4 Interlude: A review of logic and proofs


 Agnes Williams
 11 months ago
 Views:
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 ElliottMagwood 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 detours. 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
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
More informationInference 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
More informationChapter 1, Part III: Proofs
Chapter 1, Part III: Proofs Summary Valid Arguments and Rules of Inference Proof Methods Proof Strategies Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules
More informationThe 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 information1.5 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
More informationWhat 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
More informationCS 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 information1.5 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
More information1.5 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,
More informationRevisiting the Socrates Example
Section 1.6 Revisiting the Socrates Example We have the two premises: All men are mortal. Socrates is a man. And the conclusion: Socrates is mortal. How do we get the conclusion from the premises? The
More informationSummary. Valid Arguments and Rules of Inference Proof Methods Proof Strategies
Proofs 1 Summary Valid Arguments and Rules of Inference Proof Methods Proof Strategies 2 Section 1.6 3 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to
More informationdef: 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 informationCS 441 Discrete Mathematics for CS Lecture 6. Informal proofs. CS 441 Discrete mathematics for CS. Proofs
CS 441 Discrete Mathematics for CS Lecture 6 Informal proofs Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Proofs The truth value of some statements about the world are obvious and easy to assess
More informationCHAPTER 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 informationn 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
More informationp c p c p c c p p c h 1 h 2 h n c? 1. p c 2. (p c) is a tautology. Direct 3. ( p c) is a tautology. Direct 4. ( c p)is a tautology.
Proof Methods Methods of proof Direct Direct Contrapositive Contradiction p c p c p c c p p c Section 1.6 & 1.7 T T T T T F T F F F F T F T T T T F F F T T T F MSU/CSE 260 Fall 2009 1 MSU/CSE 260 Fall
More informationLOGICAL 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
More informationCSI 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
More informationAN INTRODUCTION TO LOGIC. and PROOF TECHNIQUES
i AN INTRODUCTION TO LOGIC and PROOF TECHNIQUES Michael A. Henning School of Mathematical Sciences University of KwaZuluNatal ii Contents 1 Logic 1 1.1 Introduction....................................
More informationDiscrete 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/g22.2340001/ Mailing List Subscribe at
More informationThe Logic of Compound Statements cont. Logical Arguments
The Logic of Compound Statements cont. Logical Arguments CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 Logical Arguments An argument (form) is a (finite)
More informationDefinition 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
More informationRules 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 ReRevisited Our Old Example: Suppose we have: All human beings
More informationCHAPTER 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 informationModule 1: Basic Logic. Theme 1: Propositions. English sentences are either true or false or neither. Consider the following sentences:
Module 1: Basic Logic Theme 1: Propositions English sentences are either true or false or neither. Consider the following sentences: 1. Warsaw is the capital of Poland. 2. 2+5=3. 3. How are you? The first
More informationDeductive Reasoning. Chapter 2.5, Theorems, Proofs, and Logic
Deductive Reasoning Chapter 2.5, 2.6  Theorems, Proofs, and Logic Theorems and Proofs Deductive reasoning is based on strict rules that guarantee certainty Well, a guarantee relative to the certainty
More informationPropositional 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
More informationNOTES ON PROOF TECHNIQUES (OTHER THAN INDUCTION)
NOTES ON PROOF TECHNIQUES (OTHER THAN INDUCTION) DAMIEN PITMAN Definitions & Theorems Definition: A direct proof is a valid argument that verifies the truth of an implication by assuming that the premise
More informationA declared mathematical proposition whose truth value is unknown is called a conjecture.
