Translation Guide. Not both P and Q ~(P Q) Not either P or Q (neither/nor)

Size: px
Start display at page:

Download "Translation Guide. Not both P and Q ~(P Q) Not either P or Q (neither/nor)"

Transcription

1 Translation Guide If P, then Q (P Q) P, if Q (Q P) What follows if is the antecedent of a conditional. P, provided that Q (Q P) Provided that means if. Assuming that, given that, etc., work the same way. Since they all mean if, what follows them is the antecedent of a conditional. P only if Q (P Q) This says that P is true only if Q is true. So, if P is true, so is Q. Unlike iff, it doesn t tell us that P is false only when Q is false. So, What follows only if is the consequent of a conditional. P if and only ( iff ) Q (P Q) Note that this is equivalent to the conjunction of P if Q --i.e., (Q P), and P only if Q --i.e., (P Q). P unless Q (P v Q) or (~Q P) Think: P is true, unless Q is, i.e., P is true if Q isn t, i.e., (~Q P). But (~Q P) is logically equivalent to (P v Q). While either answer is correct, it is simpler to remember that unless just means or. P just if Q (P Q) My ear tells me just if is more like only if than if and only if. But this is how the author wants it translated, and this is what is expected in Logicola. I won t put examples of just if on the test. P is sufficient for Q ( enough ) (P Q) P is necessary for Q (Q P) or (~P ~ Q) Since P is necessary for Q, Q can t be true unless P is. So, if Q is true, then it must be that P is also. Hence, P is necessary for Q means If Q, then P. You might also think: without P, no Q, i.e., (~P ~ Q). These are equivalent. P is necessary and sufficient for Q (P Q) P but Q (P. Q) P or Q, but not both ((P v Q) ~(P Q)) or ~(P Q) Not both P and Q ~(P Q) Not either P or Q (neither/nor) ~(P v Q

2 Quantified Logic All As are Bs. (Every A is a B.) Everything is A. Some As are Bs. (At least one A is a B.) Something is A. Nothing is A. (Not even one thing is A.) --Everything is non-a. No As are Bs. (Not even one A is B.) Not all As are Bs. --Some As are not Bs (are non-bs). a has property Q. (a is a Q; a Qs) the so-and-so (this names an object) (x)(ax Bx) (x)ax ( x)(ax Bx) ( x)ax ~( x)ax log. equiv. to (x)~ax ~( x)(ax Bx) log. equiv. to (x)~ (Ax Bx) ~(x)(ax Bx) log. equiv. to ( x)(ax ~Bx) Qa t (a constant) Something/everything is either A or/and B. ( x)(ax Bx) ( x)(ax Bx) (x)(ax Bx) (x) (Ax Bx) --Note that these are quantified statements. But, Both/Either something/everything is A and/or something/everything is B. (( x)ax v ( x)bx) (( x)ax ( x)bx) ((x) Ax v ( x)bx) ((x)ax ( x)bx) --Note that these are not quantified statements, but disjunctions or conjunctions (the left column are disjunction, the right column conjunctions). In general: Quantified statements begin with (x) or ( x). (You can apply drop quantifier rules only to quantified statements.) Negated quantified statements begin with ~(x) or ~( x). (You can apply reverse squiggle rules only to negated quantified statements.) If the statement begins with anything else, you cannot apply drop universal or reverse squiggle rules to it!

3 Quantified Translations With Identity a is identical to (the same things as) b. a=b or b=a (In general, the order in identity statements doesn t matter.) a isn t identical to b. Bob is a dentist. ( Being a dentist is a property Bob has.) Bob is the dentist. ( The dentist is a thing that is identical to Bob.) Someone other than Bob is a dentist. Someone in addition to Bob is a dentist. Bob alone is a dentist. (No else besides Bob is a dentist.) ~a=b or ~b=a Db b=d ( x)(dx ~x=b) (Db ( x)(dx ~x=b)) (Db ~( x)(dx ~x=b)) There is at least one dentist. ( x)dx (At least one is means something is, but says nothing about whether or not anything else is ) There is exactly one dentist. ( x)(dx ~( y)(~x=y Dx)) There is something such that it is a dentist and such that there is nothing else (i.e., other than the first thing) that is a dentist. There is something, call it x, such that it is a dentist and such that it is not the case that there is something, call it y, where x isn t identical to y and y is a dentist. Or ( x)(dx (y)(dy y=x)) There is an x such x is a dentist and is such that for all y, if y is dentist, then y is identical to x. There are at least two dentists. ( x)( y)((dx Dy) ~x=y)) (At least two are means something is and something not identical to it is, but says nothing about whether or not anything further is ) There are exactly two dentists. ( x)( y)(((dx Dy) ~x=y) ~( z)(dz (~z=x ~z=y))) There are an x and y such that x and y are dentists, are not identical to one another, and such that there is no z that is a dentist and isn t identical to either x or y. Or --similar to exactly one

