# Chapter I Logic and Proofs

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 MATH 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 170 cm. 3. You are studying in Baptist U = 3. Not propositions: 1. How are you?. Go to catch the dog. 3. I like this colour. Negation of a Proposition Let p be a proposition. The statement It is not the case that p is another proposition, called the negation of p. The negation of p is denoted by p. p and read not P: It is a sunny day. p : It is not the case that it is a sunny day., or simply It is not a sunny day. Truth Table A truth table displays the relationships between the truth values of propositions. Truth tables are especially valuable in the determination of the truth values of propositions constructed from simpler propositions.

2 MATH 1130 Discrete Structures The truth table for the negation of a proposition p T F p F T Logic Operators (Connectives) Conjunction Let p and q be propositions. The proposition p and q, denoted by true when both p and q are true and is false otherwise. The proposition conjunction of p and q. p q, is the proposition that is p q is called the The truth table for the conjunction of p and q p q p q T T T T F F F T F F F F Disjunction Let p and q be propositions. The proposition p or q, denoted by false when p and q are both false and true otherwise. The proposition disjunction of p and q. p q, is the proposition that is p q is called the The truth table for the disjunction of p and q p q p q T T T T F T F T T F F F

3 MATH Discrete Structures Exclusive Or Let p and q be propositions. The exclusive or of p and q, denoted by true when exactly one of p and q is true and is false otherwise. p q, is the proposition that is The truth table for the exclusive or of p and q p q p q T T F T F T F T T F F F Conditional Propositions Implication Let p and q be propositions. The implication p q is the proposition that is false when p is true and q is false and true otherwise. In this implication, p is called the hypothesis and q is called the conclusion. The truth table for the implication p q p q p q T T T T F F F T T F F T Remarks: I) Equivalent expressions of implication 1. if p, then q. p is sufficient for q 3. p implies q 4. p only if q 5. q is necessary for p II) Related Implications 1. q p is called the converse of p q. q p is called the contrapositive of p q

4 MATH Discrete Structures Biconditional Let p and q be propositions. The biconditional p q is the proposition that is true when p and q have the same truth values and is false otherwise. In this biconditional, p is necessary and sufficient for q, or p if and only if q. The truth table for the biconditional p q p q p q T T T T F F F T F F F T Translating English Sentences He will not be charged (c) if he is handsome (h) or he is muscular (m). h m ( ) c h m h m c c T T T F F T T T T T T F T F T F F T T F T T T F F T T F F T F T T T T F F F F T F T F F F T T F Bit String A bit string is a sequence of zero or more bits. The length of a bit string is the number of bits in the string is a 10-bit string.

5 MATH Discrete Structures Bitwise OR, bitwise AND and bitwise XOR We define the bitwise OR, bitwise AND and bitwise XOR of two strings of the same length to be the strings that have as their bits the OR, AND and XOR of the corresponding bits in the two strings, respectively bitwise OR bitwise AND bitwise XOR Logical equivalence Tautology, Contradiction and Contingency A compound proposition that is always true, no matter what the truth values of the propositions that occur in it, is called a tautology. A compound proposition that is always false is called a contradiction. Finally, a proposition that is neither a tautology nor a contradiction is called a contingency. Truth table of examples of a tautology and a contradiction p p p p p p T F T F F T T F Logically Equivalent The propositions p and q are called logically equivalent if p q denotes that p and q are logically equivalent. p q is a tautology. The notation

6 MATH Discrete Structures The following truth table shows that the compound propositions ( p q) logically equivalent. p q q p ( p q) and p q are p q p q T T T F F F F T F T F F T F F T T F T F F F F F T T T T Exercise Complete the following truth table to show that p q and p q are logically equivalent. p q p p q p q T T T F F T F F Logical Equivalences Equivalence p T p p F p p T T p F F p p p p p p Name Identity laws Domination laws Idempotent laws ( p) p Double negative law p q q p p q q p ( p q) r p ( q r) ( p q) r p ( q r) p ( q r) ( p q) ( p r) p ( q r) ( p q) ( p r) ( p q) p q ( p q) p q Commutative laws Associative laws Distributive laws De Morgan s laws

