Math 3000 Section 003 Intro to Abstract Math Homework 2

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Math 3000 Section 003 Intro to Abstract Math Homework 2"

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 non-mathematical sentences is quite difficult: for example, sentences like If Justin Bieber sang during the half-time 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 half-time 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 T-shape). 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 two-dimensional 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.

INTRODUCTORY SET THEORY

INTRODUCTORY 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 information

Mathematical Induction

Mathematical 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 information

Lecture 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 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 information

A 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 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 information

INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

More information

Math 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 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 information

Logic Appendix. Section 1 Truth Tables CONJUNCTION EXAMPLE 1

Logic 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 information

Solutions for Practice problems on proofs

Solutions 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 information

Understanding Basic Calculus

Understanding Basic Calculus Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

More information

Solutions Manual for How to Read and Do Proofs

Solutions 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 information

Mathematics 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 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 information

Section 9.5: Equations of Lines and Planes

Section 9.5: Equations of Lines and Planes Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that

More information

MATH 90 CHAPTER 1 Name:.

MATH 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

http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304

http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304 MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio

More information

MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides

More information

Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Mathematics Review for MS Finance 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 information

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v, 1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

More information

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT 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 information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

The Pointless Machine and Escape of the Clones

The Pointless Machine and Escape of the Clones MATH 64091 Jenya Soprunova, KSU The Pointless Machine and Escape of the Clones The Pointless Machine that operates on ordered pairs of positive integers (a, b) has three modes: In Mode 1 the machine adds

More information

Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

More information

Algebra and Geometry Review (61 topics, no due date)

Algebra and Geometry Review (61 topics, no due date) Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

Propositional Logic. A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both.

Propositional 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 information

This chapter is all about cardinality of sets. At first this looks like a

This chapter is all about cardinality of sets. At first this looks like a CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

More information

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

More information

Predicate 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. 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 information

FURTHER VECTORS (MEI)

FURTHER VECTORS (MEI) Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

More information

Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

More information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

MTH6120 Further Topics in Mathematical Finance Lesson 2

MTH6120 Further Topics in Mathematical Finance Lesson 2 MTH6120 Further Topics in Mathematical Finance Lesson 2 Contents 1.2.3 Non-constant interest rates....................... 15 1.3 Arbitrage and Black-Scholes Theory....................... 16 1.3.1 Informal

More information

LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

More information

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

More information

Factoring Polynomials

Factoring Polynomials Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent

More information

Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome 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 information

E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

More information

Algebra I. In this technological age, mathematics is more important than ever. When students

Algebra I. In this technological age, mathematics is more important than ever. When students In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

More information

Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I. Ronald van Luijk, 2012 Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

More information

MATH 221 FIRST SEMESTER CALCULUS. fall 2007

MATH 221 FIRST SEMESTER CALCULUS. fall 2007 MATH 22 FIRST SEMESTER CALCULUS fall 2007 Typeset:December, 2007 2 Math 22 st Semester Calculus Lecture notes version.0 (Fall 2007) This is a self contained set of lecture notes for Math 22. The notes

More information

Basic Components of an LP:

Basic Components of an LP: 1 Linear Programming Optimization is an important and fascinating area of management science and operations research. It helps to do less work, but gain more. Linear programming (LP) is a central topic

More information

5. Factoring by the QF method

5. Factoring by the QF method 5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the

More information

What are the place values to the left of the decimal point and their associated powers of ten?

What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

More information

Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

More information

MATH 21. College Algebra 1 Lecture Notes

MATH 21. College Algebra 1 Lecture Notes MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

More information

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

More information

Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

12.5 Equations of Lines and Planes

12.5 Equations of Lines and Planes Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P

More information

8-3 Dot Products and Vector Projections

8-3 Dot Products and Vector Projections 8-3 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v

More information

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

More information

Groups in Cryptography

Groups in Cryptography Groups in Cryptography Çetin Kaya Koç http://cs.ucsb.edu/~koc/cs178 koc@cs.ucsb.edu Koç (http://cs.ucsb.edu/~koc) ucsb cs 178 intro to crypto winter 2013 1 / 13 Groups in Cryptography A set S and a binary