Methods of Proofs Recall we discussed the following methods of proofs  Vacuous proof  Trivial proof  Direct proof  Indirect proof  Proof by contradiction  Proof by cases. A vacuous proof of an implication
More informationLogical 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
More information1 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
More information2. 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 information1.1 Statements and Compound Statements
Chapter 1 Logic 1.1 Statements and Compound Statements A statement or proposition is an assertion which is either true or false, though you may not know which. That is, a statement is something that has
More informationDiscrete 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 1.11.3 1 / 21 Outline 1 Propositions
More informationIntroduction to Computers and Programming. Proof by Truth Table
Introduction to Computers and Programming Prof. I. K. Lundqvist Lecture May 5 4 Proof by Truth Table Proposition x Æ y and ( x) y are logically equivalent x y xæy x ( x) y Definitions Even An integer n
More informationPredicate 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 information2. 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
More informationHarvard University, Math 101, Spring 2015
Harvard University, Math 101, Spring 2015 Lecture 1 and 2 : Introduction to propositional logic 1 Logical statements A statement is a sentence that is either true or false, but not both. Some examples:
More informationPredicate Logic & Proofs Lecture 3
Topics for Today Necessary & sufficient conditions, Only if, If and only if CPRE 310 Discrete Mathematics Predicate Logic & Proofs Lecture 3 Quantified statements: predicates, quantifiers, truth values,
More informationMath 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 informationPropositional 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
More informationReview 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
More informationRules of inference ? #1 #2. Review: Logical Implications. Section 1.5. #1 #2 Answer? F F Yes by AH. 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
More informationSection 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
More information1 Deductive Reasoning and Logical Connectives
1 Deductive Reasoning and Logical Connectives As we have seen, proofs play a central role in mathematics and they are based on deductive reasoning. Facts (or statements) can be represented using Boolean
More informationUndergraduate 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
More informationChapter 1, Part I: Propositional Logic
Chapter 1, Part I: Propositional Logic Chapter Summary Propositional Logic The Language of Propositions Applications Logical Equivalences Predicate Logic The Language of Quantifiers Logical Equivalences
More informationValid AND Invalid Arguments 2.3. Introduction to Abstract Mathematics
Valid AND Invalid Arguments 2.3 Instructor: Hayk Melikya melikyan@nccu.edu 1 Argument An argument is a sequence of propositions (statements ), and propositional forms. All statements but the final one
More informationDISCRETE 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
More informationCHAPTER 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
More informationGame Theory: Logic, Set and Summation Notation
Game Theory: Logic, Set and Summation Notation Branislav L. Slantchev Department of Political Science, University of California San Diego April 3, 2005 1 Formal Logic Refresher Here s a notational refresher:
More informationSimple Proofs in Propositional Logic
Simple Proofs in Propositional Logic We do not need to use truth tables or the shorter truth table technique in order to asses the validity of arguments in propositional form. Instead, we can show the
More informationMath 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 (
More informationLikewise, 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 informationDiscrete 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,
More informationvertex, 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,
More informationLogic 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
More informationExam study sheet for CS2742 Propositional logic:
Exam study sheet for CS2742 Propositional logic: Propositional statement: expression that has a truth value (true/false). It is a tautology if it is always true, contradiction if always false. Logic connectives:
More informationExam 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
More informationThe Logic of Compound Statements. CSE 215, Foundations of Computer Science Stony Brook University
The Logic of Compound Statements CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 Mathematical Formalization Why formalize? to remove ambiguity to represent
More informationWhat is Logic? Logic is the study of valid reasoning. Philosophy. Mathematics. Computer science
What is Logic? Definition Logic is the study of valid reasoning. Philosophy Mathematics Computer science Definition Mathematical Logic is the mathematical study of the methods, structure, and validity
More informationCS 2336 Discrete Mathematics
CS 2336 Discrete Mathematics Lecture 4 Proofs: Methods and Strategies 1 Outline What is a Proof? Methods of Proving Common Mistakes in Proofs Strategies : How to Find a Proof? 2 What is a Proof? A proof
More informationBoolean Algebra. Computability and Logic
Boolean Algebra Computability and Logic Some Useful Equivalences for Boolean Connectives Double Negation: P P Commutation: P Q Q P P Q Q P Association: P (Q R) (P Q) R P (Q R) (P Q) R Idempotence: P P
More informationPhil Course Requirements, What is logic?
Phil. 2440 Course Requirements, What is logic? To discuss today: About the class: Some general course information Who should take this class? Course mechanics What you need to do About logic: Why is it
More informationIntroduction to Logic
Introduction to Logic Why study Logic? Understand meaning of mathematical sentences. Develop the building blocks for mathematical reasoning. Write correct proofs for mathematical statements. Identify buggy
More informationPropositional Logic, Truth Tables, and Predicate Logic (Rosen, Sections 1.1, 1.2, 1.3)
Propositional Logic, Truth Tables, and Predicate Logic (Rosen, Sections 1.1, 1.2, 1.3) TOPICS Propositional Logic Logical Operations Equivalences Predicate Logic Logic? What is logic? Logic is a truthpreserving
More informationSection Summary. Tautologies, Contradictions, and Contingencies. Logical Equivalence. Normal Forms (optional, covered in exercises in text)
Section 1.3 Section Summary Tautologies, Contradictions, and Contingencies. Logical Equivalence Important Logical Equivalences Showing Logical Equivalence Normal Forms (optional, covered in exercises in
More informationTHE LOGIC OF COMPOUND STATEMENTS
CHAPTER 2 THE LOGIC OF COMPOUND STATEMENTS Copyright Cengage Learning. All rights reserved. SECTION 2.3 Valid and Invalid Arguments Copyright Cengage Learning. All rights reserved. Valid and Invalid Arguments
More informationSemantics for Business Rules Predicate Logic
Master of Science Business Information Systems Semantics for Business Rules Predicate Logic Knut Hinkelmann Semantic and Logical Foundations of Business Rules The SBVR initiative is intended to capture
More informationThe Process of Mathematical Proof
The Process of Mathematical Proof Introduction. Mathematical proofs use the rules of logical deduction that grew out of the work of Aristotle around 350 BC. In Math 213 and other courses that involve writing
More informationPreface PART I: SETTING THE STAGE
Preface PART I: SETTING THE STAGE Chapter 1: What Logic Studies A. Statements and Arguments B. Recognizing Arguments Exercises 1B C. Arguments and Explanations Exercises 1C D. Truth and Logic E. Deductive
More informationHandout #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 information3. 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 informationDISCRETE 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 informationCSE 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
More information(LMCS, p. 37) PROPOSITIONAL LOGIC. and or implies. Using the connectives and variables we can make propositional formulas like
(LMCS, p. 37) II.1 PROPOSITIONAL LOGIC The Standard Connectives: 1 true 0 false not and or implies iff Propositional Variables: P, Q, R,... Using the connectives and variables we can make propositional
More informationDisjunctive Syllogism (DS) A " B, B. and
Section 6.3 Formal Reasoning formal proof (or derivation) is a sequence of wffs, where each wff is either a premise or the result of applying a proof rule to certain previous wffs in the sequence. Basic
More informationCS 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
More informationNotes on Logic. 1 Propositional Calculus. 2 Logical Operators and Truth Tables
Notes on Logic 1 Propositional Calculus A proposition or statement is an assertion which can be determined to be either true or false (T or F). For example, zero is less than any positive number is a statement.