4 Quantified Translations, with Identity and other relations Romeo loves Juliet. Lrj Juliet loves Romeo. Ljr (the order matters!) Romeo gave the flowers to Juliet. Grfj Someone loves Romeo. ( x)lxr Romeo loves someone. ( x)lrx Everyone loves Romeo. (x)lxr Romeo loves everyone. (x)lrx No one loves Romeo. ~( x)lxr or (x)~lxr Romeo loves no one. ~( x)lrx or (x)~lrx All logicians admire Gensler. (x)(lx Axg) Some logicians admire Gensler. ( x)(lx Axg) No logician admires Gensler. (Sorry, Harry!) ~( x)(lx Axg) Gensler admires all logicians. (x)(lx Agx) Gensler admires some logicians. ( x)(lx Agx) Gensler admires no logicians. ~( x)(lx Lgx) Some logicians besides Baldner admire Gensler. ( x)((lx ~x=b) Axg) There are logicians who aren t Baldner that admire Gensler. Some logicians in addition to Baldner admire Gensler (Abg ( x)((lx ~x=b) Axg)) No logicians except Baldner admire Gensler. (Abg ~( x)((lx ~x=b) Axg)) Note on the previous three examples: Do we need to add that Baldner is a logician (thus adding Lb somewhere) to the above? This is like the is a boastful druggist in the previous chapter. It all depends on the context of the argument. Is there anything in the argument as a whole that depends upon the explicit claim that Baldner is a logician? If we wanted to add this clause to the last example, it would come out: ((Lb Abg) ~( x)((lx ~x=b) Axg)) On the test I will either, like last time, make it clear how I want it translated, or I will accept either (correct!) answer.

5 And With Multiple Relations and Mixed Quantifiers! Someone loves someone. Someone loves everyone. ( x)( y)lxy ( x)(y)lxy Everyone loves everyone. (x)(y)axy Everyone loves someone. Note: This might be understood in two distinct ways: Everyone loves someone or other, this is generally what is meant versus There is some specific person that everyone loves. These are not equivalent. Thus: Everyone loves someone or other. (x)( y)lxy Everyone is such that there is someone they love. For everyone, there is someone they love. There is some specific person who everyone loves. ( x)(y)lyx There is someone such that everyone loves that person. There is someone that is loved by everyone. Pay attention to the difference in the order of x and y in the above examples! All dentists love one another. Every banker loves some dentist. Some banker loves every dentist. (x)(y)((dx Dy) Lxy) (x)(bx ( y)(dx Lxy)) ( x)(bx (y)(dy Lxy)) Jones is a lover. ( x)ljx In the text, the author understands x is a lover as there is someone that x loves. There is an unloved lover. ( x)(~( y)lyx ( y)lxy) There is someone such that no one loves that person, but there is someone that person loves. For some x, it is false that there is a y such that y loves x and it is true that there is a y such that x loves y. Everyone loves a lover. (x)(( y)lxy (y)lyx) Everyone loves anybody who loves somebody. Everyone is such that, if there is someone they love, then they are loved by everyone. For all x, if there is a y that x loves, then, for all y, y loves x. Everybody doesn t love something, but nobody doesn t love Sara Lee. (an ad slogan from before your time!) ((x)( y)~lxy). ~( x)~lxs) It is true both that for all x there is some y such that x doesn t love y, and there is no x such that x doesn t love Sara Lee.

