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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

1 CS 441 Discrete Mathematics for CS Lecture 5 Predicate logic Milos Hauskrecht 5329 Sennott Square Negation of quantifiers English statement: Nothing is perfect. Translation: x Perfect(x) Another way to express the same meaning: Everything... 1

2 Negation of quantifiers English statement: Nothing is perfect. Translation: x Perfect(x) Another way to express the same meaning: Everything is imperfect. Translation: x Perfect(x) Conclusion: x P (x) is equivalent to x P(x) Negation of quantifiers English statement: It is not the case that all dogs are fleabags. Translation: x Dog(x) Fleabag(x) Another way to express the same meaning: There is a dog that 2

3 Negation of quantifiers English statement: It is not the case that all dogs are fleabags. Translation: x Dog(x) Fleabag(x) Another way to express the same meaning: There is a dog that is not a fleabag. Translation: x Dog(x) Fleabag(x) Logically equivalent to: x ( Dog(x) Fleabag(x) ) Conclusion: x P (x) is equivalent to x P(x) Negation of quantified statements (aka DeMorgan Laws for quantifiers) Negation x P(x) x P(x) Equivalent x P(x) x P(x) 3

4 Formal and informal proofs Theorems and proofs The truth value of some statement about the world is obvious and easy to assign The truth of other statements may not be obvious,. But it may still follow (be derived) from known facts about the world To show the truth value of such a statement following from other statements we need to provide a correct supporting argument - a proof Important questions: When is the argument correct? How to construct a correct argument, what method to use? 4

5 Theorems and proofs Theorem: a statement that can be shown to be true. Typically the theorem looks like this: (p1 p2 p3 pn ) q Premises conclusion Example: Fermat s Little theorem: If p is a prime and a is an integer not divisible by p, then: a p 1 1 mod p Theorems and proofs Theorem: a statement that can be shown to be true. Typically the theorem looks like this: (p1 p2 p3 pn ) q Premises conclusion Example: Premises (hypotheses) Fermat s Little theorem: If p is a prime and a is an integer not divisible by p, then: a p 1 1mod p conclusion 5

6 Formal proofs Proof: Provides an argument supporting the validity of the statement Proof of the theorem: shows that the conclusion follows from premises May use: Premises Axioms Results of other theorems Formal proofs: steps of the proofs follow logically from the set of premises and axioms Formal proofs Formal proofs: show that steps of the proofs follow logically from the set of hypotheses and axioms premises + axioms + proved theorems conclusion In this class we assume formal proofs in the propositional logic 6

7 Using logical equivalence rules Proofs based on logical equivalences. A proposition or its part can be transformed using a sequence of equivalence rewrites till some conclusion can be reached. Example: Show that () p is a tautology. Proof: (we must show () p <=> T) () p <=> () p Useful <=> [ p q] p DeMorgan <=> [ q p] p Commutative <=> q [ p p ] Associative <=> q [ T ] Useful Formal proofs Allow us to infer new True statements from known True statements premises + axioms + proved theorems True conclusion True? 7

8 Rules of inference Rules of inference: Allow us to infer new True statements from existing True statements Represent logically valid inference patterns Example: Modus Ponens, or the Law of Detachment Rule of inference p q Given p is true and the implication is true then q is true. Rules of inference Rules of inference: logically valid inference patterns Example; Modus Ponens, or the Law of Detachment Rule of inference p q Given p is true and the implication is true then q is true. p q False False True False True True True False False True True True 8

9 Rules of inference Rules of inference: logically valid inference patterns Example; Modus Ponens, or the Law of Detachment Rule of inference p q Given p is true and the implication is true then q is true. p q False False True False True True True False False True True True Rules of inference Rules of inference: logically valid inference patterns Example: Modus Ponens, or the Law of Detachment Rules of inference p q Given p is true and the implication is true then q is true. Tautology Form: (p ()) q 9

10 Addition p () Rules of inference p Example: It is below freezing now. Therefore, it is below freezing or raining snow. Simplification () p p Example: It is below freezing and snowing. Therefore it is below freezing. Rules of inference Modus Tollens (modus ponens for the contrapositive) [ q ()] p q p Hypothetical Syllogism [() (q r)] (p r) q r p r Disjunctive Syllogism [() p] q p q 10

