4 Interlude: A review of logic and proofs

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "4 Interlude: A review of logic and proofs"

Transcription

1 4 Interlude: A review of logic and proofs 4.1 Logic Propositions and logical connectives A proposition is a declarative sentence that is either true (T or false (F, but not both. Example 4.1 Examples of propositions: It rains every Monday. Whenever I am happy, I sing. If x 1, then x 2 x. x > y if and only if x 2 > y 2. Compound propositions can be formed from one or more propositions using logical operators. The following are the most important compound propositions. Negation: p, not p. True if and only if p is false. Conjunction: p q, p and q. True if and only if both p and q are true. Disjunction: p q, p or q (or both. True if and only if at least one of p and q is true. Implication: p q, if p, then q. (Also read: p implies q, p is sufficient for q, p only if q, q is a necessary condition for p. False if and only if p is true and q is false. Biconditional: p q, p if and only if q. (Also read: p is necessary and sufficient for q. True if and only if p and q are both true or both false. A truth table displays the relationship between the truth value of a compound proposition and the truth values of the individual propositions within it. Carefully examine the truth tables of the compound propositions in the table below. Pay particular attention to the truth values of the implication! p q p p q p q p q p q T T F T T T T T F F F T F F F T T F T T F F F T F F T T 39

2 4.1.2 Logical equivalence A compound proposition is called a tautology if it is true no matter what the truth values of the propositions within it; contradiction if it is false no matter what the truth values of the propositions within it; contingency if it can assume both true and false values. Exercise 4.2 Use truth tables to show that p p is a tautology and p p is a contradiction. Propositions p and q are called logically equivalent (denoted p q if p q is a tautology; that is, if p and q always assume the same truth value. Exercise 4.3 Some of the most important logical equivalences are shown in the table below. Here, T denotes a tautology and F denotes a contradiction. Use truth tables to prove these equivalences. Logical equivalence Name p T p Identity laws p F p p T T Domination laws p F F p p p Idempotent laws p p p ( p p Double negation law p q q p Commutative laws p q q p (p q r p (q r Associative laws (p q r p (q r p (q r (p q (p r Distributive laws p (q r (p q (p r (p q p q De Morgan s laws (p q p q p (p q p Absorption laws p (p q p p p T Negation laws p p F p q p q Implication and disjunction p q p q p q q p The contrapositive p q (p q (q p Biconditional and implication p q (p q ( p q Biconditional, conjunction, and disjunction 40

3 Exercise 4.4 Given an implication p q, the following important implications can be formed: q p, called the converse of p q; q p, called the contrapositive of p q; p q, called the inverse of p q. Using truth tables, prove that an implication is equivalent to its contrapositive, but not equivalent to its converse and inverse. Keep this result in mind whenever proving a theorem! Predicates and quantifiers A propositional function (or predicate P is a mapping that assigns to each value c in its domain (also called the universe of discourse a proposition P (c. Example 4.5 Let P (x denote the statement x > 3. As long as x is a variable (has no assigned value, this is not a proposition because it is neither true nor false. When x is assigned a value, say x = c, then P (c is a proposition, that is, a statement that is either true or false. For example, P (4 is the proposition 4 >3 (which is true, and P (2 is the proposition 2 > 3 (which is false. Another way to convert a propositional function into a proposition is to use quantifiers: The universal quantification of P (x is xp (x, read for all x (in the universe of discourse P (x (is true. The existential quantification of P (x is xp (x, read there exists x (in the universe of discourse such that P (x (is true. Carefully examine the truth values of the quantified propositions in the table below. Proposition When true? When false? xp (x P (c is true for all c in the domain P (c is false for some c in the domain xp (x P (c is true for some c in the domain P (c is false for all c in the domain xp (x P (c is false for some c in the domain P (c is true for all c in the domain x P (x xp (x P (c is false for all c in the domain P (c is true for some c in the domain x P (x 41

