and L 1 : ax + by = p and L 2 : cx + dy = q be two lines in the plane. Then L 1 L 2 if and only if ac + bd = 0. (v) Let p, q, r, s R 2 be real twodimensional vectors and L 1 : (x, y) = p + tr and L 2 : (x, y) = q + ts be two lines in the plane parametrized by the real scalar t (, ). Then L 1 L 2 if and only if r s = 0. (b) Possible definition: A number that can be expressed as the simple fraction of an integer and a positive integer is called a rational number. The same definition using more symbols: A number r is rational, denoted by r Q, if and only if there exists an integer p Z and a positive integer q N such that r = p/q (recall that the letter Q is derived from the word quotient ). The same definition using only symbols: Let the set of rational numbers be defined by Q := {r : p Z, q N : r = p/q} = {p/q : (p, q) Z N}. Possible characterizations: (i) A real number is rational if and only if it is not irrational (of course, this definition only makes sense if we already know or have defined what real and irrational numbers are). (ii) A real number is rational if and only if it has a finite or repeating decimal expansion. (iii?) The following characterization is wrong: Let p, q R be two real numbers. Then the number r = p/q is rational if and only p Z and q N. (The if direction is true but it is not difficult to find counterexample for the only if direction: r = p/q is also rational if p = q I because then r = 1 Q, among others. In other words, given the representation r = p/q, the condition (p, q) Z N is sufficient but not necessary for r Q.) Additional Exercises for Chapter Exercise 2.60: Rewrite each of the implications below using (1) only if and (2) sufficient. (a) If a function f is differentiable, then f is continuous. (b) If x = 5, then x 2 = 25. Solution: (a) A function is differentiable only if it is continuous. The differentiability of a function is sufficient for its continuity. Differentiability of a function is a sufficient condition for its continuity. (In other words, it is not possible that a function is differentiable but not continuous. This condition is not necessary, however: it is also possible that a function is continuous but not differentiable). (b) A number equals 5 only if its square equals 25 (note that the inverse is not correct: the square of a number equals 25 not only if that number is 5, but also if that number is (positive) 5.) A value of 5 is sufficient for that number s square being 25 (but it is not necessary: another possibility would be a value of (positive) 5). Please let me know if you have any questions, comments, corrections, or remarks.
AN INTRODUCTION TO LOGIC. and PROOF TECHNIQUES
i AN INTRODUCTION TO LOGIC and PROOF TECHNIQUES Michael A. Henning School of Mathematical Sciences University of KwaZuluNatal ii Contents 1 Logic 1 1.1 Introduction....................................
More information1.1 Statements and Compound Statements
Chapter 1 Logic 1.1 Statements and Compound Statements A statement or proposition is an assertion which is either true or false, though you may not know which. That is, a statement is something that has
More informationHarvard University, Math 101, Spring 2015
Harvard University, Math 101, Spring 2015 Lecture 1 and 2 : Introduction to propositional logic 1 Logical statements A statement is a sentence that is either true or false, but not both. Some examples:
More information31 is a prime number is a mathematical statement (which happens to be true).
Chapter 1 Mathematical Logic In its most basic form, Mathematics is the practice of assigning truth to welldefined statements. In this course, we will develop the skills to use known true statements to
More informationMath 3000 Running Glossary
Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (
More information1 Deductive Reasoning and Logical Connectives
1 Deductive Reasoning and Logical Connectives As we have seen, proofs play a central role in mathematics and they are based on deductive reasoning. Facts (or statements) can be represented using Boolean
More informationChapter 2. The Logic of Quantified Statements
2.1.1 Chapter 2. The Logic of Quantified Statements Predicates Quantified Statements Valid Arguments and Quantified Statements 2.1.2 Section 1. Predicates and Quantified Statements I In Chapter 1, we studied
More informationHandout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
More informationPredicate Logic & Proofs Lecture 3
Topics for Today Necessary & sufficient conditions, Only if, If and only if CPRE 310 Discrete Mathematics Predicate Logic & Proofs Lecture 3 Quantified statements: predicates, quantifiers, truth values,
More informationChapter I Logic and Proofs
MATH 1130 1 Discrete Structures Chapter I Logic and Proofs Propositions A proposition is a statement that is either true (T) or false (F), but or both. s Propositions: 1. I am a man.. I am taller than
More informationMath 0413 Supplement Logic and Proof
Math 0413 Supplement Logic and Proof January 17, 2008 1 Propositions A proposition is a statement that can be true or false. Here are some examples of propositions: 1 = 1 1 = 0 Every dog is an animal.
