Announcement. Homework 1
|
|
- Edwina Ball
- 5 years ago
- Views:
Transcription
1 Announcement I made a few small changes to the course calendar No class on Wed eb 27 th, watch the video lecture Quiz 8 will take place on Monday April 15 th We will submit assignments using Gradescope Invite Code: 9DK28Y 1Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 1 Homework 1 Problems due next Monday Jan. 28 th Quiz 1 Wednesday Jan 30 th I will choose one problem from the set of six questions 2Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 2
2 Reading Read Sections 1.4(Predicates and Quantifiers) and 1.5 (Nested Quantifiers) 3Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 3 Precedence of Operators rom the table we can see that the negation operator has a higher precedence than the conjunction operator 4Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 4
3 Propositional Logic (Section 1) Proofs involve stepping through a mathematical argument Propositional Logic provides such steps oday we will discuss the process of moving from one proposition to the next to form a mathematical argument 5Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 5 Logical Equivalence Challenge: ry to find a proposition that is equivalent to p q, but that uses only the connectives, Ù, and Ú p q p q p q p p v q p q is logically equivalent to p Ú q or p q º p Ú q 6Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 6
4 Are these equivalent? Contrapositives: p q and q p Ex. If it is noon, then I am hungry. If I am not hungry, then it is not noon. Converses: p q and q p Ex. If it is noon, then I am hungry. If I am hungry, then it is noon. Inverses: p q and p q Ex. If it is noon, then I am hungry. If it is not noon, then I am not hungry. Let s take a vote 7Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 7 Are these equivalent? Contrapositives: p q º q p? Ex. If it is noon, then I am hungry. If I am not hungry, then it is not noon. Converses: p q º q p? Ex. If it is noon, then I am hungry. If I am hungry, then it is noon. Inverses: p q º p q? Ex. If it is noon, then I am hungry. If it is not noon, then I am not hungry. YES NO NO 8Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 8
5 Definitions A tautology is a proposition that s always RUE. A contradiction is a proposition that s always ALSE. p p p Ú p p Ù p 9Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 9 Logical Equivalences Identity Domination Idempotent p Ù º p p Ú º p p Ú º p Ù º p Ú p º p p Ù p º p p p Ù 10Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 10
6 Logical Equivalences Continued Excluded Middle p Ú p º Uniqueness p Ù p º Double negation ( p) º p 11Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 11 Logical Equivalences Continued Commutativity p Ú q º p Ù q º q Ú p q Ù p Associativity (p Ú q) Ú r º (p Ù q) Ù r º p Ú (q Ú r) p Ù (q Ù r) Distributivity p Ú (q Ù r) º p Ù (q Ú r) º (p Ú q) Ù (p Ú r) (p Ù q) Ú (p Ù r) 12Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 12
7 Proof of Distributivity p Ú (q Ù r) º (p Ú q) Ù (p Ú r) p q r q Ù r p Ú (q Ù r) p Ú q p Ú r (p Ú q) Ù (p Ú r) 13Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 13 DeMorgan s De Morgan s I (p Ú q) º p Ù q De Morgan s II (p Ù q) º p Ú q 14Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 14
8 Example of De Morgan s (p Ú q) º p Ù q (p Ù q) º p Ú q p q 15Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 15 De Morgan s Continued De Morgan s II (p Ù q) º p Ú q (p Ù q) º ( p Ù q) Double negation º ( p Ú q) DeMorgan s I º ( p Ú q) Double negation 16Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 16
9 Proof equivalence if NO (blue AND NO red) OR red then (p Ù q) Ú q º p Ú q (p Ù q) Ú q º ( p Ú q) Ú q De Morgan s II º ( p Ú q) Ú q Double negation º p Ú (q Ú q) Associativity º p Ú q Idempotent 17Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 17 Another Example Show that [p Ù (p q)] q is a tautology We use º to show that [p Ù (p q)] q º [p Ù (p q)] q º [p Ù ( p Ú q)] q º [(p Ù p) Ú (p Ù q)] q º [ Ú (p Ù q)] q º (p Ù q) q º (p Ù q) Ú q º ( p Ú q) Ú q º p Ú ( q Ú q ) º p Ú º substitution for distributive uniqueness identity substitution for De Morgan s II associative excluded middle domination 18Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 18
10 Another Example Consider the newspaper headline: Legislature ails to Override Governor s Veto of Bill to Cancel Sales ax Reform Did the legislature vote in favor of or against the sales tax reform? 19Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 19 Another Example Consider the newspaper headline: Legislature ails to Override Governor s Veto of Bill to Cancel Sales ax Reform Did the legislature vote in favor of or against the sales tax reform? Let s take a vote A) he legislature DID vote in favor B) he legislature DID NO vote in favor 20Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 20