4 4.2 Methods of proof In this section, we look at what constitutes a correct mathematical argument that can be used in a proof of a theorem A theorem is a mathematical statement that can be shown to be true. A theorem is sometimes called a proposition, fact, or result. A lemma is a theorem whose main purpose is to help in the proof of another theorem. A corollary is a theorem that easily follows from another theorem. A conjecture is a proposition whose truth value is unknown; once proved (if true, it becomes a theorem. A proof is a mathematical argument that demonstrates that a theorem is true. An argument is a set of propositions in which one, called the conclusion, is claimed to follow from the others, called the hypotheses or premises (more about arguments later. The hypotheses of a proof may be axioms or postulates, which are the underlying assumptions about mathematical structures, the hypotheses of the theorem to be proved, or previously proved theorems. The steps of the proof are tied together by the rules of inference (to be discussed below, or correct reasoning Rules of inference Rules of inference are the means to draw conclusions from other assertions, thereby correctly tying together the steps of the proof. Each rule of inference for propositions (see the table below is simply a tautology. Rule of inference Tautology Name p p q p (p q Addition p q p (p q p Simplification p q ((p (q (p q Conjunction p q p p q q (p (p q q Modus ponens (law of detachment q p q p ( q (p q p Modus tollens p q q r p r ((p q (q r (p r Hypothetical syllogism p q p q ((p q ( p q Disjunctive syllogism p q p r q r ((p q ( p r (q r Resolution Exercise 4.6 Using truth tables, show that each rule of inference in the table above is a tautology. 42

5 4.2.2 Arguments As mentioned above, an argument is a set of propositions in which one (called the conclusion is claimed to follow from the others (called the hypotheses or premises. That is, an argument is a proposition of the form (p 1 p 2... p k c, where p 1, p 2,..., p k are the premises and c is the conclusion. This is also written as p 1 p 2. The argument (* is said to be valid if the proposition (p 1 p 2... p k c is a tautology; that is, if the conclusion c is true whenever the premises p 1, p 2,..., p k are all true. Validity of an argument can be verified by following the rules of inference or by using truth tables. Example 4.7 Show that the hypotheses It is not sunny this afternoon and it is colder than yesterday We will go swimming only if it is sunny If we do not go swimming, then we will take a canoe trip If we take a canoe trip, then we will be home by sunset lead to the conclusion We will be home by sunset. Exercise 4.8 Show that the hypotheses If you send me an message, then I will finish doing my homework If you do not send me an message, then I will go to bed early If I go to bed early, then I will wake up feeling refreshed lead to the conclusion If I do not finish doing my homework, then I will wake up feeling refreshed. The following rules of inference for quantified statements are used extensively in mathematical arguments. p k c ( Rule of inference xp (x P (c for an arbitrary c P (c for an arbitrary c xp (x xp (x P (c for some c P (c for some c xp (x Name Universal instantiation Universal generalization Existential instantiation Existential generalization 43

6 Example 4.9 What rules of inference are used in the following famous argument? All men are mortal. Socrates is a man. Therefore, Socrates is mortal. Exercise 4.10 What rules of inference are used in the following argument? No man is an island. Baffin is an island. Therefore, Baffin is not a man. Exercise 4.11 Show that the premises A student in this class has not read the notes Everybody in this class passed the midterm exam imply the conclusion Someone who passed the midterm exam has not read the notes. A fallacy is a common form of incorrect reasoning. Fallacies resemble rules of inference but are based on contingencies rather than tautologies. In order to write correct mathematical arguments, it is very important to recognize them. Here are the two most common fallacies: Fallacy of confirming the conclusion: ((p q q p Fallacy of denying the hypothesis: ((p q p q Example 4.12 Examine the truth tables of the propositions ((p q q p and ((p q p q to show that they are indeed contingencies. Example 4.13 Are the following arguments valid? If not, what fallacy has been committed? 1. If the weather is nice, we go skiing. We are skiing. Therefore, the weather is nice. 2. If the weather is nice, we go skiing. The weather is nice. Therefore, we are skiing. 3. If the weather is nice, we go skiing. We are not skiing. Therefore, the weather is not nice. 4. If the weather is nice, we go skiing. The weather is not nice. Therefore, we are not skiing Methods of proving theorems The table below lists some of the most important methods for proving theorems. Note that most theorems are written in the form of an implication: one needs to prove that p q is true (that is, that p implies q, and not that q itself is true. 44