11 Rules of inference Logical equivalences (discussed earlier) A <=> B A B is a tautology Example: De Morgan Law ( ) <=> p q ( ) p q is a tautology Rules of inference A valid argument is one built using the rules of inference from premises (hypotheses). When all premises are true the argument should lead us to the correct conclusion. (p1 p2 p3 pn ) q How to use the rules of inference? 11

12 Applying rules of inference Assume the following statements (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. Show that all these lead to a conclusion: We will be home by sunset. Applying rules of inference Text: (1) It is not sunny this afternoon and it is colder than yesterday. (2) We will go swimming only if it is sunny. (3) If we do not go swimming then we will take a canoe trip. (4) If we take a canoe trip, then we will be home by sunset. Propositions: p = It is sunny this afternoon, q = it is colder than yesterday, r = We will go swimming, s= we will take a canoe trip t= We will be home by sunset Translation: Assumptions: (1), (2)? We want to show: t 12

13 Applying rules of inference Approach: p = It is sunny this afternoon, q = it is colder than yesterday, r = We will go swimming, s= we will take a canoe trip t= We will be home by sunset Translation: We will go swimming only if it is sunny. Ambiguity: r p or p r? Sunny is a must before we go swimming Thus, if we indeed go swimming it must be sunny, therefore r p Applying rules of inference Text: (1) It is not sunny this afternoon and it is colder than yesterday. (2) We will go swimming only if it is sunny. (3) If we do not go swimming then we will take a canoe trip. (4) If we take a canoe trip, then we will be home by sunset. Propositions: p = It is sunny this afternoon, q = it is colder than yesterday, r = We will go swimming, s= we will take a canoe trip t= We will be home by sunset Translation: Assumptions: (1), (2) r p, (3) r s, (4) s t We want to show: t 13

14 Proofs using rules of inference Translations: Assumptions:, r p, r s, s t We want to show: t Proof: 1. Hypothesis 2. p Simplification 3. r p Hypothesis 4. r Modus tollens (step 2 and 3) 5. r s Hypothesis 6. s Modus ponens (steps 4 and 5) 7. s t Hypothesis 8. t Modus ponens (steps 6 and 7) end of proof Informal proofs Proving theorems in practice: The steps of the proofs are not expressed in any formal language as e.g. propositional logic Steps are argued less formally using English, mathematical formulas and so on One must always watch the consistency of the argument made, logic and its rules can often help us to decide the soundness of the argument if it is in question We use (informal) proofs to illustrate different methods of proving theorems 14

15 Methods of proving theorems Basic methods to prove the theorems: Direct proof is proved by showing that if p is true then q follows Indirect proof Show the contrapositive q p. If q holds then p follows Proof by contradiction Show that (p q) contradicts the assumptions Proof by cases Proofs of equivalence is replaced with () (q p) Sometimes one method of proof does not go through as nicely as the other method. Hint: try more than one approach. Direct proof is proved by showing that if p is true then q follows Example: Prove that If n is odd, then n 2 is odd. Proof: Assume the hypothesis is true, i.e. suppose n is odd. Then n = 2k + 1, where k is an integer. n 2 = (2k + 1) 2 = 4k 2 + 4k + 1 = 2(2k 2 + 2k) + 1 Therefore, n 2 is odd. 15

16 Indirect proof To show prove its contrapositive q p Why? and q p are equivalent!!! Assume q is true, show that p is true. Example: Prove If 3n + 2 is odd then n is odd. Proof: Assume n is even, that is n = 2k, where k is an integer. Then: 3n + 2 = 3(2k) + 2 = 6k + 2 = 2(3k+1) Therefore 3n + 2 is even. Indirect proof To show prove its contrapositive q p Why? and q p are equivalent!!! Assume q is true, show that p is true. Example: Prove If 3n + 2 is odd then n is odd. Proof: Assume n is even, that is n = 2k, where k is an integer. Then: 3n + 2 = 3(2k) + 2 = 6k + 2 = 2(3k+1) Therefore 3n + 2 is even. We proved n is odd 3n + 2 is odd. This is equivalent to 3n + 2 is odd n is odd. 16