11 Another Example Consider the newspaper headline: Legislature ails to Override Governor s Veto of Bill to Cancel Sales ax Reform Did the legislature vote in favor of or against the sales tax reform? Let s stand for sales tax reform Unravel one at a time. -he bill to cancel sales tax reform is s -he governors veto of the bill is s -Overriding this means s -he double negation cancel out, leaving just s herefore the legislature does not support sales tax reform. hey voted in favor of the bill (to cancel it). 21Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 21 Group Exercise I Break up into groups of 2 or 3 people Solve the following problem: Prove De Morgan s I Law by manipulating symbols (not a truth table) (p Ú q) º p Ù q Hint- Similar to the proof of De Morgan s II Law Reminder DeMorgan s II (p Ù q) º p Ú q 22Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 22
12 Predicate Logic Proposition, YES or NO? = 5 Yes X + 2 = 5 No X + 2 = 5 for all choices of X in {1, 2, 3} X + 2 = 5 for some choice of X in {1, 2, 3} Yes Yes 23Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 23 Predicates Alicia eats pizza at least once a week. Define: EP(x) = x eats pizza at least once a week. Universe of Discourse - x is a student in cs240 A predicate, or propositional function, is a function that takes some variable(s) as arguments and returns rue or alse. Note that EP(x) is not a proposition, EP(Ariel) is. 24Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 24
13 Predicates Suppose Q(x,y) = x > y Proposition, YES or NO? Q(x,y) No Q(3,4) Yes Q(x,9) Predicate, YES or NO? No Q(x,y) Yes Q(3,4) No Q(x,9) Yes 25Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 25 Universal Quantifier Another way of changing a predicate into a proposition. Suppose P(x) is a predicate on some universe of discourse. Ex. B(x) = x is carrying a backpack, x is set of cs240 students. he universal quantifier of P(x) is the proposition: P(x) is true for all x in the universe of discourse. We write it "x P(x), and say for all x, P(x) "x P(x) is RUE if P(x) is true for every single x. "x P(x) is ALSE if there is an x for which P(x) is false. "x B(x)? What is this asking? Is it true or false? 26Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 26
14 Universal Quantifier Universe of discourse is people in this room. B(x) = x is enrolled in CS240. L(x) = x is signed up for at least one class. Y(x)= x is wearing sneakers. Are either of these propositions true? a) "x (B(x) Y(x)) b) "x (Y(x) Ú L(x)) What does this proposition mean? "x (Y(x) Ù B(x)) A: only a is true B: only b is true C: both are true D: neither is true Is it true? 27Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 27 Existential Quantifier Another way of changing a predicate into a proposition. Suppose P(x) is a predicate on some universe of discourse. Ex. C(x) = x has a candy bar, x is the set of cs240 students. he existential quantifier of P(x) is the proposition: P(x) is true for some x in the universe of discourse. We write it $x P(x), and say for some x, P(x) $x P(x) is RUE if there is an x for which P(x) is true. $x P(x) is ALSE if P(x) is false for every single x. $x C(x)? What is this asking? Is it true or false? 28Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 28
15 Existential Quantifier B(x) = x is enrolled in CS240. L(x) = x is signed up for at least one class. Universe of discourse is people in this room. Are either of these propositions true? a) $x B(x) b) $x (B(x) Ù L(x)) What does this proposition mean? $ x (B(x) L(x)) Why do I have to be careful with this proposition? A: only a is true B: only b is true C: both are true D: neither is true 29Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 29 Predicate Examples L(x) = x is a lion. (x) = x is fierce. C(x) = x drinks coffee. All lions are fierce. Universe of discourse is all creatures. "x (L(x) (x)) Some lions don t drink coffee. $x (L(x) Ù C(x)) Some fierce creatures don t drink coffee. $x ((x) Ù C(x)) 30Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 30
16 More Examples B(x) = x is a hummingbird. L(x) = x is a large bird. H(x) = x lives on honey. R(x) = x is richly colored. Universe of discourse is all birds. All hummingbirds are richly colored. No large birds live on honey. Birds that do not live on honey are dully colored. "x (B(x) R(x)) $x (L(x) Ù H(x)) "x ( H(x) R(x)) 31Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 31 More Examples S(x) = x is a student in this class. C(x) = x has visited Chicago. Universe of discourse is all people. "x (S(x) Ù C(x)) "x (S(x) C(x)) All people are students in this class and all people have visited Chicago All students in this class have visited Chicago $x (S(x) C(x))? $x (S(x) Ù C(x)) Some student in this class has visited Chicago 32Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 32