More informationDiscrete Mathematics What is a proof?
Discrete Mathematics What is a proof? Saad Mneimneh 1 The pigeonhole principle The pigeonhole principle is a basic counting technique. It is illustrated in its simplest form as follows: We have n + 1 pigeons
More informationSet and element. Cardinality of a set. Empty set (null set) Finite and infinite sets. Ordered pair / ntuple. Cartesian product. Proper subset.
Set and element Cardinality of a set Empty set (null set) Finite and infinite sets Ordered pair / ntuple Cartesian product Subset Proper subset Power set Partition The cardinality of a set is the number
More informationDiscrete Mathematics Lecture 2 Logic of Quantified Statements, Methods of Proof, Set Theory, Number Theory Introduction and General Good Times
Discrete Mathematics Lecture 2 Logic of Quantified Statements, Methods of Proof, Set Theory, Number Theory Introduction and General Good Times Harper Langston New York University Predicates A predicate
More informationEven Number: An integer n is said to be even if it has the form n = 2k for some integer k. That is, n is even if and only if n divisible by 2.
MATH 337 Proofs Dr. Neal, WKU This entire course requires you to write proper mathematical proofs. All proofs should be written elegantly in a formal mathematical style. Complete sentences of explanation
More informationComputing Science 272 The Integers
Computing Science 272 The Integers Properties of the Integers The set of all integers is the set Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, and the subset of Z given by N = {0, 1, 2, 3, 4, }, is the set
More informationconditional statement conclusion Vocabulary Flash Cards Chapter 2 (p. 66) Chapter 2 (p. 69) Chapter 2 (p. 66) Chapter 2 (p. 76)
biconditional statement conclusion Chapter 2 (p. 69) conditional statement conjecture Chapter 2 (p. 76) contrapositive converse Chapter 2 (p. 67) Chapter 2 (p. 67) counterexample deductive reasoning Chapter
More informationYou are on a strange island, where some of the inhabitants are knights, and always tell the truth, and some are knaves, and always lie.
Chapter 2 Logic and proof 2.1 Knights and knaves You are on a strange island, where some of the inhabitants are knights, and always tell the truth, and some are knaves, and always lie. You come to a fork
More informationElementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
More informationNOTES ON PROOF TECHNIQUES (OTHER THAN INDUCTION)
NOTES ON PROOF TECHNIQUES (OTHER THAN INDUCTION) DAMIEN PITMAN Definitions & Theorems Definition: A direct proof is a valid argument that verifies the truth of an implication by assuming that the premise
More informationDISCRETE MATH: LECTURE 4
DISCRETE MATH: LECTURE 4 DR. DANIEL FREEMAN 1. Chapter 3.1 Predicates and Quantified Statements I A predicate is a sentence that contains a finite number of variables and becomes a statement when specific
More informationDISCRETE MATHEMATICS W W L CHEN
DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationINTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS
INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28
More informationA declared mathematical proposition whose truth value is unknown is called a conjecture.
Methods of Proofs Recall we discussed the following methods of proofs  Vacuous proof  Trivial proof  Direct proof  Indirect proof  Proof by contradiction  Proof by cases. A vacuous proof of an implication
More informationGeometry Unit 1. Basics of Geometry
Geometry Unit 1 Basics of Geometry Using inductive reasoning  Looking for patterns and making conjectures is part of a process called inductive reasoning Conjecture an unproven statement that is based
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
More informationSection 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.