### 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

### Chapter I Logic and Proofs

MATH 1130 1 Discrete Structures Chapter I Logic and Proofs Propositions A proposition is a statement that is either true (T) or false (F), but or both. s Propositions: 1. I am a man.. I am taller than

### 1.5 Rules of Inference

1.5 Rules of Inference (Inference: decision/conclusion by evidence/reasoning) Introduction Proofs are valid arguments that establish the truth of statements. An argument is a sequence of statements that

### Inference Rules and Proof Methods

Inference Rules and Proof Methods Winter 2010 Introduction Rules of Inference and Formal Proofs Proofs in mathematics are valid arguments that establish the truth of mathematical statements. An argument

### 1.5 Arguments & Rules of Inference

1.5 Arguments & Rules of Inference Tools for establishing the truth of statements Argument involving a seuence of propositions (premises followed by a conclusion) Premises 1. If you have a current password,

### LOGICAL INFERENCE & PROOFs. Debdeep Mukhopadhyay Dept of CSE, IIT Madras

LOGICAL INFERENCE & PROOFs Debdeep Mukhopadhyay Dept of CSE, IIT Madras Defn A theorem is a mathematical assertion which can be shown to be true. A proof is an argument which establishes the truth of a

### 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

### 1.5 Methods of Proof INTRODUCTION

1.5 Methods of Proof INTRODUCTION Icon 0049 Two important questions that arise in the study of mathematics are: (1) When is a mathematical argument correct? (2) What methods can be used to construct mathematical

### What is logic? Propositional Logic. Negation. Propositions. This is a contentious question! We will play it safe, and stick to:

Propositional Logic This lecture marks the start of a new section of the course. In the last few lectures, we have had to reason formally about concepts. This lecture introduces the mathematical language

### Discrete Mathematics Lecture 1 Logic of Compound Statements. Harper Langston New York University

Discrete Mathematics Lecture 1 Logic of Compound Statements Harper Langston New York University Administration Class Web Site http://cs.nyu.edu/courses/summer05/g22.2340-001/ Mailing List Subscribe at

### n logical not (negation) n logical or (disjunction) n logical and (conjunction) n logical exclusive or n logical implication (conditional)

Discrete Math Review Discrete Math Review (Rosen, Chapter 1.1 1.6) TOPICS Propositional Logic Logical Operators Truth Tables Implication Logical Equivalence Inference Rules What you should know about propositional

### CS 441 Discrete Mathematics for CS Lecture 2. Propositional logic. CS 441 Discrete mathematics for CS. Course administration

CS 441 Discrete Mathematics for CS Lecture 2 Propositional logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Course administration Homework 1 First homework assignment is out today will be posted

### CSI 2101 / Rules of Inference ( 1.5)

CSI 2101 / Rules of Inference ( 1.5) Introduction what is a proof? Valid arguments in Propositional Logic equivalence of quantified expressions Rules of Inference in Propositional Logic the rules using

### Review Name Rule of Inference

CS311H: Discrete Mathematics Review Name Rule of Inference Modus ponens φ 2 φ 2 Modus tollens φ 2 φ 2 Inference Rules for Quantifiers Işıl Dillig Hypothetical syllogism Or introduction Or elimination And

### 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

### DISCRETE MATHEMATICS W W L CHEN

DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free

### Logical Inference and Mathematical Proof

Logical Inference and Mathematical Proof CSE 191, Class Note 03: Logical Inference and Mathematical Proof Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures

### Mathematical induction & Recursion

CS 441 Discrete Mathematics for CS Lecture 15 Mathematical induction & Recursion Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Proofs Basic proof methods: Direct, Indirect, Contradiction, By Cases,

### Discrete Mathematics for Computer Science

CS 441 Discrete Mathematics for CS Discrete Mathematics for Computer Science Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Course administrivia Instructor: Milos Hauskrecht 5329 Sennott Square

### 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

### 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

### Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.

