CSC165 Mathematical Expression and Reasoning for Computer Science
|
|
|
- Myron Vernon Quinn
- 5 years ago
- Views:
Transcription
1 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 and Reasoning January 21, / 19
2 Announcements NEXT MONDAY REVIEWING SESSION: Piazza forum: vote for the troublesome topic(s) in Logical Notation; DUE DATE: This Friday before midnight on Piazza. MORNING Section: AFTERNOON Section: TOPICS: Logical Grammar, Venn Diagram, Predicates, Equivalence Conjunction, Disjunction, Negation, Quantifiers Truth Tables Implication, Logical Arithmetic Manipulation Rules Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
3 Today s Topics APPLICATION OF NEGATION TO LOGICAL SENTENCES LOGICAL ARITHMETIC MANIPULATION RULES Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
4 Implication rule P Q means: Every P is a Q Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
5 Implication rule P Q means: Every P is a Q P Q P Q T T T T F F F T T F F T Evaluate S 1: implication rule S 1 : (P Q) ( P Q) Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
6 Implication rule P Q means: Every P is a Q P Q P Q T T T T F F F T T F F T Evaluate S 1: implication rule S 1 : (P Q) ( P Q) P Q P Q P P Q S 1 T T T F T T T F F F F T F T T T T T F F T T T T Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
7 Exercise #1 Use TRUTH TABLES to evaluate the following claims. Indicate which one is a TAUTOLOGY, which one is SATISFIABLE and which one is UNSATISFIABLE. 1 P Q is equivalent to its contrapositive. T 2 P Q is equivalent to its converse. S 3 P Q is equivalent to (P Q) (Q P ). T 4 P P. U 5 P P. T Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
8 CHAPTER 2: LOGICAL NOTATION APPLICATION OF NEGATION TO LOGICAL SENTENCES Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
9 Negation Negation of a Sentence The negation of a sentence inverts its TRUTH VALUE. The negation of a sentence P is written as P, P is TRUE if P was FALSE, P is FALSE if P was TRUE. P is equal with P. Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
10 Negation Negation of a Sentence The negation of a sentence inverts its TRUTH VALUE. The negation of a sentence P is written as P, P is TRUE if P was FALSE, P is FALSE if P was TRUE. P is equal with P. Example: Claim: All employees making over 80,000 are female. The negation is: Not all employees making over 80,000 are female. Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
11 Negation over Conjunction and Disjunction: DeMorgan s Law (S 1 S 2) ( S 1 S 2) Sentence S 1 S 2 is FALSE exactly when at least one of S 1 or S 2 is FALSE. (S 1 S 2) ( S 1 S 2) Sentence S 1 S 2 is FALSE exactly when both S 1 and S 2 are FALSE. Exercise: Recall that (P Q) ( P Q). (P Q) ((P Q) (Q P )) Now use DeMorgan s law to simplify the following sentences so that only P and Q are negated. 1 (P Q). 2 (P Q) Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
12 Negation over Conjunction and Disjunction Solution: 1 2 (P Q) ( P Q) ( P Q) (P Q) (P Q) ((P Q) (Q P )) ( (P Q) (Q P )) ((P Q) (Q P )) Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
13 Negation Negation over Quantifiers Example 1: ( x D, y D, P (x, y)) x D, ( y D, P (x, y)) x D, ( y D, P (x, y)) x D, y D, P (x, y) Example 2: ( x D, y D, P (x, y)) x D, ( y D, P (x, y)) ( x D, y D, P (x, y)) x D, y D, P (x, y) Example 3: ( x D, y D, (P (x, y) Q(x, y))) ( x D, y D, ( P (x, y) Q(x, y))) Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
14 CHAPTER 2: LOGICAL NOTATION LOGICAL ARITHMETIC Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
15 Logical Arithmetic Precedence Rules Precedence DECREASES from TOP TO BOTTOM: Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
16 Logical Arithmetic Precedence Rules Precedence DECREASES from TOP TO BOTTOM: Arithmetic Rules 1 Commutative 2 Associative 3 Distributive 4 Identity 5 Idempotency 6 DeMorgan s Law 7 etc. These rules can be verified by a truth table. Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
17 Logical Arithmetic Commutative P Q Q P and P Q Q P Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
18 Logical Arithmetic Commutative P Q Q P and P Q Q P Associative P (Q R) (P Q) R and P (Q R) (P Q) R Distributive P (Q R) (P Q) (P R) and P (Q R) (P Q) (P R) Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