Invalidity in Predicate Logic

Invalidity 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 information

FACTORING TRINOMIALS IN THE FORM OF ax 2 + bx + c

FACTORING TRINOMIALS IN THE FORM OF ax 2 + bx + c Tallahassee Community College 55 FACTORING TRINOMIALS IN THE FORM OF ax 2 + bx + c This kind of trinomial differs from the previous kind we have factored because the coefficient of x is no longer "1".

More information

If an English sentence is ambiguous, it may allow for more than one adequate transcription.

If 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 information

1.2 Forms and Validity

1.2 Forms and Validity 1.2 Forms and Validity Deductive Logic is the study of methods for determining whether or not an argument is valid. In this section we identify some famous valid argument forms. Argument Forms Consider

More information

Predicate Logic Review

Predicate Logic Review Predicate Logic Review UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Grammar A term is an individual constant or a variable. An individual constant is a lowercase letter from the beginning

More information

1.4. Removing Brackets. Introduction. Prerequisites. Learning Outcomes. Learning Style

1.4. Removing Brackets. Introduction. Prerequisites. Learning Outcomes. Learning Style Removing Brackets 1. Introduction In order to simplify an expression which contains brackets it is often necessary to rewrite the expression in an equivalent form but without any brackets. This process

More information

Factoring a Difference of Two Squares. Factoring a Difference of Two Squares

Factoring a Difference of Two Squares. Factoring a Difference of Two Squares 284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this

More information

POLYNOMIALS and FACTORING

POLYNOMIALS and FACTORING POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use

More information

Boolean Algebra Part 1

Boolean Algebra Part 1 Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems

More information

Factoring - Grouping

Factoring - Grouping 6.2 Factoring - Grouping Objective: Factor polynomials with four terms using grouping. The first thing we will always do when factoring is try to factor out a GCF. This GCF is often a monomial like in

More information

Math PreCalc 20 Chapter 4 Review of Factoring. Questions to try. 2. x 2 6xy x x x x 2 y + 8xy

Math PreCalc 20 Chapter 4 Review of Factoring. Questions to try. 2. x 2 6xy x x x x 2 y + 8xy Math PreCalc 20 Chapter 4 Review of Factoring Multiplying (Expanding) Type 1: Monomial x Binomial Monomial x Trinomial Ex: 3(x + 4) = 3x + 12-2(x 2 + 2x 1) = -2x 2 4x + 2 Multiply the following: 1. 5(x

More information

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

1.5. 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 information

1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes

1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.

More information

Predicate Calculus. There are certain arguments that seem to be perfectly logical, yet they cannot be expressed by using propositional calculus.

Predicate Calculus. There are certain arguments that seem to be perfectly logical, yet they cannot be expressed by using propositional calculus. Predicate Calculus (Alternative names: predicate logic, first order logic, elementary logic, restricted predicate calculus, restricted functional calculus, relational calculus, theory of quantification,

More information

TERMS. Parts of Speech

TERMS. Parts of Speech TERMS Parts of Speech Noun: a word that names a person, place, thing, quality, or idea (examples: Maggie, Alabama, clarinet, satisfaction, socialism). Pronoun: a word used in place of a noun (examples:

More information

Math 432 HW 2.5 Solutions

Math 432 HW 2.5 Solutions Math 432 HW 2.5 Solutions Assigned: 1-10, 12, 13, and 14. Selected for Grading: 1 (for five points), 6 (also for five), 9, 12 Solutions: 1. (2y 3 + 2y 2 ) dx + (3y 2 x + 2xy) dy = 0. M/ y = 6y 2 + 4y N/

More information

Factoring. Factoring Monomials Monomials can often be factored in more than one way.

Factoring. Factoring Monomials Monomials can often be factored in more than one way. Factoring Factoring is the reverse of multiplying. When we multiplied monomials or polynomials together, we got a new monomial or a string of monomials that were added (or subtracted) together. For example,

More information

Operations with Algebraic Expressions: Multiplication of Polynomials

Operations with Algebraic Expressions: Multiplication of Polynomials Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the

More information

Handout #1: Mathematical Reasoning

Handout #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 information

Exam 1 Sample Question SOLUTIONS. y = 2x

Exam 1 Sample Question SOLUTIONS. y = 2x Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

More information

Math 55: Discrete Mathematics

Math 55: Discrete Mathematics Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 1, due Wedneday, January 25 1.1.10 Let p and q be the propositions The election is decided and The votes have been counted, respectively.

