Quantifiers are used to describe variables in statements. - The universal quantifier means for all. - The existential quantifier means there exists.
|
|
- Grace Norton
- 7 years ago
- Views:
Transcription
1 11 Quantifiers are used to describe variables in statements. - The universal quantifier means for all. - The existential quantifier means there exists. The phrases, for all x in R if x is an arbitrary element of R for every x in R let x be in R can be translated symbolically into: x R. The phrases, there exists an x in R such that for some x in R there is an x in R such that can be translated symbolically into: x R. More than one quantifier. We may say ( x E)( k Z)x = 2k or to mean the same thing, we could use commas x E, k Z, x = 2k Once a variable is quantified it is fixed for the remainder of the statement. For example, ( x R)( y R)( z R)x 2 + y 2 = z 2. This is a true statement. Proof: Let z = x 2 + y 2. The order of the quantifiers counts. ( y R)( x R)x = y 2 ( x R)( y R)x = y 2 The first is false and the second is true.
2 12 The negation of quantifiers. ( x, Q) = x, Q ( x, Q) = x, Q Example 8. Q: Every good boy does fine. Symbolic translation: x G, x does fine. Negation: x G, x does not do fine. Rewrite the negation in words: There are some good boys that do not do fine. Q Q ( P ) Notice that Q and its negation satisfy: T F F T Example 9. P: There is a broken chair in this room. Symbolic translation: x C, x is broken. Where C is the set of chairs in this room. Negation: x C, x is not broken. Rewrite the negation in words: All chairs in this room are not broken. Or, we might say, No chair in this room is broken. P P ( P ) Notice that P and its negation satisfy: T F F T Negation of double quantifiers. [ n N, x A, nx < 1] n N, [ x A, nx < 1] n N, x A, [nx < 1] n N, x A, nx 1
3 13 - Implication. P Q P Q T T T T F F F T T F F T Implication The phrases, P implies Q If P then Q Q if P Q is implied by P Q is true whenever P is true P only if Q can be translated symbolically into: P Q. In the implication P Q, P is the antecedent or hypothesis and Q is the consequence or conclusion. The Converse of P Q. is Q P. The Contrapositive of P Q. is ( Q) ( P ). Example 10. P: it is sunny Q: there is a ball game The implication. P implies Q: If it is sunny then there is a ball game. Its inverse: Q implies P: If there is a ball game then it is sunny. In this case, we could also say, There is a ball game only if it is sunny. Its contrapositive: not Q implies not P: If there is no ball game then it is not sunny. There is a ball game only if it is sunny.
4 14 The implication, P Q, and its contrapositive, Q P, have the same truth value. Lets look at the truth table. P Q P Q ( Q) ( P ) P Q T T F F T T T F F T F F F T T F T T F F T T T T Example 11. Show (P Q) has the same truth value as P ( Q). Example 12. P: If x < 0 then x = x. Negate P. Example 13. Q: f(x) = f(y) implies x = y. Negate Q.