Math 232 - Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the

### Exam 1 Answers: Logic and Proof

Q250 FALL 2012, INDIANA UNIVERSITY Exam 1 Answers: Logic and Proof September 17, 2012 Instructions: Please answer each question completely, and show all of your work. Partial credit will be awarded where

### 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

### Definition 10. A proposition is a statement either true or false, but not both.

Chapter 2 Propositional Logic Contrariwise, continued Tweedledee, if it was so, it might be; and if it were so, it would be; but as it isn t, it ain t. That s logic. (Lewis Carroll, Alice s Adventures

### Even Number: An integer n is said to be even if it has the form n = 2k for some integer k. That is, n is even if and only if n divisible by 2.

MATH 337 Proofs Dr. Neal, WKU This entire course requires you to write proper mathematical proofs. All proofs should be written elegantly in a formal mathematical style. Complete sentences of explanation

### Discrete Structures Lecture Rules of Inference

Term argument valid remise fallacy Definition A seuence of statements that ends with a conclusion. The conclusion, or final statement of the argument, must follow from the truth of the receding statements,

### 1 Proposition, Logical connectives and compound statements

Discrete Mathematics: Lecture 4 Introduction to Logic Instructor: Arijit Bishnu Date: July 27, 2009 1 Proposition, Logical connectives and compound statements Logic is the discipline that deals with the

### Propositional Logic. Definition: A proposition or statement is a sentence which is either true or false.

Propositional Logic Definition: A proposition or statement is a sentence which is either true or false. Definition:If a proposition is true, then we say its truth value is true, and if a proposition is

### Propositional Logic and Methods of Inference SEEM

Propositional Logic and Methods of Inference SEEM 5750 1 Logic Knowledge can also be represented by the symbols of logic, which is the study of the rules of exact reasoning. Logic is also of primary importance

### Mathematical Induction

Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers

### 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

### 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

### Logic, Sets, and Proofs

Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical

### Problems on Discrete Mathematics 1

Problems on Discrete Mathematics 1 Chung-Chih Li 2 Kishan Mehrotra 3 Syracuse University, New York L A TEX at January 11, 2007 (Part I) 1 No part of this book can be reproduced without permission from

### 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

### 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.

### Integers: applications, base conversions.

CS 441 Discrete Mathematics for CS Lecture 14 Integers: applications, base conversions. Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Modular arithmetic in CS Modular arithmetic and congruencies

### Logic and Proofs. Chapter 1

Section 1.0 1.0.1 Chapter 1 Logic and Proofs 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers 1.5 Rules of Inference 1.6 Introduction to Proofs

### CHAPTER 1 The Foundations: Logic and Proof, Sets, and Functions

Section 1.5 Methods of Proof 1 CHAPTER 1 The Foundations: Logic and Proof, Sets, and Functions SECTION 1.5 Methods of Proof Learning to construct good mathematical proofs takes years. There is no algorithm

### 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

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics. Introductory Notes in Discrete Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics Introductory Notes in Discrete Mathematics Marcel B. Finan c All Rights Reserved Last Updated April 6, 2016 Preface

### 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)

### Logic will get you from A to B. Imagination will take you everywhere.

Chapter 3 Predicate Logic Logic will get you from A to B. Imagination will take you everywhere. A. Einstein In the previous chapter, we studied propositional logic. This chapter is dedicated to another

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### 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

### CSE 191, Class Note 01 Propositional Logic Computer Sci & Eng Dept SUNY Buffalo

Propositional Logic CSE 191, Class Note 01 Propositional Logic Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 37 Discrete Mathematics What is Discrete

### 3.3. INFERENCE 105. Table 3.5: Another look at implication. p q p q T T T T F F F T T F F T

3.3. INFERENCE 105 3.3 Inference Direct Inference (Modus Ponens) and Proofs We concluded our last section with a proof that the sum of two even numbers is even. That proof contained several crucial ingredients.