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

NSM100 Introduction to Algebra Chapter 5 Notes Factoring Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

More information

Factoring Quadratic Expressions

Factoring Quadratic Expressions Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

More information

In the above, the number 19 is an example of a number because its only positive factors are one and itself.

In the above, the number 19 is an example of a number because its only positive factors are one and itself. Math 100 Greatest Common Factor and Factoring by Grouping (Review) Factoring Definition: A factor is a number, variable, monomial, or polynomial which is multiplied by another number, variable, monomial,

More information

The Handshake Problem

The Handshake Problem The Handshake Problem Tamisha is in a Geometry class with 5 students. On the first day of class her teacher asks everyone to shake hands and introduce themselves to each other. Tamisha wants to know how

More information

CSL105: Discrete Mathematical Structures. Ragesh Jaiswal, CSE, IIT Delhi

CSL105: Discrete Mathematical Structures. Ragesh Jaiswal, CSE, IIT Delhi Propositional Logic: logical operators Negation ( ) Conjunction ( ) Disjunction ( ). Exclusive or ( ) Conditional statement ( ) Bi-conditional statement ( ): Let p and q be propositions. The biconditional

More information

Section 6.1 Joint Distribution Functions

Section 6.1 Joint Distribution Functions Section 6.1 Joint Distribution Functions We often care about more than one random variable at a time. DEFINITION: For any two random variables X and Y the joint cumulative probability distribution function

More information

UNDERSTANDING YOUR ONLINE FOOTPRINTS: HOW TO PROTECT YOUR PERSONAL INFORMATION ON THE INTERNET

UNDERSTANDING YOUR ONLINE FOOTPRINTS: HOW TO PROTECT YOUR PERSONAL INFORMATION ON THE INTERNET UNDERSTANDING YOUR ONLINE FOOTPRINTS: HOW TO PROTECT YOUR PERSONAL INFORMATION ON THE INTERNET SPEAKING NOTES FOR GRADES 4 TO 6 PRESENTATION SLIDE (1) Title Slide SLIDE (2) Key Points It can be really

More information

Factoring Trinomials of the Form x 2 bx c

Factoring Trinomials of the Form x 2 bx c 4.2 Factoring Trinomials of the Form x 2 bx c 4.2 OBJECTIVES 1. Factor a trinomial of the form x 2 bx c 2. Factor a trinomial containing a common factor NOTE The process used to factor here is frequently

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

10.4 Traditional Subject Predicate Propositions

10.4 Traditional Subject Predicate Propositions M10_COPI1396_13_SE_C10.QXD 10/22/07 8:42 AM Page 445 10.4 Traditional Subject Predicate Propositions 445 Continuing to assume the existence of at least one individual, we can say, referring to this square,

More information

COLLEGE ALGEBRA. Paul Dawkins

COLLEGE ALGEBRA. Paul Dawkins COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5

More information

A Short Course in Logic Example 8

A Short Course in Logic Example 8 A Short ourse in Logic xample 8 I) Recognizing Arguments III) valuating Arguments II) Analyzing Arguments valuating Arguments with More than one Line of Reasoning valuating If then Premises Independent

More information

Factoring Polynomials and Solving Quadratic Equations

Factoring Polynomials and Solving Quadratic Equations Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3

More information

Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai

Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for

More information

6.4 Special Factoring Rules

6.4 Special Factoring Rules 6.4 Special Factoring Rules OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor a sum of cubes. By reversing the rules for multiplication

More information

Recitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere

Recitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x

More information

Being a Guarantor. Financial Series. in Alberta. What is a Guarantor? June 2011. Has someone you know asked you to be a Guarantor?