7 MATH Discrete Structures p p T p p F p q p q Predicates and Quantifiers Predicates In statements involving variables, there are two parts the variable (is the subject of the statement) and predicate (refers to a property that the subject can have). In the statement: x > 3 (x is greater than 3) x is the variable and is greater than 3 is the predicate. Let P ( x) denote the statement x > 3. The value of P ( 4) is true and the value of ( ) P is false. Universe of Discourse Many mathematical statements assert that a property is true for all values of a variable in a particular domain, called the universe of discourse. Universal Quantification and Universal Quantifier The universal quantification of P ( x) is the proposition: ( x) universe of discourse, and is denoted by x P ( x) for all (every) x P ( x) the universal quantifier. P is true for all values of x in the. Here, is called Existential Quantification and Existential Quantifier The existential quantification of P ( x) is the proposition: There exists an element x in the universe of discourse such that P ( x) is true, and is denoted by x P ( x) for some x P ( x). Here, is called existential quantifier. Statement When True? When False? x P ( x) ( x) P is true for every x. There is an x for which ( x) x P ( x) There is an x for which P ( x) is true. P ( x) is false for every x. P is false.

8 MATH Discrete Structures s Translating logical statements into English I) x ( C ( x) y( C( y) F( x, y) )) where C ( x) is x has a computer, ( x y) F, is x and y are friends, and the universe of discourse for both x and y is the set of all students in this class. Every student in your school has a computer or has a friend who has a computer. II) x y z( (( F( x, y) F( x, z) ( y z) ) F( y, z) )) where ( a b) F, means a and b are friends and the universe of discourse for x, y and z is the set of all students in your school. There is a student none of whose friends are also friends with each other. Translating sentences into logical expressions III) Some student in this class has visited Mexico. Let M ( x) be the statement x has visited Mexico. xm ( x), the universe of discourse for x is the set of all the students in this class. IV) Every student in this class has visited either Canada or Mexico. Let M ( x) be the statement x has visited Mexico and ( x) Canada. x C x M ( ( ) ( x) ) C be the statement x has visited, the universe of discourse for x is the set of all the students in this class. V) If somebody is female and is a parent, then this person is someone s mother Let F ( x) be the statement x is female, P ( x) be the statement x is a parent, and M ( x, y) be the statement x is the mother of y. x F x P x ym x, y, the universe of discourse for x and y is the set of all people. (( ( ) ( )) ( ))

9 MATH Discrete Structures VI) f ( x) = L lim (For every real number ε > 0, there exists a real number δ > 0 such that x a ( x) L < ε f whenever 0 < x a < δ. ( < x a < δ f ( x) L ε ) ε δ x 0 <, the universe of discourse for ε and δ is the set of positive real numbers, and that of x is the set of real numbers, and a is a real constant. Negations The negation of a universal quantification is an existential quantification. ( x) x P( x) xp The negation of an existential quantification is a universal quantification. ( x) x Q( x) xq Method of Proofs Theorems A theorem is a statement that can be shown to be true. Proofs We demonstrate that a theorem is true with a sequence of statements that form an argument, called a proof. Axioms and Postulates Statements used in a proof include axioms and postulates, which are the underlying assumptions about mathematical structures, the hypotheses of the theorem to be proved, and previously proved theorems. Lemmas A lemma is a simple theorem used in the proof of other theorems. Corollaries A corollary is a proposition that can be established directly from a theorem that has been proved. Conjectures A conjecture is a statement whose truth value is unknown.

10 MATH Discrete Structures Remark: When a proof of a conjecture is found, the conjecture becomes a theorem. Many times conjectures are shown to be false, so they are not theorems. Rules of inference The rules of inference, which are the means used to draw conclusions from other assertions, tie together the steps of a proof. Rule of Inference Tautology Name p p q p ( p q) Addition p q p p q p q p p q q q p q p Simplification (( p) ( q) ) ( p q) ( p ( p q) ) q p q p p q q r p r p q p q Conjunction Modus ponens ( q ( p q) ) p Modus tollens (( p q) ( q r) ) ( p r) (( p q) p) q Hypothetical syllogism Disjunctive syllogism