7 Type of Proof To Prove Approach Direct proof p q Assume p is true. Using the rules of inference show q is true. Indirect proof p q Prove q p (the contrapositive. Vacuous proof p q Show p is false. Trivial proof p q Show q is true. Proof by p Show p F is true. contradiction Proof by cases (p 1 p 2... p k q Prove p i q for each i = 1, 2,..., k. Proof of equivalence p q Prove both p q and q p. Proof of equivalence p 1 p 2... p k Prove p 1 p 2, p 2 p 3,..., and p k p 1. Existence proof xp (x Find c such that P (c is true. constructive Existence proof xp (x Prove xp (x without finding c such that nonconstructive P (c is true (e.g. by contradiction. Uniqueness proof!xp (x Prove xp (x. Prove (P (x P (y (x = y. Proof by xp (x Find c such that P (c is false. counterexample Exercise 4.14 Give an example for each of the types of proofs in the table above Mistakes in proofs Here are some common errors made in mathematical proofs: Using incorrect rules of inference or committing a fallacy (in particular, the fallacy of denying the hypothesis or the fallacy of affirming the conclusion. Proving q p instead of p q. Using q p instead of p q as a hypothesis of the theorem. Circular reasoning (also called begging the question: To prove proposition p, a proposition equivalent to p is used. Committing an error in arithmetic or basic algebra (e.g. dividing by x where x might be 0. Omitting a case in a proof by cases. 45

8 4.2.5 Paul Elliott-Magwood s advice on writing proofs 1. If you assume what you are trying to prove then of course the result will follow (circular reasoning!. Unfortunately, this is not a valid proof. 2. An assumed statement which implies (through a sequence of logical statements a true statement is not necessarily true (that is, p q being true does not make p true this is the fallacy of affirming the hypothesis. However, an assumed statement which leads to a contradiction is false (that is, if p F is true, then p must be false. 3. In terms of style, try to avoid writing unproven statements, even if you later follow with the proof. If you do wish to do this, write them as a separate lemma or claim. 4. A proof must be a sequence of logical steps, each one logically following from the previous (using the rules of inference. Each step should use a single idea (i.e. precisely applying a definition, precisely applying a theorem, appealing to an earlier assumption, or making a single observation. A general discussion is not a proof. Also, I (Mateja would like to add that a proof is not an essay and you should not try to write down all that you know on the topic. Rather, present only the ingredients that (using correct logical reasoning lead from the assumptions to the conclusion. In other words, your proof should be as direct a route from p to q as possible, without de-tours. 5. Proofs must be rigorous and must take into account every possible case. 6. Definitions are important. Be sure to correctly use existing mathematical language and notation (i.e. two vertices being adjacent is not the same as connected. Do not make up new words or new notation unless you write a precise definition of your new word or notation. Paul also mentions that on Assignment 1, there were lots of difficulties with concepts 1, 2, and 3 in Question 2, and with concepts 4, 5, and 6 in Question Mathematical induction The table below gives templates for the most commonly used forms of mathematical induction. Here are some additional recommendations for writing induction proofs on graphs. (This is written for the basic form of induction; try to rewrite the induction step to fit the format of Strong Induction. Carefully choose the graph parameter on which to use the induction. This can be the number of vertices, number of edges, number of connected components,... Below, I ll call this the size of the graph. The statement to prove should be of the following form: P (n is true for all n n 0, 46

9 where P (n is a statement of the form P (n: property P holds for all graphs (in a given domain of size n. Basis of induction: Prove P (n 0 holds for all graphs of size n 0. Induction Step: Assume P (n holds for all graphs of size n, for some n n 0 (Induction Hypothesis. Take an arbitrary graph G of size n + 1. Now prove property P holds for G. To use the induction hypothesis, remove a vertex, edge, a connected component,... to obtain a graph G of size n from the graph G. Be sure that G lies in the same domain as G so that the induction hypothesis really applies to it. Type of Statement You Must Prove Induction to Prove Mathematical P (n for all Basis Step: P (1 Induction n Z + Induction Step: P (k P (k + 1 for all k Z + Mathematical P (n for all Basis Step: P (n 0 Induction n Z, n n 0 Induction Step: P (k P (k + 1 for all k Z, k n 0 a generalization Strong P (n for all Basis Step: P (1 Induction n Z + Induction Step: (P (1 P (2... P (k P (k + 1 for all k Z + Strong P (n for all Basis Step: P (n 0 Induction n Z, n n 0 Induction Step: (P (n 0 P (n P (k P (k + 1 a generalization for all k Z, k n 0 References for Chapter 4: Rosen 47