Being a Guarantor. Financial Series. in Alberta. What is a Guarantor? June 2011. Has someone you know asked you to be a Guarantor? Financial Series June 2011 Being a Guarantor in Alberta Has someone you know asked you to be a Guarantor? Are you already a Guarantor and worried about what comes next, or what is already occurring? This

More information

Using the ac Method to Factor

Using the ac Method to Factor 4.6 Using the ac Method to Factor 4.6 OBJECTIVES 1. Use the ac test to determine factorability 2. Use the results of the ac test 3. Completely factor a trinomial In Sections 4.2 and 4.3 we used the trial-and-error

More information

First-Order Logics and Truth Degrees

First-Order Logics and Truth Degrees First-Order Logics and Truth Degrees George Metcalfe Mathematics Institute University of Bern LATD 2014, Vienna Summer of Logic, 15-19 July 2014 George Metcalfe (University of Bern) First-Order Logics

More information

SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen

SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen A. GENERALITIES. When a given quadratic equation can be factored, there are 2 best methods

More information

Module 6: How to Write the Book

Module 6: How to Write the Book Module 6: How to Write the Book Okay, guys. Katrina Starzhynskaya here, and lesson 6, how to actually write the book: time frame, where to get your ideas from, and how to get inspired. Let s begin. Time

More information

Grammar Unit: Pronouns

Grammar Unit: Pronouns Name: Miss Phillips Period: Grammar Unit: Pronouns Unit Objectives: 1. Students will identify personal, indefinite, and possessive pronouns and recognize antecedents of pronouns. 2. Students will demonstrate

More information

Three Attributes of Every Successful Merchant Services Program-20140604 1602-1

Three Attributes of Every Successful Merchant Services Program-20140604 1602-1 Three Attributes of Every Successful Merchant Services Program-20140604 1602-1 [Start of recorded material] [Starts Mid Sentence] thank everyone that s joined the call today. I know everybody is busy with

More information

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1 5.7 Factoring ax 2 bx c (5-49) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.

More information

Factoring Flow Chart

Factoring Flow Chart Factoring Flow Chart greatest common factor? YES NO factor out GCF leaving GCF(quotient) how many terms? 4+ factor by grouping 2 3 difference of squares? perfect square trinomial? YES YES NO NO a 2 -b

More information

1.4 Variable Expressions

1.4 Variable Expressions 1.4 Variable Expressions Now that we can properly deal with all of our numbers and numbering systems, we need to turn our attention to actual algebra. Algebra consists of dealing with unknown values. These

More information

Rotation Matrices and Homogeneous Transformations

Rotation Matrices and Homogeneous Transformations Rotation Matrices and Homogeneous Transformations A coordinate frame in an n-dimensional space is defined by n mutually orthogonal unit vectors. In particular, for a two-dimensional (2D) space, i.e., n

More information

Reducing Customer Churn

Reducing Customer Churn Reducing Customer Churn A Love Story smarter customer contact Breaking up is hard to do The old adage that it s cheaper (and better) to hold onto an existing customer than to acquire a new one isn t just

More information

Don t Mark My Paper, Help Me Get an A

Don t Mark My Paper, Help Me Get an A ONE Don t Mark My Paper, Help Me Get an A Garry Ridge AS I SHARE with you how we successfully implemented our Don t Mark My Paper, Help Me Get an A philosophy into our performance review system, we ll

More information

Kant s deontological ethics

Kant s deontological ethics Michael Lacewing Kant s deontological ethics DEONTOLOGY Deontologists believe that morality is a matter of duty. We have moral duties to do things which it is right to do and moral duties not to do things

More information

AP CALCULUS AB 2009 SCORING GUIDELINES

AP CALCULUS AB 2009 SCORING GUIDELINES AP CALCULUS AB 2009 SCORING GUIDELINES Question 3 Mighty Cable Company manufactures cable that sells for $120 per meter. For a cable of fixed length, the cost of producing a portion of the cable varies

More information

Particular Solutions. y = Ae 4x and y = 3 at x = 0 3 = Ae 4 0 3 = A y = 3e 4x

Particular Solutions. y = Ae 4x and y = 3 at x = 0 3 = Ae 4 0 3 = A y = 3e 4x Particular Solutions If the differential equation is actually modeling something (like the cost of milk as a function of time) it is likely that you will know a specific value (like the fact that milk