M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider
More informationChapter 1, Part III: Proofs
Chapter 1, Part III: Proofs Summary Valid Arguments and Rules of Inference Proof Methods Proof Strategies Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules
More informationInference Rules and Proof Methods
Inference Rules and Proof Methods Winter 2010 Introduction Rules of Inference and Formal Proofs Proofs in mathematics are valid arguments that establish the truth of mathematical statements. An argument
More informationThe Process of Mathematical Proof
The Process of Mathematical Proof Introduction. Mathematical proofs use the rules of logical deduction that grew out of the work of Aristotle around 350 BC. In Math 213 and other courses that involve writing
More informationMAT2400 Analysis I. A brief introduction to proofs, sets, and functions
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take
More informationThe Real Numbers and the Integers
The Real Numbers and the Integers PRIMITIVE TERMS To avoid circularity, we cannot give every term a rigorous mathematical definition; we have to accept some things as undefined terms. For this course,
More information1 SET THEORY CHAPTER 1.1 SETS
CHAPTER 1 SET THEORY 1.1 SETS The main object of this book is to introduce the basic algebraic systems (mathematical systems) groups, ring, integral domains, fields, and vector spaces. By an algebraic
More informationCS 2336 Discrete Mathematics
CS 2336 Discrete Mathematics Lecture 4 Proofs: Methods and Strategies 1 Outline What is a Proof? Methods of Proving Common Mistakes in Proofs Strategies : How to Find a Proof? 2 What is a Proof? A proof
More informationIt is time to prove some theorems. There are various strategies for doing
CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it
More informationMATH CSE20 Test 2 Review Sheet Test Tuesday October 29 in lecture: CENTER 115, 3:30pm
MATH CSE20 Test 2 Review Sheet Test Tuesday October 29 in lecture: CENTER 115, 3:30pm Textbook sections: Unit Lo Sections 1 and 2 (1) All questions from Homeworks 3 and 4 (2) (Lo Review Question 4) The
More informationLOGIC & SET THEORY AMIN WITNO
LOGIC & SET THEORY AMIN WITNO.. w w w. w i t n o. c o m Logic & Set Theory Revision Notes and Problems Amin Witno Preface These notes are for students of Math 251 as a revision workbook
More informationDiscrete Mathematics Lecture 1 Logic of Compound Statements. Harper Langston New York University
Discrete Mathematics Lecture 1 Logic of Compound Statements Harper Langston New York University Administration Class Web Site http://cs.nyu.edu/courses/summer05/g22.2340001/ Mailing List Subscribe at
More informationNotes on Logic. 1 Propositional Calculus. 2 Logical Operators and Truth Tables
Notes on Logic 1 Propositional Calculus A proposition or statement is an assertion which can be determined to be either true or false (T or F). For example, zero is less than any positive number is a statement.
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationMAT Mathematical Concepts and Applications
MAT.1180  Mathematical Concepts and Applications Chapter (Aug, 7) Number Theory: Prime and Composite Numbers. The set of Natural numbers, aka, Counting numbers, denoted by N, is N = {1,,, 4,, 6,...} If
More information1 Proposition, Logical connectives and compound statements
Discrete Mathematics: Lecture 4 Introduction to Logic Instructor: Arijit Bishnu Date: July 27, 2009 1 Proposition, Logical connectives and compound statements Logic is the discipline that deals with the
More informationCS 441 Discrete Mathematics for CS Lecture 6. Informal proofs. CS 441 Discrete mathematics for CS. Proofs
CS 441 Discrete Mathematics for CS Lecture 6 Informal proofs Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Proofs The truth value of some statements about the world are obvious and easy to assess
More informationThe Foundations: Logic and Proofs. Chapter 1, Part III: Proofs
The Foundations: Logic and Proofs Chapter 1, Part III: Proofs Rules of Inference Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments
More information4 Interlude: A review of logic and proofs
4 Interlude: A review of logic and proofs 4.1 Logic 4.1.1 Propositions and logical connectives A proposition is a declarative sentence that is either true (T or false (F, but not both. Example 4.1 Examples
More informationLogic, Sets, and Proofs
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical
More informationChapter One. Logic and Sets
Chapter One Logic and Sets 1.1 INTRODUCTION Given positive integers m and n, we say that m is a factor of n provided n = mq for some positive integer q. In particular, n is a factor of itself, since n
More information2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.
2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then
More informationReview for Final Exam
Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More information=. Rewriting this as ( ) m which satisfies x 6m. is true we select an arbitrary x U from the universe, and then prove the assertion P( x ) is true.
1 Section 1.5: : Purpose of Section: The theorems in Section 1.4 included quantifiers although the theorems were not stated explicitly in the language of predicate logic. In this section we state theorems
More informationKey Concepts: Fundamentals of Logic and Techniques for Mathematical Proofs
Key Concepts: Fundamentals of Logic and Techniques for Mathematical Proofs Samvel Atayan and Brent Hickman August 11, 2009 Additional Readings: Analysis with an Introduction to Proof 3rd ed. by Steven
More informationSummary. Valid Arguments and Rules of Inference Proof Methods Proof Strategies
Proofs 1 Summary Valid Arguments and Rules of Inference Proof Methods Proof Strategies 2 Section 1.6 3 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to
More information(i) Every natural number is a whole number. True, since the collection of whole numbers contains all natural numbers.
Exercise 1.1 1. Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q 0? Yes. Zero is a rational number as it can be represented as 0/1 or 0/2. 2. Find six rational
More information1.1. Basic Concepts. Write sets using set notation. Write sets using set notation. Write sets using set notation. Write sets using set notation.