### Inference in propositional logic

C 1571 Introduction to AI Lecture 13 Inference in propositional logic Milos Hauskrecht milos@cs.pitt.edu 5329 ennott quare Logical inference problem Logical inference problem: Given: a knowledge base KB

### Integers and division

CS 441 Discrete Mathematics for CS Lecture 12 Integers and division Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Symmetric matrix Definition: A square matrix A is called symmetric if A = A T.

### Chapter 1 LOGIC AND PROOF

Chapter 1 LOGIC AND PROOF To be able to understand mathematics and mathematical arguments, it is necessary to have a solid understanding of logic and the way in which known facts can be combined to prove

### MATH 55: HOMEWORK #2 SOLUTIONS

MATH 55: HOMEWORK # SOLUTIONS ERIC PETERSON * 1. SECTION 1.5: NESTED QUANTIFIERS 1.1. Problem 1.5.8. Determine the truth value of each of these statements if the domain of each variable consists of all

### 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

### Direct Proofs. CS 19: Discrete Mathematics. Direct Proof: Example. Indirect Proof: Example. Proofs by Contradiction and by Mathematical Induction

Direct Proofs CS 19: Discrete Mathematics Amit Chakrabarti Proofs by Contradiction and by Mathematical Induction At this point, we have seen a few examples of mathematical proofs. These have the following

### Problems on Discrete Mathematics 1

Problems on Discrete Mathematics 1 Chung-Chih Li 2 Kishan Mehrotra 3 L A TEX at July 18, 2007 1 No part of this book can be reproduced without permission from the authors 2 Illinois State University, Normal,

### 31 is a prime number is a mathematical statement (which happens to be true).

Chapter 1 Mathematical Logic In its most basic form, Mathematics is the practice of assigning truth to welldefined statements. In this course, we will develop the skills to use known true statements to

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

### vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

### 1 Propositional Logic - Axioms and Inference Rules

1 Propositional Logic - Axioms and Inference Rules Axioms Axiom 1.1 [Commutativity] (p q) (q p) (p q) (q p) (p q) (q p) Axiom 1.2 [Associativity] p (q r) (p q) r p (q r) (p q) r Axiom 1.3 [Distributivity]

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 1

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 1 Course Outline CS70 is a course on Discrete Mathematics and Probability for EECS Students. The purpose of the course

### Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.

M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider

### WUCT121. Discrete Mathematics. Logic

WUCT121 Discrete Mathematics Logic 1. Logic 2. Predicate Logic 3. Proofs 4. Set Theory 5. Relations and Functions WUCT121 Logic 1 Section 1. Logic 1.1. Introduction. In developing a mathematical theory,

### 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

### Mathematical Induction

Mathematical Induction MAT30 Discrete Mathematics Fall 016 MAT30 (Discrete Math) Mathematical Induction Fall 016 1 / 19 Outline 1 Mathematical Induction Strong Mathematical Induction MAT30 (Discrete Math)

### 3.3 MATHEMATICAL INDUCTION

Section 3.3 Mathematical Induction 3.3.1 3.3 MATHEMATICAL INDUCTION From modus ponens: p p q q basis assertion conditional assertion conclusion we can easily derive double modus ponens : p 0 p 0 p 1 p

### Discrete Mathematics, Chapter : Propositional Logic

Discrete Mathematics, Chapter 1.1.-1.3: Propositional Logic Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.1-1.3 1 / 21 Outline 1 Propositions

### 2.4 Mathematical Induction

2.4 Mathematical Induction What Is (Weak) Induction? The Principle of Mathematical Induction works like this: What Is (Weak) Induction? The Principle of Mathematical Induction works like this: We want

### Oh Yeah? Well, Prove It.

Oh Yeah? Well, Prove It. MT 43A - Abstract Algebra Fall 009 A large part of mathematics consists of building up a theoretical framework that allows us to solve problems. This theoretical framework is built

### Artificial Intelligence Automated Reasoning

Artificial Intelligence Automated Reasoning Andrea Torsello Automated Reasoning Very important area of AI research Reasoning usually means deductive reasoning New facts are deduced logically from old ones