11 MATH Discrete Structures s I) 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, and If we take a canoe trip, then we will be home by sunset lead to the conclusion We will be home by sunset. Let 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 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 p q r p r s s t Conclusion We will be home by sunset t Step Reason 1. p q Hypothesis. p Simplification using Step 1 3. r p Hypothesis 4. r Modus tollens using Steps and 3 5. r s Hypothesis 6. s Modus ponens using Steps 4 and 5 7. s t Hypothesis 8. t Modus ponens using Steps 6 and 7

12 MATH Discrete Structures II) Show that the hypotheses If you send me an message, then I will finish writing the program, If you do not send me an message, then I will go to sleep early, and If I go to sleep early, then I will wake up feeling refreshed lead to the conclusion If I do not finish writing the program, then I will wake up feeling refreshed. Let p: You send me an message q: I will finish writing the program r: I will go to sleep early s: I will wake up feeling refreshed Hypotheses If you send me an message, then I will finish writing the program If you do not send me an message, then I will go to sleep early If I go to sleep early, then I will wake up feeling refreshed p q p r s r Conclusion If I do not finish writing the program, then I will wake up feeling refreshed q s Step Reason 1. p q Hypothesis. q p Contrapositive of Step 1 3. p r Hypothesis 4. q r Hypothetical syllogism using Steps and 3 5. r s Hypothesis 6. q s Hypothetical syllogism using Steps 4 and 5 Fallacies Some common forms of incorrect reasoning are called fallacies. I) Fallacy of affirming the conclusion (( p q) q) p (( F T) T) F is False

13 MATH Discrete Structures p: it is an apple q: it is red. It is red but it may not be an apple. II) Fallacy of denying the hypothesis (( p q) p) q (( F T) T) F is False p: it is an apple q: it is red. It is not an apple but it may be red. III) Circular reasoning ( p p) p ( F F) F is False p: it is an apple If it is an apple, then it is an apple is always true, even though it is not an apple. Rules of Inference for quantified statements U is the universal of discourse Rule of Inference P P xp( x) P() c if c U () c for an arbitrary c U xp( x) xp( x) P() c for some element c U () c for some element c U xp( x) Name Universal instantiation Universal generalization Existential instantiation Existential generalization

14 MATH Discrete Structures Methods of Proving Theorems Direct Proof A proof that the implication true. p q is true that proceeds by showing that q must be true when p is Show that if n is odd, then n is odd. Suppose n is odd. I.e. = k + 1 is an odd number. n for some integer k. = ( k + 1) = 4k + 4k + 1 = 4k( k + 1) + 1 n, Indirect Proof A proof that the implication false. p q is true that proceeds by showing that p must be false when q is Show that if n is odd, then n is odd. Suppose n is even. I.e. n k = for some integer k. ( ) n = k = 4k, is an even number. Vacuous Proof A proof that the implication p q is the based on the fact that p is false. Show that if 3 > 5, then 5 > 9. Since 3 > 5 is false, the statement if 3 > 5, then 5 > 9 is true.

15 MATH Discrete Structures Trivial Proof A proof that the implication p q is true based on the fact that q is true. Show that if 5 > 3, then 9 > 5. Since 9 > 5 is always true, the statement if 5 > 3, then 9 > 5 is true. Proof by Contradiction A proof that a proposition p is true based on the truth of the implication contradiction p q where q is a Show that the sum of m + 1 and n 1 is even. Assume the sum of m + 1 and 1 contradiction. n is odd. ( m + n), a multiple of, is odd. This is a Proof by Cases A proof of an implication where the hypothesis is a disjunction of propositions that shows that each hypothesis separately implies the conclusion Show that if n > 3 or n <, then n n 6 > 0. Case 1 Suppose n > 3. n n 6 = ( n 3)( n + ) > ( 3 3)( 3 + ) = 0. Case Suppose n <. n > 4. n n 6 > 4 ( ) 6 = 0. Therefore, if n > 3 or n <, then n n 6 > 0. Existence Proofs A proof of a proposition of the form xp( x) is called an existence proof. Constructive Existence Proofs An existence proof of xp( x) given by finding an element a such that ( a) constructive existence proof. P is true is called a