More information

First Order Non-Linear Equations

First Order Non-Linear Equations First Order Non-Linear Equations We will briefly consider non-linear equations. In general, these may be much more difficult to solve than linear equations, but in some cases we will still be able to solve

More information

Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

More information

CHAPTER 7 ARGUMENTS WITH DEFIITIONAL AND MISSING PREMISES

CHAPTER 7 ARGUMENTS WITH DEFIITIONAL AND MISSING PREMISES CHAPTER 7 ARGUMENTS WITH DEFIITIONAL AND MISSING PREMISES What You ll Learn in this Chapter In Chapters -5, we developed a skill set that s sufficient for the recognition, analysis, evaluation and construction

More information

6.3 FACTORING ax 2 bx c WITH a 1

6.3 FACTORING ax 2 bx c WITH a 1 290 (6 14) Chapter 6 Factoring e) What is the approximate maximum revenue? f) Use the accompanying graph to estimate the price at which the revenue is zero. y Revenue (thousands of dollars) 300 200 100

More information

7-6. Choosing a Factoring Model. Extension: Factoring Polynomials with More Than One Variable IN T RO DUC E T EACH. Standards for Mathematical Content

7-6. Choosing a Factoring Model. Extension: Factoring Polynomials with More Than One Variable IN T RO DUC E T EACH. Standards for Mathematical Content 7-6 Choosing a Factoring Model Extension: Factoring Polynomials with More Than One Variable Essential question: How can you factor polynomials with more than one variable? What is the connection between

More information

Where's Gone? LEAD GENERATION PRINTABLE WORKBOOK

Where's Gone? LEAD GENERATION PRINTABLE WORKBOOK Have you ever stopped to think why you are in business? Good question, isn t it? But before we take a closer look at this, spend a few moments now thinking about what you believe your reasons to be. Jot

More information

Filename: P4P 016 Todd: Kim: Todd: Kim:

Filename: P4P 016 Todd: Kim: Todd: Kim: Filename: P4P 016 Todd: [0:00:18] Hey everybody, welcome to another edition of The Prosperity Podcast, this is No BS Money Guy Todd Strobel. Once again, we have my cohost, bestselling financial author

More information

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

More information

Web Conferencing Overview

Web Conferencing Overview Web Conferencing Overview Market Definition Webcasting or web-conferencing products are well know and have been traditionally relegated to purchases within a line of business (LOB) to perform a specific

More information

Kim: Thank you Todd, I m delighted to be here today and totally looking forward to our conversation.

Kim: Thank you Todd, I m delighted to be here today and totally looking forward to our conversation. Filename: P4P 019 The Facts of Life Insurance Todd: [0:00:18] Hey everybody, welcome to another edition of The Prosperity Podcast, this is No BS Money Guy Todd Strobel. Once again, we re lucky enough to

More information

A Primer in the Semantics of English

A Primer in the Semantics of English A Primer in the Semantics of English Some Nuts and Bolts 2000-2010 by Terence Parsons A Primer in the Semantics of English Some Nuts and Bolts Chapter 0: Preliminary Issues 0.1 What Sentences Do 0.2 Standing-for

More information

Mathematical Induction

Mathematical 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 information

CROSS EXAMINATION OF AN EXPERT WITNESS IN A CHILD SEXUAL ABUSE CASE. Mark Montgomery

CROSS EXAMINATION OF AN EXPERT WITNESS IN A CHILD SEXUAL ABUSE CASE. Mark Montgomery CROSS EXAMINATION OF AN EXPERT WITNESS IN A CHILD SEXUAL ABUSE CASE Mark Montgomery Post Office Box 161 Durham, NC 27702 (919) 680-6249 mark.montgomery@mindspring.com Opinion Testimony by a Pediatrician/Nurse/Counselor/Social

More information

Relations: their uses in programming and computational specifications

Relations: their uses in programming and computational specifications PEPS Relations, 15 December 2008 1/27 Relations: their uses in programming and computational specifications Dale Miller INRIA - Saclay & LIX, Ecole Polytechnique 1. Logic and computation Outline 2. Comparing