### Midterm Examination 1 with Solutions - Math 574, Frank Thorne Thursday, February 9, 2012

Midterm Examination 1 with Solutions - Math 574, Frank Thorne (thorne@math.sc.edu) Thursday, February 9, 2012 1. (3 points each) For each sentence below, say whether it is logically equivalent to the sentence

### Discrete Mathematics

Slides for Part IA CST 2014/15 Discrete Mathematics Prof Marcelo Fiore Marcelo.Fiore@cl.cam.ac.uk What are we up to? Learn to read and write, and also work with, mathematical

### Fundamentals of Mathematics Lecture 6: Propositional Logic

Fundamentals of Mathematics Lecture 6: Propositional Logic Guan-Shieng Huang National Chi Nan University, Taiwan Spring, 2008 1 / 39 Connectives Propositional Connectives I 1 Negation: (not A) A A T F

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### Computing Science 272 Solutions to Midterm Examination I Tuesday February 8, 2005

Computing Science 272 Solutions to Midterm Examination I Tuesday February 8, 2005 Department of Computing Science University of Alberta Question 1. 8 = 2+2+2+2 pts (a) How many 16-bit strings contain exactly

### 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

### MATHS 315 Mathematical Logic

MATHS 315 Mathematical Logic Second Semester, 2006 Contents 2 Formal Statement Logic 1 2.1 Post production systems................................. 1 2.2 The system L.......................................

### 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

### Chapter 1. Logic and Proof

Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known

### APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

### 2. Propositional Equivalences

2. PROPOSITIONAL EQUIVALENCES 33 2. Propositional Equivalences 2.1. Tautology/Contradiction/Contingency. Definition 2.1.1. A tautology is a proposition that is always true. Example 2.1.1. p p Definition

### 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

### CS 2710 Foundations of AI. Lecture 12. Inference in FOL. CS 2710 Foundations of AI. Optimization competition winners.

Lecture 12 Inference in FOL Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Optimization competition winners CS2710 Homework 3 Simulated annealing 1. Valko Michal 2. Zhang Yue 3. Villamarin Ricardo

### Sets and set operations: cont. Functions.

CS 441 Discrete Mathematics for CS Lecture 8 Sets and set operations: cont. Functions. Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Set Definition: set is a (unordered) collection of objects.

### 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

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### 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.

### Mathematical Induction. Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition)

Mathematical Induction Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition) Mathmatical Induction Mathmatical induction can be used to prove statements that assert that P(n) is true for all positive

### CSI 2101 Discrete Structures Winter 2012

CSI 2101 Discrete Structures Winter 2012 Prof. Lucia Moura University of Ottawa Homework Assignment #1 (100 points, weight 5%) Due: Thursday Feb 9, at 1:00 p.m. (in lecture); assignments with lateness

### Students in their first advanced mathematics classes are often surprised

CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are

### Definition of Statement: A group words or symbols that can be classified as true or false.

Logic Math 116 Section 3.1 Logic and Statements Statements Definition of Statement: A group words or symbols that can be classified as true or false. Examples of statements Violets are blue Five is a natural

### Announcements. CS243: Discrete Structures. More on Cryptography and Mathematical Induction. Agenda for Today. Cryptography

Announcements CS43: Discrete Structures More on Cryptography and Mathematical Induction Işıl Dillig Class canceled next Thursday I am out of town Homework 4 due Oct instead of next Thursday (Oct 18) Işıl

### 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

### 22C:19 Discrete Math. So. What is it? Why discrete math? Fall 2009 Hantao Zhang

22C:19 Discrete Math Fall 2009 Hantao Zhang So. What is it? Discrete mathematics is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the

### 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

### We now explore a third method of proof: proof by contradiction.

CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement

### Resolution in Propositional and First-Order Logic

Resolution in Propositional and First-Order Logic Inference rules Logical inference creates new sentences that logically follow from a set of sentences (KB) An inference rule is sound if every sentence

### CS 2336 Discrete Mathematics

CS 2336 Discrete Mathematics Lecture 2 Logic: Predicate Calculus 1 Outline Predicates Quantifiers Binding Applications Logical Equivalences 2 Predicates In mathematics arguments, we will often see sentences