17 Quantifier Negation Not all large birds live on honey. "x (L(x) H(x)) "x P(x) means P(x) is true for every x. What about "x P(x)? Not [ P(x) is true for every x. ] here is an x for which P(x) is not true. $x P(x) So, "x P(x) is the same as $x P(x). $x (L(x) H(x)) 33Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 33 Quantifier Negation No large birds live on honey. $x (L(x) Ù H(x)) $x P(x) means P(x) is true for some x. What about $x P(x)? Not [ P(x) is true for some x. ] P(x) is not true for all x. "x P(x) So, $x P(x) is the same as "x P(x). "x (L(x) Ù H(x)) 34Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 34
18 Quantifier Negation So, "x P(x) is the same as $x P(x). So, $x P(x) is the same as "x P(x). General rule: to negate a quantifier, move negation to the right, changing quantifiers as you go. 35Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 35
1.3 Predicates and Quantifiers
1.3 Predicates and Quantifiers INTRODUCTION Statements x>3, x=y+3 and x + y=z are not propositions, if the variables are not specified. In this section we discuss the ways of producing propositions from
More informationIntroduction to Sets and Logic (MATH 1190)
Introduction to Sets Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics Statistics York University Sept 18, 2014 Outline 1 2 Tautologies Definition A tautology is a compound proposition
More informationPredicates and Quantifiers. CS 231 Dianna Xu
Predicates and Quantifiers CS 231 Dianna Xu 1 Predicates Consider P(x) = x < 5 P(x) has no truth values (x is not given a value) P(1) is true 1< 5 is true P(10) is false 10 < 5 is false Thus, P(x) will
More informationCS 220: Discrete Structures and their Applications. Predicate Logic Section in zybooks
CS 220: Discrete Structures and their Applications Predicate Logic Section 1.6-1.10 in zybooks From propositional to predicate logic Let s consider the statement x is an odd number Its truth value depends
More informationCSC165 Mathematical Expression and Reasoning for Computer Science
CSC165 Mathematical Expression and Reasoning for Computer Science Lisa Yan Department of Computer Science University of Toronto January 21, 2015 Lisa Yan (University of Toronto) Mathematical Expression
More informationReview. Propositional Logic. Propositions atomic and compound. Operators: negation, and, or, xor, implies, biconditional.
Review Propositional Logic Propositions atomic and compound Operators: negation, and, or, xor, implies, biconditional Truth tables A closer look at implies Translating from/ to English Converse, inverse,
More informationDiscrete Structures for Computer Science
Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #4: Predicates and Quantifiers Based on materials developed by Dr. Adam Lee Topics n Predicates n
More informationPropositional Logic Not Enough
Section 1.4 Propositional Logic Not Enough If we have: All men are mortal. Socrates is a man. Does it follow that Socrates is mortal? Can t be represented in propositional logic. Need a language that talks
More informationLogical Operators. Conjunction Disjunction Negation Exclusive Or Implication Biconditional
Logical Operators Conjunction Disjunction Negation Exclusive Or Implication Biconditional 1 Statement meaning p q p implies q if p, then q if p, q when p, q whenever p, q q if p q when p q whenever p p
More informationPredicate Logic. CSE 191, Class Note 02: Predicate Logic Computer Sci & Eng Dept SUNY Buffalo
Predicate Logic CSE 191, Class Note 02: Predicate Logic Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 22 Outline 1 From Proposition to Predicate
More informationMACM 101 Discrete Mathematics I. Exercises on Predicates and Quantifiers. Due: Tuesday, October 13th (at the beginning of the class)
MACM 101 Discrete Mathematics I Exercises on Predicates and Quantifiers. Due: Tuesday, October 13th (at the beginning of the class) Reminder: the work you submit must be your own. Any collaboration and
More information2/2/2018. CS 103 Discrete Structures. Chapter 1. Propositional Logic. Chapter 1.1. Propositional Logic
CS 103 Discrete Structures Chapter 1 Propositional Logic Chapter 1.1 Propositional Logic 1 1.1 Propositional Logic Definition: A proposition :is a declarative sentence (that is, a sentence that declares
More informationExample. Logic. Logical Statements. Outline of logic topics. Logical Connectives. Logical Connectives
Logic Logic is study of abstract reasoning, specifically, concerned with whether reasoning is correct. Logic focuses on relationship among statements as opposed to the content of any particular statement.