16 MATH Discrete Structures Show that there exists an irrational number between any two rational numbers. Let α and β be two rational numbers. With loss of generality, we may assume α < β. Let β α 1 ε ε = β α. Clearly, ε is also a rational number and = 1 > > 0 α < + α < β. ε Nonconstructive Existence Proofs An existence proof of xp( x) given by not finding an element a such that ( a) nonconstructive existence proof. P is true is called a Show that for every positive integer n there is a prime greater than n. Let n be a positive integer. To show there is a prime greater than n, we may consider the integer n!+1. One possibility is that n!+ 1 is already prime. Otherwise, n!+ 1 is divisible by a prime number. Clearly, n!+ 1 is not divisible by any number less than or equal to n. Thus, the prime factor of n!+ 1 is greater than n. Counterexample Suppose a statement of the form xp( x) is false. We find an element a such that ( a) I.e. x P( x) is true. The element a for which P ( a) is false is called a counterexample. P is false. Show that 3 n + 1 is odd for all integers n is false. For n = 3, 3 () = 10 is even. Therefore, 3 n + 1 is odd for all integers n is false.

17 MATH Discrete Structures Mathematical Induction To use prove that a statement is true for all natural numbers n by mathematical induction, we have the following three steps: 1. To prove the statement is true for the first (few) case(s).. To assume the statement is true for n k for some k. 3. With the induction hypothesis in step, to prove the statement is true for n = k + 1. I) Euler s Formula If G is a connected plane graph, then v + f = ε + where v, f and e are the numbers of vertices, faces and edges of G respectively. The formula can be shown by induction on the number of faces of G. If f = 1, then edge of G is a cut edge (a bridge) of G and so G is a tree. In this case, v = ε + 1, and so the theorem holds. We may suppose that the theorem holds for all connected plane graphs with f k, and consider a connected plane graph G having k + 1 faces. Since G has more than one face, there should be an edge e in G that is separating two faces. By taking away e from G, we may obtain ( G e) v( G), f ( G e) = f ( G) 1 and ε ( G e) = ε ( G) 1. Then v( G e) + f ( G e) = ε ( G e) +, and so ( G) + f ( G) = ( G) + v = v ε. G e with IIa) Four Colour Theorem Any map can be coloured using no more than 4 colours. Remark: Any map can be represented by a simple connected plane graph.

18 MATH Discrete Structures IIb) Five Colour Theorem This theorem can be proved by induction on the number of vertices of G. If v 5, the theorem holds. Suppose the theorem holds for all graphs of v k and consider a graph G of k + 1 vertices. From Euler s formula, we may deduce that the minimum degree of G (any plane graph), δ ( G), is less than or equal to 5. Choose a vertex u in G of the minimum degree. In the case d ( u) 4, G u is a graph of k vertices. By the induction hypothesis, G u can be coloured with five colours, and so we may add u back and put on u a colour which is not on the adjacent vertices of u. In the case d ( u) = 5, there should be two adjacent vertices, s and t, of u which are not adjacent. It is because the 5-complete graph is not planar. Then we may merge s and t to form G u s, t of k 1 vertices. Clearly, G u s, t is 5-colourable by the induction hypothesis, and the any colourings on G u s, t are also valid on G u with s and t of the same colour. Thus, on any colourings of G u, there are at most four colours on the adjacent vertices of u in G. Then we may put on u the a colour other than those on its adjacent vertices.

### 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

### 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

### 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)

### 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

### 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

### 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

### 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

### 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

### 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,

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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 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

### 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.

### 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

### 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,

### 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

### 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.

### 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

### 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

### 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

### Foundations of Computing Discrete Mathematics Solutions to exercises for week 2

Foundations of Computing Discrete Mathematics Solutions to exercises for week 2 Agata Murawska (agmu@itu.dk) September 16, 2013 Note. The solutions presented here are usually one of many possiblities.

### Math 3000 Running Glossary

Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (

### 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,

### 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

### 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)

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

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

### 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

### 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

### 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

### 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

### 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

### Rules of inference ? #1 #2. Review: Logical Implications. Section 1.5. #1 #2 Answer? F F Yes by A-H. Esfahanian. All Rights Reserved.