More information

Subject-Verb Agreement A grammar help worksheet by Abbie Potter Henry

Subject-Verb Agreement A grammar help worksheet by Abbie Potter Henry Subject-Verb Agreement A grammar help worksheet by Abbie Potter Henry (Subjects are in bold typeface and verbs are underlined) Subject-Verb Agreement means that subjects and verbs must always agree in

More information

THERE ARE SEVERAL KINDS OF PRONOUNS:

THERE ARE SEVERAL KINDS OF PRONOUNS: PRONOUNS WHAT IS A PRONOUN? A Pronoun is a word used in place of a noun or of more than one noun. Example: The high school graduate accepted the diploma proudly. She had worked hard for it. The pronoun

More information

FACTORING POLYNOMIALS

FACTORING POLYNOMIALS 296 (5-40) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated

More information

www.arden Fumigation.com (408) 279-2040 1

www.arden Fumigation.com (408) 279-2040 1 1 FREE REPORT 5 Crucial Things You Should Know Before Choosing a Pest Control Service 2 We all know how difficult it is when we need to get rid of pests in our house. It is even more difficult to do it

More information

1.3 Polynomials and Factoring

1.3 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 information

Algebra 1 Chapter 08 review

Algebra 1 Chapter 08 review Name: Class: Date: ID: A Algebra 1 Chapter 08 review Multiple Choice Identify the choice that best completes the statement or answers the question. Simplify the difference. 1. (4w 2 4w 8) (2w 2 + 3w 6)

More information

Kickass Offline Profits Imran Naseem http://www.imrannaseem.com

Kickass Offline Profits Imran Naseem http://www.imrannaseem.com Kickass Offline Profits Imran Naseem http://www.imrannaseem.com Let s get right into with this report. Everyone knows that there is a TON of money to be made in the offline niche. I am also going to assume

More information

Factoring Trinomials: The ac Method

Factoring Trinomials: The ac Method 6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For

More information

The Classes P and NP

The Classes P and NP The Classes P and NP We now shift gears slightly and restrict our attention to the examination of two families of problems which are very important to computer scientists. These families constitute the

More information

Factoring Polynomials

Factoring Polynomials UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

More information

Ultraproducts and Applications I

Ultraproducts and Applications I Ultraproducts and Applications I Brent Cody Virginia Commonwealth University September 2, 2013 Outline Background of the Hyperreals Filters and Ultrafilters Construction of the Hyperreals The Transfer

More information

Factoring Polynomials

Factoring Polynomials Factoring Polynomials 4-1-2014 The opposite of multiplying polynomials is factoring. Why would you want to factor a polynomial? Let p(x) be a polynomial. p(c) = 0 is equivalent to x c dividing p(x). Recall

More information

Learning mathematics Some hints from the psychologists Examples:

Learning mathematics Some hints from the psychologists Examples: Learning mathematics Some hints from the psychologists In your degree course you will learn, and be examined on: 1. Facts (= knowing that ) 2. Skills (= knowing how ). Examples: An example of a fact to

More information

The Adventures of Leo Rahula Leads The Way

The Adventures of Leo Rahula Leads The Way The Adventures of Leo Rahula Leads The Way by S. Dhammika & Susan Harmer [Screen Quality] E-mail: bdea@buddhanet.net Web site: www.buddhanet.net Buddha Dharma Education Association Inc. Copyright Buddha

More information

SCRIPT FOR PROVIDER/ACO PHONE INQUIRIES. What is an ACO?

SCRIPT FOR PROVIDER/ACO PHONE INQUIRIES. What is an ACO? SCRIPT FOR PROVIDER/ACO PHONE INQUIRIES What is an ACO? An Accountable Care Organization (ACO) is a group of doctors and other healthcare providers who agree to work together with Medicare to give you

More information

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

More information

CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 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 information

In algebra, factor by rewriting a polynomial as a product of lower-degree polynomials

In algebra, factor by rewriting a polynomial as a product of lower-degree polynomials Algebra 2 Notes SOL AII.1 Factoring Polynomials Mrs. Grieser Name: Date: Block: Factoring Review Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 3 In algebra, factor by rewriting

More information

Last time we had arrived at the following provisional interpretation of Aquinas second way:

Last time we had arrived at the following provisional interpretation of Aquinas second way: Aquinas Third Way Last time we had arrived at the following provisional interpretation of Aquinas second way: 1. 2. 3. 4. At least one thing has an efficient cause. Every causal chain must either be circular,

More information

We will learn the Python programming language. Why? Because it is easy to learn and many people write programs in Python so we can share.