More informationPropositional Logic. Syntax of Propositional Logic
Propositional Logic Propositional logic is a subset of the predicate logic.! Syntax! Semantics! Models! Inference Rules! Complexity 9 Syntax of Propositional Logic! symbols! logical constants,! propositional
More informationconditional 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
More informationLogic and Discrete Math Lecture notes 3
CSE 240 Logic and Discrete Math Lecture notes 3 Weixiong Zhang Washington University in St. Louis http://www.cse.wustl.edu/~zhang/teaching /cse240/spring10/index.html 1 Today Refresher: Chapter 1.2 Chapter
More informationDiscrete 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 information31 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
More informationArtificial 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
More informationBoolean logic. Lecture 12
Boolean logic Lecture 12 Contents Propositions Logical connectives and truth tables Compound propositions Disjunctive normal form (DN) Logical equivalence Laws of logic Predicate logic Post's unctional
More informationAnnouncements. CS311H: Discrete Mathematics. Propositional Logic II. Inverse of an Implication. Converse of a Implication
Announcements CS311H: Discrete Mathematics Propositional Logic II Instructor: Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Tuesday Instructor: Işıl Dillig,
More informationCS 441 Discrete Mathematics for CS Lecture 3. Predicate logic. CS 441 Discrete mathematics for CS. Propositional logic: review
CS 441 Discrete Mathematics for CS Lecture 3 Predicate logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Propositional logic: review Propositional logic: a formal language for making logical
More informationFirst Order Logic (1A) Young W. Lim 11/9/13
Copyright (c) 2013. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software
More informationLogic, 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
More informationFormal Proofs. Computability and Logic
Formal Proofs Computability and Logic General Idea of Proof A sequence of statements, starting with premises, followed by intermediate results, and ended by the conclusion, where each of the intermediate
More informationPropositional Logic, Predicate Logic, and Logic Programming
Propositional Logic, Predicate Logic, and Logic Programming 1 Propositional Logic DE: A proposition is a statement that is either true or false (but not both). In propositional logic, we assume a collection
More informationSolve problems/calculate solutions. Develop models that describe real world situations. Offer definitions. Propose conjectures
How to Prove Things Stuart Gluck, Ph.D. Director, Institutional Research Johns Hopkins University Center for Talented Youth (CTY) Carlos Rodriguez Assistant Director, Academic Programs Johns Hopkins University
More informationPropositional Logic: Part I  Semantics 120
Propositional Logic: Part I  Semantics 120 Outline What is propositional logic? Logical connectives Semantics of propositional logic Tautologies & Logical equivalence Applications: 1. Building the world
More informationPredicate 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 informationSystems of Formal Deduction
Systems of Formal Deduction Inference Rules Systems of formal deduction (or formal derivation, or formal proof) consist of a set of formal inference rules. The inference rules are formal in the sense that
More informationPredicate Logic. Knut Hinkelmann
Predicate Logic Knut Hinkelmann An Excursion into Logic The semantics of SBVR is defined by a mapping to predicate logic A predicate calculus consists of formation rules (i.e. definitions for forming wellformed
More informationChapter 2: Proof Rules for Predicate Logic. 2.1 Introduction Proof Situations and Proofs
Chapter 2: Proof Rules for Predicate Logic 2.1 Introduction Mathematical activity can be classified mainly as proving, solving, or simplifying. Techniques for solving heavily depend on the structure of
More informationLinguistics 680 September 11, Formal Deductive Systems and Model Theory
Linguistics 680 September 11, 2004 Formal Deductive Systems and Model Theory This handout is intended as an overview and omits a number of details: For more information, see the discussion of inference
More information