More information! Predicates! Variables! Quantifiers. ! Universal Quantifier! Existential Quantifier. ! Negating Quantifiers. ! De Morgan s Laws for Quantifiers
Sec$on Summary (K. Rosen notes for Ch. 1.4, 1.5 corrected and extended by A.Borgida)! Predicates! Variables! Quantifiers! Universal Quantifier! Existential Quantifier! Negating Quantifiers! De Morgan s
More information4 Quantifiers and Quantified Arguments 4.1 Quantifiers
4 Quantifiers and Quantified Arguments 4.1 Quantifiers Recall from Chapter 3 the definition of a predicate as an assertion containing one or more variables such that, if the variables are replaced by objects
More informationPropositional and Predicate Logic
Propositional and Predicate Logic CS 536-05: Science of Programming This is for Section 5 Only: See Prof. Ren for Sections 1 4 A. Why Reviewing/overviewing logic is necessary because we ll be using it
More informationMat 243 Exam 1 Review
OBJECTIVES (Review problems: on next page) 1.1 Distinguish between propositions and non-propositions. Know the truth tables (i.e., the definitions) of the logical operators,,,, and Write truth tables for
More informationPropositional and Predicate Logic
8/24: pp. 2, 3, 5, solved Propositional and Predicate Logic CS 536: Science of Programming, Spring 2018 A. Why Reviewing/overviewing logic is necessary because we ll be using it in the course. We ll be
More informationPredicate Logic Thursday, January 17, 2013 Chittu Tripathy Lecture 04
Predicate Logic Today s Menu Predicate Logic Quantifiers: Universal and Existential Nesting of Quantifiers Applications Limitations of Propositional Logic Suppose we have: All human beings are mortal.
More informationAnnouncements CompSci 102 Discrete Math for Computer Science
Announcements CompSci 102 Discrete Math for Computer Science Read for next time Chap. 1.4-1.6 Recitation 1 is tomorrow Homework will be posted by Friday January 19, 2012 Today more logic Prof. Rodger Most
More informationSection Summary. Predicate logic Quantifiers. Negating Quantifiers. Translating English to Logic. Universal Quantifier Existential Quantifier
Section 1.4 Section Summary Predicate logic Quantifiers Universal Quantifier Existential Quantifier Negating Quantifiers De Morgan s Laws for Quantifiers Translating English to Logic Propositional Logic
More informationSection Summary. Section 1.5 9/9/2014
Section 1.5 Section Summary Nested Quantifiers Order of Quantifiers Translating from Nested Quantifiers into English Translating Mathematical Statements into Statements involving Nested Quantifiers Translated
More informationDepartment of Computer Science & Software Engineering Comp232 Mathematics for Computer Science
Department of Computer Science & Software Engineering Comp232 Mathematics for Computer Science Fall 2018 Assignment 1. Solutions 1. For each of the following statements use a truth table to determine whether
More informationLecture Predicates and Quantifiers 1.5 Nested Quantifiers
Lecture 4 1.4 Predicates and Quantifiers 1.5 Nested Quantifiers Predicates The statement "x is greater than 3" has two parts. The first part, "x", is the subject of the statement. The second part, "is
More informationDiscrete Mathematics and Its Applications
Discrete Mathematics and Its Applications Lecture 1: The Foundations: Logic and Proofs (1.3-1.5) MING GAO DASE @ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Sep. 19, 2017 Outline 1 Logical
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 Wed 09/13 Remember: Due before
More informationCS100: DISCRETE STRUCTURES. Lecture 5: Logic (Ch1)
CS100: DISCREE SRUCURES Lecture 5: Logic (Ch1) Lecture Overview 2 Statement Logical Connectives Conjunction Disjunction Propositions Conditional Bio-conditional Converse Inverse Contrapositive Laws of
More informationFormal Logic: Quantifiers, Predicates, and Validity. CS 130 Discrete Structures
Formal Logic: Quantifiers, Predicates, and Validity CS 130 Discrete Structures Variables and Statements Variables: A variable is a symbol that stands for an individual in a collection or set. For example,
More informationCSC Discrete Math I, Spring Propositional Logic
CSC 125 - Discrete Math I, Spring 2017 Propositional Logic Propositions A proposition is a declarative sentence that is either true or false Propositional Variables A propositional variable (p, q, r, s,...)
More informationFirst order Logic ( Predicate Logic) and Methods of Proof
First order Logic ( Predicate Logic) and Methods of Proof 1 Outline Introduction Terminology: Propositional functions; arguments; arity; universe of discourse Quantifiers Definition; using, mixing, negating
More informationIntroduction to Decision Sciences Lecture 2
Introduction to Decision Sciences Lecture 2 Andrew Nobel August 24, 2017 Compound Proposition A compound proposition is a combination of propositions using the basic operations. For example (p q) ( p)
More informationMath.3336: Discrete Mathematics. Nested Quantifiers
Math.3336: Discrete Mathematics Nested Quantifiers Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu Fall 2018
More informationToday. Proof using contrapositive. Compound Propositions. Manipulating Propositions. Tautology
1 Math/CSE 1019N: Discrete Mathematics for Computer Science Winter 2007 Suprakash Datta datta@cs.yorku.ca Office: CSEB 3043 Phone: 416-736-2100 ext 77875 Course page: http://www.cs.yorku.ca/course/1019
More informationCSCI-2200 FOUNDATIONS OF COMPUTER SCIENCE
1 CSCI-2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 February 5, 2015 2 Announcements Homework 1 is due now. Homework 2 will be posted on the web site today. It is due Thursday, Feb. 12 at 10am in class.