The Foundations: Logic and Proofs. Chapter 1, Part III: Proofs

The Foundations: Logic and Proofs. Chapter 1, Part III: Proofs The Foundations: Logic and Proofs Chapter 1, Part III: Proofs Rules of Inference Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments

More information

CS 441 Discrete Mathematics for CS Lecture 5. Predicate logic. CS 441 Discrete mathematics for CS. Negation of quantifiers

CS 441 Discrete Mathematics for CS Lecture 5. Predicate logic. CS 441 Discrete mathematics for CS. Negation of quantifiers CS 441 Discrete Mathematics for CS Lecture 5 Predicate logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Negation of quantifiers English statement: Nothing is perfect. Translation: x Perfect(x)

More information

def: 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.

def: 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 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

CHAPTER 1. Logic, Proofs Propositions

CHAPTER 1. Logic, Proofs Propositions CHAPTER 1 Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: Paris is in France (true), London

More information

Rules of Inference Friday, January 18, 2013 Chittu Tripathy Lecture 05

Rules of Inference Friday, January 18, 2013 Chittu Tripathy Lecture 05 Rules of Inference Today s Menu Rules of Inference Quantifiers: Universal and Existential Nesting of Quantifiers Applications Old Example Re-Revisited Our Old Example: Suppose we have: All human beings

More information

2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. 2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

More 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

Likewise, we have contradictions: formulas that can only be false, e.g. (p p).

Likewise, we have contradictions: formulas that can only be false, e.g. (p p). CHAPTER 4. STATEMENT LOGIC 59 The rightmost column of this truth table contains instances of T and instances of F. Notice that there are no degrees of contingency. If both values are possible, the formula

More information

3. Mathematical Induction

3. 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 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

DISCRETE MATH: LECTURE 3

DISCRETE 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 information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

Discrete 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 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

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

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

Math 3000 Section 003 Intro to Abstract Math Homework 2

Math 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 information

1 Propositional Logic

1 Propositional Logic 1 Propositional Logic Propositions 1.1 Definition A declarative sentence is a sentence that declares a fact or facts. Example 1 The earth is spherical. 7 + 1 = 6 + 2 x 2 > 0 for all real numbers x. 1 =

More information

Predicate logic Proofs Artificial intelligence. Predicate logic. SET07106 Mathematics for Software Engineering

Predicate logic Proofs Artificial intelligence. Predicate logic. SET07106 Mathematics for Software Engineering Predicate logic SET07106 Mathematics for Software Engineering School of Computing Edinburgh Napier University Module Leader: Uta Priss 2010 Copyright Edinburgh Napier University Predicate logic Slide 1/24

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

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

Geometry: 2.1-2.3 Notes

Geometry: 2.1-2.3 Notes Geometry: 2.1-2.3 Notes NAME 2.1 Be able to write all types of conditional statements. Date: Define Vocabulary: conditional statement if-then form hypothesis conclusion negation converse inverse contrapositive

More information

Introduction to Proofs

Introduction to Proofs Chapter 1 Introduction to Proofs 1.1 Preview of Proof This section previews many of the key ideas of proof and cites [in brackets] the sections where they are discussed thoroughly. All of these ideas are

More information

Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

More information

Predicate Logic. For example, consider the following argument:

Predicate Logic. For example, consider the following argument: Predicate Logic The analysis of compound statements covers key aspects of human reasoning but does not capture many important, and common, instances of reasoning that are also logically valid. For example,

More information

The Mathematics of GIS. Wolfgang Kainz

The Mathematics of GIS. Wolfgang Kainz The Mathematics of GIS Wolfgang Kainz Wolfgang Kainz Department of Geography and Regional Research University of Vienna Universitätsstraße 7, A-00 Vienna, Austria E-Mail: wolfgang.kainz@univie.ac.at Version.

More information

Florida State University Course Notes MAD 2104 Discrete Mathematics I

Florida State University Course Notes MAD 2104 Discrete Mathematics I Florida State University Course Notes MAD 2104 Discrete Mathematics I Florida State University Tallahassee, Florida 32306-4510 Copyright c 2011 Florida State University Written by Dr. John Bryant and Dr.