Rules of inference Section 1.5 MSU/CSE 260 Fall 2009 1 Review: Logical Imlications? #1 #2 #1 #2 Answer? T T Yes T F No F T Yes F F Yes MSU/CSE 260 Fall 2009 2 Terminology Axiom or Postulate: An underlying

### 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

### DISCRETE MATH: LECTURE 4

DISCRETE MATH: LECTURE 4 DR. DANIEL FREEMAN 1. Chapter 3.1 Predicates and Quantified Statements I A predicate is a sentence that contains a finite number of variables and becomes a statement when specific

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Discrete Mathematics (2009 Spring) Induction and Recursion (Chapter 4, 3 hours)

Discrete Mathematics (2009 Spring) Induction and Recursion (Chapter 4, 3 hours) Chih-Wei Yi Dept. of Computer Science National Chiao Tung University April 17, 2009 4.1 Mathematical Induction 4.1 Mathematical

### 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

### 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

### Mathematical Induction

MCS-236: Graph Theory Handout #A5 San Skulrattanakulchai Gustavus Adolphus College Sep 15, 2010 Mathematical Induction The following three principles governing N are equivalent. Ordinary Induction Principle.

### 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

### 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

### 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 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

### conditional statement conclusion Vocabulary Flash Cards Chapter 2 (p. 66) Chapter 2 (p. 69) Chapter 2 (p. 66) Chapter 2 (p. 76)

biconditional statement conclusion Chapter 2 (p. 69) conditional statement conjecture Chapter 2 (p. 76) contrapositive converse Chapter 2 (p. 67) Chapter 2 (p. 67) counterexample deductive reasoning Chapter

### 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,

### 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

### Sample Problems in Discrete Mathematics

Sample Problems in Discrete Mathematics This handout lists some sample problems that you should be able to solve as a pre-requisite to Computer Algorithms Try to solve all of them You should also read

### 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

### 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.

### 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

### Section 3 Sequences and Limits

Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence.

### 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

### 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 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

### 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.

### 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

### Section 1.3: Predicate Logic

1 Section 1.3: Purpose of Section: To introduce predicate logic (or first-order logic) which the language of mathematics. We see how predicate logic extends the language of sentential calculus studied

### 3. Recurrence Recursive Definitions. To construct a recursively defined function:

3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a

### Quantifiers are used to describe variables in statements. - The universal quantifier means for all. - The existential quantifier means there exists.

11 Quantifiers are used to describe variables in statements. - The universal quantifier means for all. - The existential quantifier means there exists. The phrases, for all x in R if x is an arbitrary

### 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

### Geometry Topic 5: Conditional statements and converses page 1 Student Activity Sheet 5.1; use with Overview

Geometry Topic 5: Conditional statements and converses page 1 Student Activity Sheet 5.1; use with Overview 1. REVIEW Complete this geometric proof by writing a reason to justify each statement. Given:

### 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)

### Review for Final Exam

Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters

### 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

### 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

### 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

### Predicate Logic. Lucia Moura. Winter Predicates and Quantifiers Nested Quantifiers Using Predicate Calculus

Predicate Logic Winter 2010 Predicates A Predicate is a declarative sentence whose true/false value depends on one or more variables. The statement x is greater than 3 has two parts: the subject: x is

### Predicates and Quantifiers. Niloufar Shafiei

Predicates and Quantifiers Niloufar Shafiei Review Proposition: 1. It is a sentence that declares a fact. 2. It is either true or false, but not both. Examples: 2 + 1 = 3. True Proposition Toronto is the

### 8.7 Mathematical Induction

8.7. MATHEMATICAL INDUCTION 8-135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture

### 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:

### A counterexample to a result on the tree graph of a graph

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 63(3) (2015), Pages 368 373 A counterexample to a result on the tree graph of a graph Ana Paulina Figueroa Departamento de Matemáticas Instituto Tecnológico

### 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

### 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

### Chapter 1, Part I: Propositional Logic. With Question/Answer Animations

Chapter 1, Part I: Propositional Logic With Question/Answer Animations Chapter Summary Propositional Logic The Language of Propositions Applications Logical Equivalences Predicate Logic The Language of

### 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.......................................