1.1 Basic Concepts Write sets using set notation. Objectives A set is a collection of objects called the elements or members of the set. 1 2 3 4 5 6 7 Write sets using set notation. Use number lines. Know
More informationMidterm Examination 1 with Solutions  Math 574, Frank Thorne Thursday, February 9, 2012
Midterm Examination 1 with Solutions  Math 574, Frank Thorne (thorne@math.sc.edu) Thursday, February 9, 2012 1. (3 points each) For each sentence below, say whether it is logically equivalent to the sentence
More informationCHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
More informationFundamentals Part 1 of Hammack
Fundamentals Part 1 of Hammack Dr. Doreen De Leon Math 111, Fall 2014 1 Sets 1.1 Introduction to Sets A set is a collection of things called elements. Sets are sometimes represented by a commaseparated
More informationWe now explore a third method of proof: proof by contradiction.
CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement
More informationVocabulary: Accentuate the Negative
Vocabulary: Accentuate the Negative Concept Integers: The set of whole numbers and their opposites. (Opposites are also called additive inverses.) Opposites: Numbers which are on opposite sides of zero
More informationWeek 5: Quantifiers. Number Sets. Quantifier Symbols. Quantity has a quality all its own. attributed to Carl von Clausewitz
Week 5: Quantifiers Quantity has a quality all its own. attributed to Carl von Clausewitz Number Sets Many people would say that mathematics is the science of numbers. This is a common misconception among
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationDiscrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University
Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called
More informationMA 274: EQUIVALENCE RELATIONS
MA 274: EQUIVALENCE RELATIONS 1. EQUIVALENCE RELATIONS Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that: (1) is reflexive. That is, (2) is symmetric. That is, x A,x
More informationChapter 1: Number Systems and Fundamental Concepts of Algebra. If n is negative, the number is small; if n is positive, the number is large
Final Exam Review Chapter 1: Number Systems and Fundamental Concepts of Algebra Scientific Notation: Numbers written as a x 10 n where 1 < a < 10 and n is an integer If n is negative, the number is small;
More informationCHAPTER 1 NUMBER SYSTEMS. 1.1 Introduction
NUMBER SYSTEMS NUMBER SYSTEMS CHAPTER. Introduction In your earlier classes, you have learnt about the number line and how to represent various types of numbers on it (see Fig..). Fig.. : The number line
More information1.3. Properties of Real Numbers Properties by the Pound. My Notes ACTIVITY
Properties of Real Numbers SUGGESTED LEARNING STRATEGIES: Create Representations, Activating Prior Knowledge, Think/Pair/Share, Interactive Word Wall The local girls track team is strength training by
More informationMultiple Choice Questions for Review
Review Questions Multiple Choice Questions for Review In each case there is one correct answer (given at the end of the problem set). Try to work the problem first without looking at the answer. Understand
More informationLecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties
Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics. Introductory Notes in Discrete Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics Introductory Notes in Discrete Mathematics Marcel B. Finan c All Rights Reserved Last Updated April 6, 2016 Preface
More information1. Propositional logic and equivalences (1.1 and 1.2)
COT300 Practice Problems for Exam. These problems are only meant to help you prepare the first exam. It is not guaranteed that the exam questions will be similar to these problems. There will be five problems
More informationModule 1: Basic Logic. Theme 1: Propositions. English sentences are either true or false or neither. Consider the following sentences:
Module 1: Basic Logic Theme 1: Propositions English sentences are either true or false or neither. Consider the following sentences: 1. Warsaw is the capital of Poland. 2. 2+5=3. 3. How are you? The first
More informationPredicate Logic. Example: All men are mortal. Socrates is a man. Socrates is mortal.
Predicate Logic Example: All men are mortal. Socrates is a man. Socrates is mortal. Note: We need logic laws that work for statements involving quantities like some and all. In English, the predicate is
More informationDeductive Reasoning. Chapter 2.5, Theorems, Proofs, and Logic
Deductive Reasoning Chapter 2.5, 2.6  Theorems, Proofs, and Logic Theorems and Proofs Deductive reasoning is based on strict rules that guarantee certainty Well, a guarantee relative to the certainty
More information2.) 5000, 1000, 200, 40, 3.) 1, 12, 123, 1234, 4.) 1, 4, 9, 16, 25, Draw the next figure in the sequence. 5.)