More information

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. 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 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

Examination paper for MA0301 Elementær diskret matematikk

Examination paper for MA0301 Elementær diskret matematikk Department of Mathematical Sciences Examination paper for MA0301 Elementær diskret matematikk Academic contact during examination: Iris Marjan Smit a, Sverre Olaf Smalø b Phone: a 9285 0781, b 7359 1750

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

Logic in general. Inference rules and theorem proving

Logic in general. Inference rules and theorem proving Logical Agents Knowledge-based agents Logic in general Propositional logic Inference rules and theorem proving First order logic Knowledge-based agents Inference engine Knowledge base Domain-independent

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 Mid-Session Test Summer Session 008-00

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

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

6.080/6.089 GITCS Feb 12, 2008. Lecture 3

6.080/6.089 GITCS Feb 12, 2008. Lecture 3 6.8/6.89 GITCS Feb 2, 28 Lecturer: Scott Aaronson Lecture 3 Scribe: Adam Rogal Administrivia. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my

More information

Applications of Methods of Proof

Applications of Methods of Proof CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

More information

WHAT IS PHILOSOPHY? ARGUMENTATIO N. Intro to Philosophy, Summer 2011 Benjamin Visscher Hole IV

WHAT IS PHILOSOPHY? ARGUMENTATIO N. Intro to Philosophy, Summer 2011 Benjamin Visscher Hole IV WHAT IS PHILOSOPHY? ARGUMENTATIO N Intro to Philosophy, Summer 2011 Benjamin Visscher Hole IV PHILOSOPHY CONSISTS IN CRITICAL ENGAGEMENT The philosophers we re reading have developed arguments to try to

More information

Vocabulary. Term Page Definition Clarifying Example. biconditional statement. conclusion. conditional statement. conjecture.

Vocabulary. 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 information

Philosophical argument

Philosophical argument Michael Lacewing Philosophical argument At the heart of philosophy is philosophical argument. Arguments are different from assertions. Assertions are simply stated; arguments always involve giving reasons.

More information

1.1 Logical Form and Logical Equivalence 1

1.1 Logical Form and Logical Equivalence 1 Contents Chapter I The Logic of Compound Statements 1.1 Logical Form and Logical Equivalence 1 Identifying logical form; Statements; Logical connectives: not, and, and or; Translation to and from symbolic

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

Generalized Modus Ponens

Generalized Modus Ponens Generalized Modus Ponens This rule allows us to derive an implication... True p 1 and... p i and... p n p 1... p i-1 and p i+1... p n implies p i implies q implies q allows: a 1 and... a i and... a n implies

More information

Hypothetical Syllogisms 1

Hypothetical Syllogisms 1 Phil 2302 Intro to Logic Dr. Naugle Hypothetical Syllogisms 1 Compound syllogisms are composed of different kinds of sentences in their premises and conclusions (not just categorical propositions, statements

More information

136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 CHAPTER 4. INDUCTION, GRAPHS AND TREES 136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics

More information

Semantics for the Predicate Calculus: Part I

Semantics for the Predicate Calculus: Part I Semantics for the Predicate Calculus: Part I (Version 0.3, revised 6:15pm, April 14, 2005. Please report typos to hhalvors@princeton.edu.) The study of formal logic is based on the fact that the validity

More information

Day 2: Logic and Proof

Day 2: Logic and Proof Day 2: Logic and Proof George E. Hrabovsky MAST Introduction This is the second installment of the series. Here I intend to present the ideas and methods of proof. Logic and proof To begin with, I will

More information

(LMCS, p. 317) V.1. First Order Logic. This is the most powerful, most expressive logic that we will examine.

(LMCS, p. 317) V.1. First Order Logic. This is the most powerful, most expressive logic that we will examine. (LMCS, p. 317) V.1 First Order Logic This is the most powerful, most expressive logic that we will examine. Our version of first-order logic will use the following symbols: variables connectives (,,,,

More information

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called

More information

Elementary 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. 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 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

A Guide to Proof-Writing

A Guide to Proof-Writing by Ron Morash, University of Michigan Dearborn PW-1 At the end of Section 1.7, the text states, We have not given a procedure that can be used for proving theorems in mathematics. It is a deep theorem

More information

The last three chapters introduced three major proof techniques: direct,

The 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 information

Properties of Real Numbers

Properties 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 information

A Few Basics of Probability

A Few Basics of Probability A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study