More informationFor all For every For each For any There exists at least one There exists There is Some
Section 1.3 Predicates and Quantifiers Assume universe of discourse is all the people who are participating in this course. Also let us assume that we know each person in the course. Consider the following
More information3. The Logic of Quantified Statements Summary. Aaron Tan August 2017
3. The Logic of Quantified Statements Summary Aaron Tan 28 31 August 2017 1 3. The Logic of Quantified Statements 3.1 Predicates and Quantified Statements I Predicate; domain; truth set Universal quantifier,
More informationLogic Overview, I. and T T T T F F F T F F F F
Logic Overview, I DEFINITIONS A statement (proposition) is a declarative sentence that can be assigned a truth value T or F, but not both. Statements are denoted by letters p, q, r, s,... The 5 basic logical
More informationPredicate Logic & Quantification
Predicate Logic & Quantification Things you should do Homework 1 due today at 3pm Via gradescope. Directions posted on the website. Group homework 1 posted, due Tuesday. Groups of 1-3. We suggest 3. In
More informationICS141: Discrete Mathematics for Computer Science I
ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Originals slides by Dr. Baek and Dr. Still, adapted by J. Stelovsky Based on slides Dr. M. P. Frank and Dr. J.L. Gross
More informationDiscrete Mathematical Structures. Chapter 1 The Foundation: Logic
Discrete Mathematical Structures Chapter 1 he oundation: Logic 1 Lecture Overview 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Quantifiers l l l l l Statement Logical Connectives Conjunction
More informationFormal (Natural) Deduction for Predicate Calculus
Formal (Natural) Deduction for Predicate Calculus Lila Kari University of Waterloo Formal (Natural) Deduction for Predicate Calculus CS245, Logic and Computation 1 / 42 Formal deducibility for predicate
More informationPROBLEM SET 3: PROOF TECHNIQUES
PROBLEM SET 3: PROOF TECHNIQUES CS 198-087: INTRODUCTION TO MATHEMATICAL THINKING UC BERKELEY EECS FALL 2018 This homework is due on Monday, September 24th, at 6:30PM, on Gradescope. As usual, this homework
More informationSection Summary. Predicate logic Quantifiers. Negating Quantifiers. Translating English to Logic. Universal Quantifier Existential Quantifier
Section 1.4 Section Summary Predicate logic Quantifiers Universal Quantifier Existential Quantifier Negating Quantifiers De Morgan s Laws for Quantifiers Translating English to Logic Propositional Logic
More informationMath Fundamentals of Higher Math
Lecture 9-1/29/2014 Math 3345 ematics Ohio State University January 29, 2014 Course Info Instructor - webpage https://people.math.osu.edu/broaddus.9/3345 office hours Mondays and Wednesdays 10:10am-11am
More informationLogic and Proof. Aiichiro Nakano
Logic and Proof Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Department of Computer Science Department of Physics & Astronomy Department of Chemical Engineering & Materials Science
More informationPropositional Functions. Quantifiers. Assignment of values. Existential Quantification of P(x) Universal Quantification of P(x)
Propositional Functions Rosen (6 th Ed.) 1.3, 1.4 Propositional functions (or predicates) are propositions that contain variables. Ex: P(x) denote x > 3 P(x) has no truth value until the variable x is
More informationTest 1 Solutions(COT3100) (1) Prove that the following Absorption Law is correct. I.e, prove this is a tautology:
Test 1 Solutions(COT3100) Sitharam (1) Prove that the following Absorption Law is correct. I.e, prove this is a tautology: ( q (p q) (r p)) r Solution. This is Modus Tollens applied twice, with transitivity
More informationPropositional and First-Order Logic
Propositional and irst-order Logic 1 Propositional Logic 2 Propositional logic Proposition : A proposition is classified as a declarative sentence which is either true or false. eg: 1) It rained yesterday.