5 P Q is said to be vacuously true when P is false. For example, If a snowball survives in hell, I will give everyone an A. Example 14. Use a truth table to prove that ((P Q) (Q R)) (P R) For the following two examples, let O be the odd integers and E be the even integers. Example 15. Determine if the following is true or false. xy O (x O y O) Example 16. Determine if the following is true or false. xy E (x E y E) If and only if statements. - If and only if. P Q P Q T T T T F F F T F F F T The phrases, P if and only if Q If P then Q and if Q then P P Q is the only if part Q P is the if part can be translated symbolically into: P Q. 15
6 16 Some word problems: Example 17. The following sentence appeared on a restaurant menu: Please note that every alternative may not be available at this time. Describe the unintended meaning. Rewrite the sentence to state the intended meaning clearly. Unintended meanings: Rewrite: Please note that some alternatives may not be available at this time. Not every option is available currently Please note that not every alternative may be available at this time. Every offered alternative will be available later. Please note that alternative items are subject to availability. Example 18. Give an example of an English sentence that has different meanings depending on inflection, pronunciation, or context. Everyone is wrong some of the time I lie here often. Example 19. From outside mathematics, give an example of statements A, B, C such that A and B together imply C, but such that neither A nor B alone implies C. A: I love you. B: You love me. C: We love each other. A: Dog is hungry. B: Food is put out. C: Dog is eating. A: Person x is a natural born US Citizen. B: Person x is 38 years old C: Person x can be president of the US A: It is below 32 degrees. B: there is precipitation. C: It will snow. A: She is a good teacher. B: She is a good mathematician. C: She is a good math teacher. Example 20. Negate the statement no slow learners attend this school is:
7 Example 21. A fraternity has a rule for new members: each must always tell the truth or always lie. They know who does which. If I meet three of them on the street and they make the statements below, which ones (if any) should I believe? A) Says: All three of us are liars. B) Says: Exactly two of us are liars C) Says: The other two are liars. 17
8 18 Examples from 1.4, 1.5, and Express each of the following statements as a conditional statement in if-then form or as a universally quantified statement. Also write the negation. (1) Every odd number is prime. ( x odd integers ) x is prime. Negation: ( x odd integers ) x is not prime. (2) The sum of the angles of a triangle is 180 degrees. If x, y, z are the angles of a triangle, then their sum is 180 degrees. Negation: x, y, z are the angles of a triangle and their sum is not 180 degrees. (3) Passing the test requires solving all the problems. If you passed the test then you solved all the problems. Negation: You passed the test and you did not solve all the problems. (4) Being first in line guarantees getting a good seat. If you are first in line, then you will get a good seat. Negation: You are first in line and you do not get a good seat. (5) Lockers must be turned in by the last day of class. ( x lockers) x is turned in by last day of class. Negation: ( x lockers ) x is not turned in by the last day of class. (6) Haste makes waste. ( x haste) x makes waste Negation: ( x haste ) x does not make waste. (7) I get mad whenever you do that. If you do that then I get mad. Sometimes you do that and I do not get mad. (8) I won t say that unless I mean it. If I mean that then I will say that. Sometimes I won t say that and I mean that. 2. Which of these statements are believable? (1) All of my 5-legged dogs can fly. true (2) I have no 5-legged dog that cannot fly. true (3) Some of my 5-legged dogs cannot fly. false (4) I have a 5-legged dog that cannot fly. false 3. Prove that if x and y are distinct real numbers, then (x + 1) 2 = (y + 1) 2 if and only if x + y = 2.
9 Proof: We start with (x + 1) 2 = (y + 1) 2. Expanding both sides, we get x 2 + 2x + 1 = y 2 + 2y + 1. Canceling the 1 s and rearranging terms, we get x 2 y 2 = 2y 2x. We factor each side to get (x + y)(x y) = 2(y x) = 2(x y). As we are given x y, we can divide both sides by x y to get x + y = 2. As equations preserve equality and these steps are reversible, x + y = 2 implies (x + 1) 2 = (y + 1) 2, so the dependence holds both ways. 4. Given a real number x, let A be the statement 1/2 < x < 5/2, let B be the statement x Z, let C be the statement x 2 = 1, and let D be the statement x = 2. Which statements below are true for all x R. (1) A C. False. For a counterexample, let x = 2. (2) B C. False. For a counterexample, let x = 2. (3) (A B) C. False. For a counterexample, let x = 2. (4) (A B) (C D). True. If A and B are true, then x = 1 or x = 2. If x = 1 then C is true. If x = 2 then D is true. So in either case, either C or D is true. (5) C (A B). False. For a counterexample let x = 1. (6) D [A B ( C)]. True. If x = 2 then A is true, B is true, and x 2 1 so C is true. (7) (A C) B. True. A C means that x = 1, 2, or 1. So, B is true. 5. Let x, y be integers. Determine the truth value of each statement below. (1) xy is odd if and only if x and y are odd. TRUE. If x is odd and y is odd then x = 2a + 1 for some a and y = 2b + 1 for some b. Then xy = (2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1 which is odd. Suppose it is not true that both x and y are odd. Then at least one is even. Without loss of generality, suppose x is even. Then x = 2a for some a. xy = 2ay which is even. (2) xy is even if and only if x and y are even. FALSE. For a counterexample to the only if part, let x = 3 and y = 4. Then, xy = 12 which is even. 19
CHAPTER 2. Logic. 1. Logic Definitions. Notation: Variables are used to represent propositions. The most common variables used are p, q, and r.