More information

List the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated

List the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible

More information

An Innocent Investigation

An 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 information

EVERY DAY COUNTS CALENDAR MATH 2005 correlated to

EVERY DAY COUNTS CALENDAR MATH 2005 correlated to EVERY DAY COUNTS CALENDAR MATH 2005 correlated to Illinois Mathematics Assessment Framework Grades 3-5 E D U C A T I O N G R O U P A Houghton Mifflin Company YOUR ILLINOIS GREAT SOURCE REPRESENTATIVES:

More information

Tennessee Mathematics Standards 2009-2010 Implementation. Grade Six Mathematics. Standard 1 Mathematical Processes

Tennessee Mathematics Standards 2009-2010 Implementation. Grade Six Mathematics. Standard 1 Mathematical Processes Tennessee Mathematics Standards 2009-2010 Implementation Grade Six Mathematics Standard 1 Mathematical Processes GLE 0606.1.1 Use mathematical language, symbols, and definitions while developing mathematical

More information

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

More information

Grade 6 Mathematics Assessment. Eligible Texas Essential Knowledge and Skills

Grade 6 Mathematics Assessment. Eligible Texas Essential Knowledge and Skills Grade 6 Mathematics Assessment Eligible Texas Essential Knowledge and Skills STAAR Grade 6 Mathematics Assessment Mathematical Process Standards These student expectations will not be listed under a separate

More information

Performance Level Descriptors Grade 6 Mathematics

Performance Level Descriptors Grade 6 Mathematics Performance Level Descriptors Grade 6 Mathematics Multiplying and Dividing with Fractions 6.NS.1-2 Grade 6 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with

More information

AMBIGUOUS CLASSES IN QUADRATIC FIELDS

AMBIGUOUS CLASSES IN QUADRATIC FIELDS MATHEMATICS OF COMPUTATION VOLUME, NUMBER 0 JULY 99, PAGES -0 AMBIGUOUS CLASSES IN QUADRATIC FIELDS R. A. MOLLIN Dedicated to the memory ofd. H. Lehmer Abstract. We provide sufficient conditions for the

More information

NIM with Cash. Abstract. loses. This game has been well studied. For example, it is known that for NIM(1, 2, 3; n)

NIM with Cash. Abstract. loses. This game has been well studied. For example, it is known that for NIM(1, 2, 3; n) NIM with Cash William Gasarch Univ. of MD at College Park John Purtilo Univ. of MD at College Park Abstract NIM(a 1,..., a k ; n) is a -player game where initially there are n stones on the board and the

More information

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m) Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

More information

The positive minimum degree game on sparse graphs

The positive minimum degree game on sparse graphs The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University

More information

Some Lecture Notes and In-Class Examples for Pre-Calculus:

Some Lecture Notes and In-Class Examples for Pre-Calculus: Some Lecture Notes and In-Class Examples for Pre-Calculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax

More information

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space 11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of

More information

You know from calculus that functions play a fundamental role in mathematics.

You know from calculus that functions play a fundamental role in mathematics. CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.

More information

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

More information

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 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 information

Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School

Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School Middle School 111.B. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School Statutory Authority: The provisions of this Subchapter B issued under the Texas Education

More information

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

More information

Review Sheet for Test 1

Review Sheet for Test 1 Review Sheet for Test 1 Math 261-00 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And

More information

Factoring Patterns in the Gaussian Plane

Factoring Patterns in the Gaussian Plane Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood

More information

1.1 Identify Points, Lines, and Planes

1.1 Identify Points, Lines, and Planes 1.1 Identify Points, Lines, and Planes Objective: Name and sketch geometric figures. Key Vocabulary Undefined terms - These words do not have formal definitions, but there is agreement aboutwhat they mean.

More information

Successful completion of Math 7 or Algebra Readiness along with teacher recommendation.

Successful completion of Math 7 or Algebra Readiness along with teacher recommendation. MODESTO CITY SCHOOLS COURSE OUTLINE COURSE TITLE:... Basic Algebra COURSE NUMBER:... RECOMMENDED GRADE LEVEL:... 8-11 ABILITY LEVEL:... Basic DURATION:... 1 year CREDIT:... 5.0 per semester MEETS GRADUATION

More information

Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.

Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle. Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.

More information

9 Multiplication of Vectors: The Scalar or Dot Product

9 Multiplication of Vectors: The Scalar or Dot Product Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation

More information

WOLLONGONG COLLEGE AUSTRALIA. Diploma in Information Technology

WOLLONGONG COLLEGE AUSTRALIA. Diploma in Information Technology First Name: Family Name: Student Number: Class/Tutorial: WOLLONGONG COLLEGE AUSTRALIA A College of the University of Wollongong Diploma in Information Technology Final Examination Spring Session 2008 WUCT121

More information

CHAPTER 7 GENERAL PROOF SYSTEMS

CHAPTER 7 GENERAL PROOF SYSTEMS CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes

More information

Domain of a Composition

Domain of a Composition Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such

More information

Tiers, Preference Similarity, and the Limits on Stable Partners

Tiers, Preference Similarity, and the Limits on Stable Partners Tiers, Preference Similarity, and the Limits on Stable Partners KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke February 7, 2010 Preliminary and incomplete. Do not circulate. Abstract We consider

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

More information

ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION

ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION Aldrin W. Wanambisi 1* School of Pure and Applied Science, Mount Kenya University, P.O box 553-50100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,

More information

SAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3?

SAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3? SAT Math Hard Practice Quiz Numbers and Operations 5. How many integers between 10 and 500 begin and end in 3? 1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

1 Sets and Set Notation.

1 Sets and Set Notation. LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

More information

Functions and their Graphs

Functions and their Graphs Functions and their Graphs Functions All of the functions you will see in this course will be real-valued functions in a single variable. A function is real-valued if the input and output are real numbers

More information

Notes on Factoring. MA 206 Kurt Bryan

Notes on Factoring. MA 206 Kurt Bryan The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

More information

3 1. Note that all cubes solve it; therefore, there are no more

3 1. Note that all cubes solve it; therefore, there are no more Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if

More information

GRADES 7, 8, AND 9 BIG IDEAS

GRADES 7, 8, AND 9 BIG IDEAS Table 1: Strand A: BIG IDEAS: MATH: NUMBER Introduce perfect squares, square roots, and all applications Introduce rational numbers (positive and negative) Introduce the meaning of negative exponents for

More information

1 Short Introduction to Time Series

1 Short Introduction to Time Series ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The

More information

discuss how to describe points, lines and planes in 3 space.

discuss how to describe points, lines and planes in 3 space. Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position

More information

Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

More information

11 Ideals. 11.1 Revisiting Z

11 Ideals. 11.1 Revisiting Z 11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

More information

What to Expect on the Compass

What to Expect on the Compass What to Expect on the Compass What is the Compass? COMPASS is a set of untimed computer adaptive tests created by the American College Test (ACT) Program. Because COMPASS tests are "computer adaptive,"

More information

Read this syllabus very carefully. If there are any reasons why you cannot comply with what I am requiring, then talk with me about this at once.

Read this syllabus very carefully. If there are any reasons why you cannot comply with what I am requiring, then talk with me about this at once. LOGIC AND CRITICAL THINKING PHIL 2020 Maymester Term, 2010 Daily, 9:30-12:15 Peabody Hall, room 105 Text: LOGIC AND RATIONAL THOUGHT by Frank R. Harrison, III Professor: Frank R. Harrison, III Office:

More information

LINES AND PLANES IN R 3

LINES AND PLANES IN R 3 LINES AND PLANES IN R 3 In this handout we will summarize the properties of the dot product and cross product and use them to present arious descriptions of lines and planes in three dimensional space.

More information

Florida Math for College Readiness

Florida Math for College Readiness Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

More information

Math 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction

Math 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a

More information

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

More information

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

More information

Math 166 - Week in Review #4. A proposition, or statement, is a declarative sentence that can be classified as either true or false, but not both.

Math 166 - Week in Review #4. A proposition, or statement, is a declarative sentence that can be classified as either true or false, but not both. Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166 - Week in Review #4 Sections A.1 and A.2 - Propositions, Connectives, and Truth Tables A proposition, or statement, is a declarative sentence that

More information