More informationMAT2345 Discrete Math
Fall 2013 General Syllabus Schedule (note exam dates) Homework, Worksheets, Quizzes, and possibly Programs & Reports Academic Integrity Do Your Own Work Course Web Site: www.eiu.edu/~mathcs Course Overview
More informationCOMP 182 Algorithmic Thinking. Proofs. Luay Nakhleh Computer Science Rice University
COMP 182 Algorithmic Thinking Proofs Luay Nakhleh Computer Science Rice University 1 Reading Material Chapter 1, Section 3, 6, 7, 8 Propositional Equivalences The compound propositions p and q are called
More information1.1 Language and Logic
c Oksana Shatalov, Spring 2018 1 1.1 Language and Logic Mathematical Statements DEFINITION 1. A proposition is any declarative sentence (i.e. it has both a subject and a verb) that is either true or false,
More information2.2: Logical Equivalence: The Laws of Logic
Example (2.7) For primitive statement p and q, construct a truth table for each of the following compound statements. a) p q b) p q Here we see that the corresponding truth tables for two statement p q
More information3/29/2017. Logic. Propositions and logical operations. Main concepts: propositions truth values propositional variables logical operations
Logic Propositions and logical operations Main concepts: propositions truth values propositional variables logical operations 1 Propositions and logical operations A proposition is the most basic element
More informationProposi'onal Logic Not Enough
Section 1.4 Proposi'onal Logic Not Enough If we have: All men are mortal. Socrates is a man. Socrates is mortal Compare to: If it is snowing, then I will study discrete math. It is snowing. I will study
More informationAnnouncements. Exam 1 Review
Announcements Quiz today Exam Monday! You are allowed one 8.5 x 11 in cheat sheet of handwritten notes for the exam (front and back of 8.5 x 11 in paper) Handwritten means you must write them by hand,
More informationSolutions to Homework I (1.1)
Solutions to Homework I (1.1) Problem 1 Determine whether each of these compound propositions is satisable. a) (p q) ( p q) ( p q) b) (p q) (p q) ( p q) ( p q) c) (p q) ( p q) (a) p q p q p q p q p q (p
More informationPredicate Calculus lecture 1
Predicate Calculus lecture 1 Section 1.3 Limitation of Propositional Logic Consider the following reasoning All cats have tails Gouchi is a cat Therefore, Gouchi has tail. MSU/CSE 260 Fall 2009 1 MSU/CSE
More informationSolutions to Exercises (Sections )
s to Exercises (Sections 1.1-1.10) Section 1.1 Exercise 1.1.1: Identifying propositions (a) Have a nice day. : Command, not a proposition. (b) The soup is cold. : Proposition. Negation: The soup is not
More informationPredicate Calculus - Syntax
Predicate Calculus - Syntax Lila Kari University of Waterloo Predicate Calculus - Syntax CS245, Logic and Computation 1 / 26 The language L pred of Predicate Calculus - Syntax L pred, the formal language
More information2/18/14. What is logic? Proposi0onal Logic. Logic? Propositional Logic, Truth Tables, and Predicate Logic (Rosen, Sections 1.1, 1.2, 1.
Logic? Propositional Logic, Truth Tables, and Predicate Logic (Rosen, Sections 1.1, 1.2, 1.3) TOPICS Propositional Logic Logical Operations Equivalences Predicate Logic CS160 - Spring Semester 2014 2 What
More informationPacket #2: Set Theory & Predicate Calculus. Applied Discrete Mathematics
CSC 224/226 Notes Packet #2: Set Theory & Predicate Calculus Barnes Packet #2: Set Theory & Predicate Calculus Applied Discrete Mathematics Table of Contents Full Adder Information Page 1 Predicate Calculus
More informationCSE 311: Foundations of Computing. Lecture 3: Digital Circuits & Equivalence
CSE 311: Foundations of Computing Lecture 3: Digital Circuits & Equivalence Homework #1 You should have received An e-mail from [cse311a/cse311b] with information pointing you to look at Canvas to submit
More informationProving Arguments Valid in Predicate Calculus
Proving Arguments Valid in Predicate Calculus Lila Kari University of Waterloo Proving Arguments Valid in Predicate Calculus CS245, Logic and Computation 1 / 22 Predicate calculus - Logical consequence
More informationSection Summary. Predicates Variables Quantifiers. Negating Quantifiers. Translating English to Logic Logic Programming (optional)
Predicate Logic 1 Section Summary Predicates Variables Quantifiers Universal Quantifier Existential Quantifier Negating Quantifiers De Morgan s Laws for Quantifiers Translating English to Logic Logic Programming
More informationPredicates and Quantifiers
Predicates and Quantifiers Lecture 9 Section 3.1 Robb T. Koether Hampden-Sydney College Wed, Jan 29, 2014 Robb T. Koether (Hampden-Sydney College) Predicates and Quantifiers Wed, Jan 29, 2014 1 / 32 1
More informationChapter 1: The Logic of Compound Statements. January 7, 2008
Chapter 1: The Logic of Compound Statements January 7, 2008 Outline 1 1.1 Logical Form and Logical Equivalence 2 1.2 Conditional Statements 3 1.3 Valid and Invalid Arguments Central notion of deductive
More informationcse 311: foundations of computing Fall 2015 Lecture 6: Predicate Logic, Logical Inference
cse 311: foundations of computing Fall 2015 Lecture 6: Predicate Logic, Logical Inference quantifiers x P(x) P(x) is true for every x in the domain read as for all x, P of x x P x There is an x in the
More information1 Predicates and Quantifiers
1 Predicates and Quantifiers We have seen how to represent properties of objects. For example, B(x) may represent that x is a student at Bryn Mawr College. Here B stands for is a student at Bryn Mawr College
More informationProf. Girardi Exam 1 Math 300 MARK BOX
NAME: Prof. Girardi 09.27.11 Exam 1 Math 300 problem MARK BOX points 1 40 2 5 3 10 4 5 5 10 6 10 7 5 8 5 9 8 10 2 total 100 Problem Inspiration (1) Quiz 1 (2) Exam 1 all 10 Number 3 (3) Homework and Study
More informationBefore you get started, make sure you ve read Chapter 1, which sets the tone for the work we will begin doing here.