CHAPTER 2 Logic 1. Logic Definitions 1.1. Propositions. Definition 1.1.1. A proposition is a declarative sentence that is either true (denoted either T or 1) or false (denoted either F or 0). Notation:
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. 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 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 information3. Logical Reasoning in Mathematics
3. Logical Reasoning in Mathematics Many state standards emphasize the importance of reasoning. We agree disciplined mathematical reasoning is crucial to understanding and to properly using mathematics.
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 informationMath 3000 Section 003 Intro to Abstract Math Homework 2
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
More informationIf n is odd, then 3n + 7 is even.
Proof: Proof: We suppose... that 3n + 7 is even. that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. that 3n + 7 is even. Since n is odd, there exists an integer k so that
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More informationSolutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014
Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the
More information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationBasic Proof Techniques
Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document
More informationLogic is a systematic way of thinking that allows us to deduce new information
CHAPTER 2 Logic Logic is a systematic way of thinking that allows us to deduce new information from old information and to parse the meanings of sentences. You use logic informally in everyday life and
More informationThe last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
More informationMathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
More informationWRITING PROOFS. Christopher Heil Georgia Institute of Technology
WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this
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 informationReasoning and Proof Review Questions
www.ck12.org 1 Reasoning and Proof Review Questions Inductive Reasoning from Patterns 1. What is the next term in the pattern: 1, 4, 9, 16, 25, 36, 49...? (a) 81 (b) 64 (c) 121 (d) 56 2. What is the next
More informationLecture Notes in Discrete Mathematics. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Lecture Notes in Discrete Mathematics Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 Preface This book is designed for a one semester course in discrete mathematics for sophomore or junior
More informationSYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me
SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBIN-CAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationDigitalCommons@University of Nebraska - Lincoln
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-007 Pythagorean Triples Diane Swartzlander University
More informationShow all work for credit. Attach paper as needed to keep work neat & organized.
Geometry Semester 1 Review Part 2 Name Show all work for credit. Attach paper as needed to keep work neat & organized. Determine the reflectional (# of lines and draw them in) and rotational symmetry (order
More informationalternate interior angles
alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationSOLUTIONS FOR PROBLEM SET 2
SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such
More informationVocabulary. Term Page Definition Clarifying Example. biconditional statement. conclusion. conditional statement. conjecture.
CHAPTER Vocabulary The table contains important vocabulary terms from Chapter. As you work through the chapter, fill in the page number, definition, and a clarifying example. biconditional statement conclusion
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
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 informationChapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications
More informationSECTION 10-2 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
More informationIntroduction. Appendix D Mathematical Induction D1
Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to
More informations = 1 + 2 +... + 49 + 50 s = 50 + 49 +... + 2 + 1 2s = 51 + 51 +... + 51 + 51 50 51. 2
1. Use Euler s trick to find the sum 1 + 2 + 3 + 4 + + 49 + 50. s = 1 + 2 +... + 49 + 50 s = 50 + 49 +... + 2 + 1 2s = 51 + 51 +... + 51 + 51 Thus, 2s = 50 51. Therefore, s = 50 51. 2 2. Consider the sequence
More informationMathematical Induction
Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How
More informationClifton High School Mathematics Summer Workbook Algebra 1
1 Clifton High School Mathematics Summer Workbook Algebra 1 Completion of this summer work is required on the first day of the school year. Date Received: Date Completed: Student Signature: Parent Signature:
More informationSolutions Manual for How to Read and Do Proofs
Solutions Manual for How to Read and Do Proofs An Introduction to Mathematical Thought Processes Sixth Edition Daniel Solow Department of Operations Weatherhead School of Management Case Western Reserve
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationStupid Divisibility Tricks
Stupid Divisibility Tricks 101 Ways to Stupefy Your Friends Appeared in Math Horizons November, 2006 Marc Renault Shippensburg University Mathematics Department 1871 Old Main Road Shippensburg, PA 17013
More information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More informationp: I am elected q: I will lower the taxes
Implication Conditional Statement p q (p implies q) (if p then q) is the proposition that is false when p is true and q is false and true otherwise. Equivalent to not p or q Ex. If I am elected then I
More informationStudent Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain
More informationThe Deadly Sins of Algebra
The Deadly Sins of Algebra There are some algebraic misconceptions that are so damaging to your quantitative and formal reasoning ability, you might as well be said not to have any such reasoning ability.