More information

CS510 Software Engineering

CS510 Software Engineering CS510 Software Engineering Propositional Logic Asst. Prof. Mathias Payer Department of Computer Science Purdue University TA: Scott A. Carr Slides inspired by Xiangyu Zhang http://nebelwelt.net/teaching/15-cs510-se

More information

Announcements. CompSci 230 Discrete Math for Computer Science Sets. Introduction to Sets. Sets

Announcements. CompSci 230 Discrete Math for Computer Science Sets. Introduction to Sets. Sets CompSci 230 Discrete Math for Computer Science Sets September 12, 2013 Prof. Rodger Slides modified from Rosen 1 nnouncements Read for next time Chap. 2.3-2.6 Homework 2 due Tuesday Recitation 3 on Friday

More information

Chapter 1. Use the following to answer questions 1-5: In the questions below determine whether the proposition is TRUE or FALSE

Chapter 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 information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate 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 information

SJÄLVSTÄNDIGA ARBETEN I MATEMATIK

SJÄLVSTÄNDIGA ARBETEN I MATEMATIK SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Automated Theorem Proving av Tom Everitt 2010 - No 8 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM

More information

Beyond Propositional Logic Lukasiewicz s System

Beyond Propositional Logic Lukasiewicz s System Beyond Propositional Logic Lukasiewicz s System Consider the following set of truth tables: 1 0 0 1 # # 1 0 # 1 1 0 # 0 0 0 0 # # 0 # 1 0 # 1 1 1 1 0 1 0 # # 1 # # 1 0 # 1 1 0 # 0 1 1 1 # 1 # 1 Brandon

More information

BASIC COMPOSITION.COM USING LOGIC

BASIC COMPOSITION.COM USING LOGIC BASIC COMPOSITION.COM USING LOGIC As we have noted, Aristotle advocated that persuasion comes from the use of different kinds of support, including natural and artificial forms of support. We have discussed

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

WRITING PROOFS. Christopher Heil Georgia Institute of Technology

WRITING 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 information

Symbolic Logic on the TI-92

Symbolic Logic on the TI-92 Symbolic Logic on the TI-92 Presented by Lin McMullin The Sixth Conference on the Teaching of Mathematics June 20 & 21, 1997 Milwaukee, Wisconsin 1 Lin McMullin Mathematics Department Representative, Burnt

More information

Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

More information

Introduction to Logic: Argumentation and Interpretation. Vysoká škola mezinárodních a veřejných vztahů PhDr. Peter Jan Kosmály, Ph.D. 9. 3.

Introduction to Logic: Argumentation and Interpretation. Vysoká škola mezinárodních a veřejných vztahů PhDr. Peter Jan Kosmály, Ph.D. 9. 3. Introduction to Logic: Argumentation and Interpretation Vysoká škola mezinárodních a veřejných vztahů PhDr. Peter Jan Kosmály, Ph.D. 9. 3. 2016 tests. Introduction to Logic: Argumentation and Interpretation

More information

Incenter Circumcenter

Incenter Circumcenter TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is

More information

Foundations of Logic and Mathematics

Foundations of Logic and Mathematics Yves Nievergelt Foundations of Logic and Mathematics Applications to Computer Science and Cryptography Birkhäuser Boston Basel Berlin Contents Preface Outline xiii xv A Theory 1 0 Boolean Algebraic Logic

More information

Mathematical induction. Niloufar Shafiei

Mathematical induction. Niloufar Shafiei Mathematical induction Niloufar Shafiei Mathematical induction Mathematical induction is an extremely important proof technique. Mathematical induction can be used to prove results about complexity of

More information

CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 Solutions CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

More information

Midterm Practice Problems

Midterm Practice Problems 6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator

More information

Geometry Chapter 2 Study Guide

Geometry Chapter 2 Study Guide Geometry Chapter 2 Study Guide Short Answer ( 2 Points Each) 1. (1 point) Name the Property of Equality that justifies the statement: If g = h, then. 2. (1 point) Name the Property of Congruence that justifies

More information

MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction. MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

More information

Foundations of Geometry 1: Points, Lines, Segments, Angles

Foundations of Geometry 1: Points, Lines, Segments, Angles Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.