Chapter 2 Mathematics and Logic Before you get started, make sure you ve read Chapter 1, which sets the tone for the work we will begin doing here. 2.1 A Taste of Number Theory In this section, we will
More informationsoftware design & management Gachon University Chulyun Kim
Gachon University Chulyun Kim 2 Outline Propositional Logic Propositional Equivalences Predicates and Quantifiers Nested Quantifiers Rules of Inference Introduction to Proofs 3 1.1 Propositional Logic
More informationIntroduction to Predicate Logic Part 1. Professor Anita Wasilewska Lecture Notes (1)
Introduction to Predicate Logic Part 1 Professor Anita Wasilewska Lecture Notes (1) Introduction Lecture Notes (1) and (2) provide an OVERVIEW of a standard intuitive formalization and introduction to
More informationCOMP 2600: Formal Methods for Software Engineeing
COMP 2600: Formal Methods for Software Engineeing Dirk Pattinson Semester 2, 2013 What do we mean by FORMAL? Oxford Dictionary in accordance with convention or etiquette or denoting a style of writing
More informationReview: Potential stumbling blocks
Review: Potential stumbling blocks Whether the negation sign is on the inside or the outside of a quantified statement makes a big difference! Example: Let T(x) x is tall. Consider the following: x T(x)
More informationDiscrete Mathematics. Instructor: Sourav Chakraborty. Lecture 4: Propositional Logic and Predicate Lo
gic Instructor: Sourav Chakraborty Propositional logic and Predicate Logic Propositional logic and Predicate Logic Every statement (or proposition) is either TRUE or FALSE. Propositional logic and Predicate
More informationDiscrete Mathematics Recitation Course 張玟翔
Discrete Mathematics Recitation Course 1 2013.03.07 張玟翔 Acknowledge 鄭安哲 TA 2012 About Myself English Name : Zak Chinese Name : 張玟翔 Mail:o0000032@yahoo.com.tw Lab: ED612 1-1 Propositional Logic 1-1 Ex.2
More informationIntroduction. Predicates and Quantifiers. Discrete Mathematics Andrei Bulatov
Introduction Predicates and Quantifiers Discrete Mathematics Andrei Bulatov Discrete Mathematics Predicates and Quantifiers 7-2 What Propositional Logic Cannot Do We saw that some declarative sentences
More informationDiscrete Mathematics and Applications COT3100
Discrete Mathematics and Applications CO3100 Dr. Ungor Sources: Slides are based on Dr. G. Bebis material. uesday, January 7, 2014 oundations of Logic: Overview Propositional logic: (Sections 1.1-1.3)
More informationNegations of Quantifiers
Negations of Quantifiers Lecture 10 Section 3.2 Robb T. Koether Hampden-Sydney College Thu, Jan 31, 2013 Robb T. Koether (Hampden-Sydney College) Negations of Quantifiers Thu, Jan 31, 2013 1 / 20 1 Negations
More informationLogic of Sentences (Propositional Logic) is interested only in true or false statements; does not go inside.