We will learn the Python programming language. Why? Because it is easy to learn and many people write programs in Python so we can share. LING115 Lecture Note Session #4 Python (1) 1. Introduction As we have seen in previous sessions, we can use Linux shell commands to do simple text processing. We now know, for example, how to count words.

More information

Jaakko Hintikka Boston University. and. Ilpo Halonen University of Helsinki INTERPOLATION AS EXPLANATION

Jaakko Hintikka Boston University. and. Ilpo Halonen University of Helsinki INTERPOLATION AS EXPLANATION Jaakko Hintikka Boston University and Ilpo Halonen University of Helsinki INTERPOLATION AS EXPLANATION INTERPOLATION AS EXPLANATION In the study of explanation, one can distinguish two main trends. On

More information

To give it a definition, an implicit function of x and y is simply any relationship that takes the form:

To give it a definition, an implicit function of x and y is simply any relationship that takes the form: 2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to

More information

THE GREAT DEBATE: Is GPS Tracking Really Beneficial for Fleets? Fear of Employee Pushback. Fleet Intelligence for Your Business GPS INSIGHT

THE GREAT DEBATE: Is GPS Tracking Really Beneficial for Fleets? Fear of Employee Pushback. Fleet Intelligence for Your Business GPS INSIGHT THE GREAT DEBATE: Is GPS Tracking Really Beneficial for Fleets? Since GPS tracking was first introduced to the fleet industry, there has been a debate whether this technology is really beneficial for fleets

More information

Polynomials. 4-4 to 4-8

Polynomials. 4-4 to 4-8 Polynomials 4-4 to 4-8 Learning Objectives 4-4 Polynomials Monomials, binomials, and trinomials Degree of a polynomials Evaluating polynomials functions Polynomials Polynomials are sums of these "variables

More information

Your guide to. Communicating with people with a learning disability

Your guide to. Communicating with people with a learning disability Your guide to Communicating with people with a learning disability About this guide This guide is designed to provide a brief introduction to communication, and the problems faced by someone with a learning

More information

salary / wages employer employee tax earn

salary / wages employer employee tax earn Worksheet 1 salary / wages employer employee tax earn freelancer overtime contract National Insurance National Insurance no. Worksheet 2 Money you get every month or week for doing your job. The person

More information

Efficient database auditing

Efficient database auditing Topicus Fincare Efficient database auditing And entity reversion Dennis Windhouwer Supervised by: Pim van den Broek, Jasper Laagland and Johan te Winkel 9 April 2014 SUMMARY Topicus wants their current

More information

Business Loan. This document sets out your loan s terms and conditions. Contents of these terms and conditions. Terms and Conditions

Business Loan. This document sets out your loan s terms and conditions. Contents of these terms and conditions. Terms and Conditions Business Loan Terms and Conditions This document sets out your loan s terms and conditions In this document we ve explained the terms and conditions applying to your ANZ Business Loan. It includes key

More information

Factoring, Solving. Equations, and Problem Solving REVISED PAGES

Factoring, Solving. Equations, and Problem Solving REVISED PAGES 05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring

More information

Critical analysis. Be more critical! More analysis needed! That s what my tutors say about my essays. I m not really sure what they mean.

Critical analysis. Be more critical! More analysis needed! That s what my tutors say about my essays. I m not really sure what they mean. Critical analysis Be more critical! More analysis needed! That s what my tutors say about my essays. I m not really sure what they mean. I thought I had written a really good assignment this time. I did

More information

LIFETIME MORTGAGE LUMP SUM

LIFETIME MORTGAGE LUMP SUM LIFETIME MORTGAGE LUMP SUM Terms and Conditions (version 5) This is an important document. Please keep it in a safe place. LV= Lifetime Mortgage lump sum Terms and Conditions Welcome to LV=, and thank

More information