More informationdef: An axiom is a statement that is assumed to be true, or in the case of a mathematical system, is used to specify the system.
Section 1.5 Methods of Proof 1.5.1 1.5 METHODS OF PROOF Some forms of argument ( valid ) never lead from correct statements to an incorrect. Some other forms of argument ( fallacies ) can lead from true
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What
More information1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]
1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,
More informationSolutions for Practice problems on proofs
Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some
More informationApplications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
More informationFull and Complete Binary Trees
Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full
More informationMental Questions. Day 1. 1. What number is five cubed? 2. A circle has radius r. What is the formula for the area of the circle?
Mental Questions 1. What number is five cubed? KS3 MATHEMATICS 10 4 10 Level 8 Questions Day 1 2. A circle has radius r. What is the formula for the area of the circle? 3. Jenny and Mark share some money
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationAdvanced GMAT Math Questions
Advanced GMAT Math Questions Version Quantitative Fractions and Ratios 1. The current ratio of boys to girls at a certain school is to 5. If 1 additional boys were added to the school, the new ratio of
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationUmmmm! Definitely interested. She took the pen and pad out of my hand and constructed a third one for herself:
Sum of Cubes Jo was supposed to be studying for her grade 12 physics test, but her soul was wandering. Show me something fun, she said. Well I wasn t sure just what she had in mind, but it happened that
More informationIf an English sentence is ambiguous, it may allow for more than one adequate transcription.
Transcription from English to Predicate Logic General Principles of Transcription In transcribing an English sentence into Predicate Logic, some general principles apply. A transcription guide must be
More informationA Brief Introduction to Mathematical Writing
A Brief Introduction to Mathematical Writing William J. Turner August 26, 2009 1 Introduction Mathematics is not just about computation. It is about ideas, knowledge, learning, understanding, and perception.
More informationLINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL
Chapter 6 LINEAR INEQUALITIES 6.1 Introduction Mathematics is the art of saying many things in many different ways. MAXWELL In earlier classes, we have studied equations in one variable and two variables
More informationLogic Appendix. Section 1 Truth Tables CONJUNCTION EXAMPLE 1
Logic Appendix T F F T Section 1 Truth Tables Recall that a statement is a group of words or symbols that can be classified collectively as true or false. The claim 5 7 12 is a true statement, whereas
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More informationInvalidity in Predicate Logic
Invalidity in Predicate Logic So far we ve got a method for establishing that a predicate logic argument is valid: do a derivation. But we ve got no method for establishing invalidity. In propositional
More informationAPPENDIX 1 PROOFS IN MATHEMATICS. A1.1 Introduction 286 MATHEMATICS
286 MATHEMATICS APPENDIX 1 PROOFS IN MATHEMATICS A1.1 Introduction Suppose your family owns a plot of land and there is no fencing around it. Your neighbour decides one day to fence off his land. After
More informationChapter 7. Functions and onto. 7.1 Functions
Chapter 7 Functions and onto This chapter covers functions, including function composition and what it means for a function to be onto. In the process, we ll see what happens when two dissimilar quantifiers
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 informationMathematics Georgia Performance Standards
Mathematics Georgia Performance Standards K-12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by
More information25 Integers: Addition and Subtraction
25 Integers: Addition and Subtraction Whole numbers and their operations were developed as a direct result of people s need to count. But nowadays many quantitative needs aside from counting require numbers
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationDay 1. 1. What number is five cubed? 2. A circle has radius r. What is the formula for the area of the circle?