You are a mathematician if 1.1 Overview you say to a car dealer, I ll take the red car or the blue one, but then you feel the need to add, but not both. --- 1. Logic and Mathematical Notation (not in the
More informationPREDICATE LOGIC. Schaum's outline chapter 4 Rosen chapter 1. September 11, ioc.pdf
PREDICATE LOGIC Schaum's outline chapter 4 Rosen chapter 1 September 11, 2018 margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, 2018 1 / 25 Contents 1 Predicates and quantiers 2 Logical equivalences
More informationECOM Discrete Mathematics
ECOM 2311- Discrete Mathematics Chapter # 1 : The Foundations: Logic and Proofs Fall, 2013/2014 ECOM 2311- Discrete Mathematics - Ch.1 Dr. Musbah Shaat 1 / 85 Outline 1 Propositional Logic 2 Propositional
More informationChapter 2. Logical Notation. Bahar Aameri. Department of Computer Science University of Toronto. Jan 09, Mathematical Expression and Reasoning 1
Chapter 2 Logical Notation Bahar Aameri Department of Computer Science University of Toronto Jan 09, 2015 Mathematical Expression and Reasoning 1 General Info Instructors: Bahar Aameri Email: bahar at
More informationBoolean Logic. CS 231 Dianna Xu
Boolean Logic CS 231 Dianna Xu 1 Proposition/Statement A proposition is either true or false but not both The sky is blue Lisa is a Math major x == y Not propositions: Are you Bob? x := 7 2 Boolean variables
More informationRecitation Week 3. Taylor Spangler. January 23, 2012
Recitation Week 3 Taylor Spangler January 23, 2012 Questions about Piazza, L A TEX or lecture? Questions on the homework? (Skipped in Recitation) Let s start by looking at section 1.1, problem 15 on page
More informationChapter 2. Logical Notation. Bahar Aameri. Department of Computer Science University of Toronto. Jan 09, Mathematical Expression and Reasoning 1
Chapter 2 Logical Notation Bahar Aameri Department of Computer Science University of Toronto Jan 09, 2015 Mathematical Expression and Reasoning 1 Announcements Tutorials: Locations and times are posted
More informationLogic. Logic is a discipline that studies the principles and methods used in correct reasoning. It includes:
Logic Logic is a discipline that studies the principles and methods used in correct reasoning It includes: A formal language for expressing statements. An inference mechanism (a collection of rules) to
More information2. Use quantifiers to express the associative law for multiplication of real numbers.
1. Define statement function of one variable. When it will become a statement? Statement function is an expression containing symbols and an individual variable. It becomes a statement when the variable
More informationSymbolising Quantified Arguments
Symbolising Quantified Arguments 1. (i) Symbolise the following argument, given the universe of discourse is U = set of all animals. Animals are either male or female. Not all Cats are male, Therefore,
More informationAnnouncements. CS243: Discrete Structures. Propositional Logic II. Review. Operator Precedence. Operator Precedence, cont. Operator Precedence Example
Announcements CS243: Discrete Structures Propositional Logic II Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Tuesday 09/11 Weilin s Tuesday office hours are
More informationCS 173: Discrete Mathematical Structures, Spring 2008 Homework 1 Solutions
CS 173: Discrete Mathematical Structures, Spring 2008 Homework 1 Solutions 1. [10 points] Translate the following sentences into propositional logic, making the meaning of your propositional variables
More informationChapter 3. The Logic of Quantified Statements
Chapter 3. The Logic of Quantified Statements 3.1. Predicates and Quantified Statements I Predicate in grammar Predicate refers to the part of a sentence that gives information about the subject. Example:
More informationPredicate Logic: Introduction and Translations
Predicate Logic: Introduction and Translations Alice Gao Lecture 10 Based on work by J. Buss, L. Kari, A. Lubiw, B. Bonakdarpour, D. Maftuleac, C. Roberts, R. Trefler, and P. Van Beek 1/29 Outline Predicate
More informationIII. Elementary Logic
III. Elementary Logic The Language of Mathematics While we use our natural language to transmit our mathematical ideas, the language has some undesirable features which are not acceptable in mathematics.
More informationLecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook)
Lecture 2 Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits Reading (Epp s textbook) 2.1-2.4 1 Logic Logic is a system based on statements. A statement (or
More informationWe last time we began introducing equivalency laws.
Monday, January 14 MAD2104 Discrete Math 1 Course website: www/mathfsuedu/~wooland/mad2104 Today we will continue in Course Notes Chapter 22 We last time we began introducing equivalency laws Today we
More informationChapter Summary. Propositional Logic. Predicate Logic. Proofs. The Language of Propositions (1.1) Applications (1.2) Logical Equivalences (1.
Chapter 1 Chapter Summary Propositional Logic The Language of Propositions (1.1) Applications (1.2) Logical Equivalences (1.3) Predicate Logic The Language of Quantifiers (1.4) Logical Equivalences (1.4)
More informationDISCRETE MATHEMATICS BA202
TOPIC 1 BASIC LOGIC This topic deals with propositional logic, logical connectives and truth tables and validity. Predicate logic, universal and existential quantification are discussed 1.1 PROPOSITION
More informationCSE 20: Discrete Mathematics
Spring 2018 Summary Last time: Today: Logical connectives: not, and, or, implies Using Turth Tables to define logical connectives Logical equivalences, tautologies Some applications Proofs in propositional
More informationLogical Structures in Natural Language: First order Logic (FoL)
Logical Structures in Natural Language: First order Logic (FoL) Raffaella Bernardi Università degli Studi di Trento e-mail: bernardi@disi.unitn.it Contents 1 How far can we go with PL?................................
More information