Chapter 2 Geometry Notes 2.1/2.2 Patterns and Inductive Reasoning and Conditional Statements Inductive reasoning: looking at numbers and determining the next one Conjecture: sometimes thought of as an
More informationMATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
More informationvertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws
Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,
More informationGeometry  Chapter 2 Review
Name: Class: Date: Geometry  Chapter 2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine if the conjecture is valid by the Law of Syllogism.
More informationChapter 1 Introductory Information and Review
SECTION 1.1 Numbers Chapter 1 Introductory Information and Review Section 1.1: Numbers Types of Numbers Order on a Number Line Types of Numbers Natural Numbers: MATH 1300 Fundamentals of Mathematics 1
More informationThe set consisting of all natural numbers that are in A and are in B is the set f1; 3; 5g;
Chapter 5 Set Theory 5.1 Sets and Operations on Sets Preview Activity 1 (Set Operations) Before beginning this section, it would be a good idea to review sets and set notation, including the roster method
More informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More informationDefinition of Subtraction x  y = x + 1y2. Subtracting Real Numbers
Algebra Review Numbers FRACTIONS Addition and Subtraction i To add or subtract fractions with the same denominator, add or subtract the numerators and keep the same denominator ii To add or subtract fractions
More informationp c p c p c c p p c h 1 h 2 h n c? 1. p c 2. (p c) is a tautology. Direct 3. ( p c) is a tautology. Direct 4. ( c p)is a tautology.
Proof Methods Methods of proof Direct Direct Contrapositive Contradiction p c p c p c c p p c Section 1.6 & 1.7 T T T T T F T F F F F T F T T T T F F F T T T F MSU/CSE 260 Fall 2009 1 MSU/CSE 260 Fall
More informationA Primer on Mathematical Proof
A Primer on Mathematical Proof A proof is an argument to convince your audience that a mathematical statement is true. It can be a calculation, a verbal argument, or a combination of both. In comparison
More informationCS103X: Discrete Structures Homework Assignment 1: Solutions
CS103X: Discrete Structures Homework Assignment 1: Solutions Due January, 008 Exercise 1 (10 Points). Prove or give a counterexample for each of the following: (a) If A B and B C, then A C. (b) If A B
More informationBasic Properties of Rings
LECTURE 15 Basic Properties of Rings Theorem 15.1. For any element a in a ring R, the equation a + x 0 R has a unique solution. We know that a + x 0 R has at least one solution u R by Axiom (5) in the
More informationLogic, Proofs, and Sets
Logic, Proofs, and Sets JWR Tuesday August 29, 2000 1 Logic A statement of form if P, then Q means that Q is true whenever P is true. The converse of this statement is the related statement if Q, then
More informationProof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.
Math 232  Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the
More informationCHAPTER 1. Logic, Proofs Propositions
CHAPTER 1 Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: Paris is in France (true), London
More informationLogic will get you from A to B. Imagination will take you everywhere.
Chapter 3 Predicate Logic Logic will get you from A to B. Imagination will take you everywhere. A. Einstein In the previous chapter, we studied propositional logic. This chapter is dedicated to another
More informationAs usual, Burton refers to the Seventh Edition of the course text by Burton (the page numbers for the Sixth Edition may be off slightly).
Math 5 Spring 00 R. Schultz SOLUTIONS TO EXERCISES FROM math5exercises0.pdf As usual, Burton refers to the Seventh Edition of the course text by Burton (the page numbers for the Sixth Edition may be off
More informationIntroduction to Computers and Programming. Proof by Truth Table
Introduction to Computers and Programming Prof. I. K. Lundqvist Lecture May 5 4 Proof by Truth Table Proposition x Æ y and ( x) y are logically equivalent x y xæy x ( x) y Definitions Even An integer n
More information3.1 Trivial and Vacuous Proofs
CH3: DIRECT PROOF AND PROOF BY CONTRAPOSITIVE Lemma = is a mathematical result that is useful in verifying the truth of another result. Theorem/ Proposition = a true mathematical statement (that are especially
More informationGeometry 1A. Homework 1.6 and 1.7
Geometry 1A Name Homework 1.6 and 1.7 Identify the hypothesis and conclusion of each conditional. 1. If you want to be fit, then get plenty of exercise. 2. If x + 20 = 32, then x = 12. 3. If a triangle
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationOPERATIONS AND PROPERTIES
CHAPTER OPERATIONS AND PROPERTIES Jesse is fascinated by number relationships and often tries to find special mathematical properties of the fivedigit number displayed on the odometer of his car. Today
More information