Mental Arithmetic Questions 1. What number is five cubed? KS3 MATHEMATICS 10 4 10 Level 7 Questions Day 1 2. A circle has radius r. What is the formula for the area of the circle? 3. Jenny and Mark share
More informationCHAPTER 2: METHODS OF PROOF
CHAPTER 2: METHODS OF PROOF Section 2.1: BASIC PROOFS WITH QUANTIFIERS Existence Proofs Our first goal is to prove a statement of the form ( x) P (x). There are two types of existence proofs: Constructive
More informationBlack Problems - Prime Factorization, Greatest Common Factor and Simplifying Fractions
Black Problems Prime Factorization, Greatest Common Factor and Simplifying Fractions A natural number n, such that n >, can t be written as the sum of two more consecutive odd numbers if and only if n
More informationPrime Factorization 0.1. Overcoming Math Anxiety
0.1 Prime Factorization 0.1 OBJECTIVES 1. Find the factors of a natural number 2. Determine whether a number is prime, composite, or neither 3. Find the prime factorization for a number 4. Find the GCF
More informationPropositional Logic. A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both.
irst Order Logic Propositional Logic A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both. Are the following sentences propositions? oronto
More informationVerbal Phrases to Algebraic Expressions
Student Name: Date: Contact Person Name: Phone Number: Lesson 13 Verbal Phrases to s Objectives Translate verbal phrases into algebraic expressions Solve word problems by translating sentences into equations
More informationDirect Translation is the process of translating English words and phrases into numbers, mathematical symbols, expressions, and equations.
Section 1 Mathematics has a language all its own. In order to be able to solve many types of word problems, we need to be able to translate the English Language into Math Language. is the process of translating
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationDay 1. Mental Arithmetic Questions. 1. What number is five cubed? 2. A circle has radius r. What is the formula for the area of the circle?
Mental Arithmetic Questions 1. What number is five cubed? KS3 MATHEMATICS 10 4 10 Level 6 Questions Day 1 2. A circle has radius r. What is the formula for the area of the circle? 3. Jenny and Mark share
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 informationSample Induction Proofs
Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given
More informationMaths Workshop for Parents 2. Fractions and Algebra
Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)
More informationChapter 1. Use the following to answer questions 1-5: In the questions below determine whether the proposition is TRUE or FALSE
Use the following to answer questions 1-5: Chapter 1 In the questions below determine whether the proposition is TRUE or FALSE 1. 1 + 1 = 3 if and only if 2 + 2 = 3. 2. If it is raining, then it is raining.
More informationDISCRETE MATH: LECTURE 3
DISCRETE MATH: LECTURE 3 DR. DANIEL FREEMAN 1. Chapter 2.2 Conditional Statements If p and q are statement variables, the conditional of q by p is If p then q or p implies q and is denoted p q. It is false
More information2.1. Inductive Reasoning EXAMPLE A
CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers
More information26 Integers: Multiplication, Division, and Order
26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue
More informationnorth seattle community college
INTRODUCTION TO FRACTIONS If we divide a whole number into equal parts we get a fraction: For example, this circle is divided into quarters. Three quarters, or, of the circle is shaded. DEFINITIONS: The
More information1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style
Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with
More informationWelcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013
Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move
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 informationAn Innocent Investigation
An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number
More informationCommutative Property Grade One
Ohio Standards Connection Patterns, Functions and Algebra Benchmark E Solve open sentences and explain strategies. Indicator 4 Solve open sentences by representing an expression in more than one way using
More informationMathematical Induction. Mary Barnes Sue Gordon
Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by
More informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the
More informationI remember that when I
8. Airthmetic and Geometric Sequences 45 8. ARITHMETIC AND GEOMETRIC SEQUENCES Whenever you tell me that mathematics is just a human invention like the game of chess I would like to believe you. But I
More informationThe thing that started it 8.6 THE BINOMIAL THEOREM
476 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers (b) Based on your results for (a), guess the minimum number of moves required if you start with an arbitrary number of n disks.
More informationPythagorean Theorem: Proof and Applications
Pythagorean Theorem: Proof and Applications Kamel Al-Khaled & Ameen Alawneh Department of Mathematics and Statistics, Jordan University of Science and Technology IRBID 22110, JORDAN E-mail: kamel@just.edu.jo,
More informationMATH 90 CHAPTER 1 Name:.
MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.
More information