More information

Reasoning and Proof Review Questions

Reasoning 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 information

3. Logical Reasoning in Mathematics

3. 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 information

DEDUCTIVE & INDUCTIVE REASONING

DEDUCTIVE & INDUCTIVE REASONING DEDUCTIVE & INDUCTIVE REASONING Expectations 1. Take notes on inductive and deductive reasoning. 2. This is an information based presentation -- I simply want you to be able to apply this information to

More information

This asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.

This asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements. 3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but

More information

L - Standard Letter Grade P - Pass/No Pass Repeatability: N - Course may not be repeated

L - Standard Letter Grade P - Pass/No Pass Repeatability: N - Course may not be repeated Course: MATH 26 Division: 10 Also Listed As: 200930, INACTIVE COURSE Short Title: Full Title: DISCRETE MATHEMATIC Discrete Mathematics Contact Hours/Week Lecture: 4 Lab: 0 Other: 0 Total: 4 4 Number of

More information

Geometry Chapter 2: Geometric Reasoning Lesson 1: Using Inductive Reasoning to Make Conjectures Inductive Reasoning:

Geometry Chapter 2: Geometric Reasoning Lesson 1: Using Inductive Reasoning to Make Conjectures Inductive Reasoning: Geometry Chapter 2: Geometric Reasoning Lesson 1: Using Inductive Reasoning to Make Conjectures Inductive Reasoning: Conjecture: Advantages: can draw conclusions from limited information helps us to organize

More information

alternate interior angles

alternate 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 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

Odd induced subgraphs in graphs of maximum degree three

Odd induced subgraphs in graphs of maximum degree three Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A long-standing

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

The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.

The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors. The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,

More information

Mathematics Georgia Performance Standards

Mathematics 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 information

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

Solutions Q1, Q3, Q4.(a), Q5, Q6 to INTLOGS16 Test 1

Solutions Q1, Q3, Q4.(a), Q5, Q6 to INTLOGS16 Test 1 Solutions Q1, Q3, Q4.(a), Q5, Q6 to INTLOGS16 Test 1 Prof S Bringsjord 0317161200NY Contents I Problems 1 II Solutions 3 Solution to Q1 3 Solutions to Q3 4 Solutions to Q4.(a) (i) 4 Solution to Q4.(a)........................................

More information

Mathematical Induction. Lecture 10-11

Mathematical Induction. Lecture 10-11 Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach

More information

Perfect being theology and modal truth

Perfect being theology and modal truth Perfect being theology and modal truth Jeff Speaks February 9, 2016 Perfect being theology is the attempt to use the principle that God is the greatest possible being to derive claims about the divine

More information

Automated Theorem Proving - summary of lecture 1

Automated Theorem Proving - summary of lecture 1 Automated Theorem Proving - summary of lecture 1 1 Introduction Automated Theorem Proving (ATP) deals with the development of computer programs that show that some statement is a logical consequence of

More information

13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385

13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385 mcs-ftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite

More information

Schedule. Logic (master program) Literature & Online Material. gic. Time and Place. Literature. Exercises & Exam. Online Material

Schedule. Logic (master program) Literature & Online Material. gic. Time and Place. Literature. Exercises & Exam. Online Material OLC mputational gic Schedule Time and Place Thursday, 8:15 9:45, HS E Logic (master program) Georg Moser Institute of Computer Science @ UIBK week 1 October 2 week 8 November 20 week 2 October 9 week 9

More information

Conjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true)

Conjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true) Mathematical Sentence - a sentence that states a fact or complete idea Open sentence contains a variable Closed sentence can be judged either true or false Truth value true/false Negation not (~) * Statement

More information

Fuzzy Implication Rules. Adnan Yazıcı Dept. of Computer Engineering, Middle East Technical University Ankara/Turkey

Fuzzy Implication Rules. Adnan Yazıcı Dept. of Computer Engineering, Middle East Technical University Ankara/Turkey Fuzzy Implication Rules Adnan Yazıcı Dept. of Computer Engineering, Middle East Technical University Ankara/Turkey Fuzzy If-Then Rules Remember: The difference between the semantics of fuzzy mapping rules

More information

Math 55: Discrete Mathematics

Math 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 information

4 Domain Relational Calculus

4 Domain Relational Calculus 4 Domain Relational Calculus We now present two relational calculi that we will compare to RA. First, what is the difference between an algebra and a calculus? The usual story is that the algebra RA is

More information

p: I am elected q: I will lower the taxes

p: 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 information