19 Logical Arithmetic Commutative P Q Q P and P Q Q P Associative P (Q R) (P Q) R and P (Q R) (P Q) R Distributive P (Q R) (P Q) (P R) and P (Q R) (P Q) (P R) Identity P (Q Q) P P (Q Q) Idempotency P P P P P Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
20 Logical Arithmetic: implication, bi-implication, with,, and Implication: P Q P Q Negate implication (use DeMorgan s law): (P Q) ( P Q) ( P Q) (P Q) Bi-implication: P Q Written with,, and : (P Q) (( P Q) ( Q P )) ((P Q) ( P Q)) Negate bi-implication (use DeMorgan s law): (P Q) ((P Q) ( P Q)) (( P Q) (P Q)) Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
21 Transitivity of Universally-quantified Implication ((P (x) Q(x)) (Q(x) R(x))) (P (x) R(x)) x D, (P (x) (Q(x) R(x))) x D, ((P (x) Q(x)) R(x)) Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
22 Mixed Multiple Quantifiers Everybody has somebody who respects him/her.... Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
23 Mixed Multiple Quantifiers Everybody has somebody who respects him/her.... y is respected by somebody x P, r(x, y) Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
24 Mixed Multiple Quantifiers y is respected by somebody x P, r(x, y) Everybody has somebody who respects him/her. y P, x P, r(x, y) Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
25 Summary of manipulation rules - 1 The following is a summary of the basic laws and rules we use for manipulating formal statements. identity laws idempotency laws commutative laws P (Q Q) P P (Q Q) P P P P P P P P Q Q P P Q Q P (P Q) (Q P ) associative laws (P Q) R P (Q R) (P Q) R P (Q R) distributive laws P (Q R) (P Q) (P R) P (Q R) (P Q) (P R) contrapositive P Q Q P implication P Q P Q equivalence (P Q) (P Q) (Q P ) Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
26 Summary of manipulation rules - 2 double negation ( P ) P DeMorgan s laws (P Q) P Q (P Q) P Q implication negation (P Q) P Q equivalence negation (P Q) (P Q) (Q P ) quantifier negation quantifier distributive laws (where R does not contain variable x) variable renaming (where y does not appear in P (x)) ( x D, P (x)) x D, P (x) ( x D, P (x)) x D, P (x) x D, P (x) Q(x) ( x D, P (x)) ( x D, Q x D, P (x) Q(x) ( x D, P (x)) ( x D, Q x D, R Q(x) R ( x D, Q(x)) x D, R Q(x) R ( x D, Q(x)) x D, R Q(x) R ( x D, Q(x)) x D, R Q(x) R ( x D, Q(x)) x D, P (x) y D, P (y) x D, P (x) y D, P (y) Lisa Yan (University of Toronto) Mathematical Expression and Reasoning January 21, / 19
2.2: 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
CSC 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,...)
Chapter 2. Logical Notation. Bahar Aameri. Department of Computer Science University of Toronto. Jan 16, Mathematical Expression and Reasoning 1
Chapter 2 Logical Notation Bahar Aameri Department of Computer Science University of Toronto Jan 16, 2015 Mathematical Expression and Reasoning 1 Announcements Assignment 1: Assignment 1 will be posted
We 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
Compound Propositions
Discrete Structures Compound Propositions Producing new propositions from existing propositions. Logical Operators or Connectives 1. Not 2. And 3. Or 4. Exclusive or 5. Implication 6. Biconditional Truth
3.1 Universal quantification and implication again. Claim 1: If an employee is male, then he makes less than 55,000.
Chapter 3 Logical Connectives 3.1 Universal quantification and implication again So far we have considered an implication to be universal quantication in disguise: Claim 1: If an employee is male, then
Announcements. 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
Introduction 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
Chapter 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
What is Logic? Introduction to Logic. Simple Statements. Which one is statement?
What is Logic? Introduction to Logic Peter Lo Logic is the study of reasoning It is specifically concerned with whether reasoning is correct Logic is also known as Propositional Calculus CS218 Peter Lo
What is the decimal (base 10) representation of the binary number ? Show your work and place your final answer in the box.
Question 1. [10 marks] Part (a) [2 marks] What is the decimal (base 10) representation of the binary number 110101? Show your work and place your final answer in the box. 2 0 + 2 2 + 2 4 + 2 5 = 1 + 4
Logic: Propositional Logic (Part I)
Logic: Propositional Logic (Part I) Alessandro Artale Free University of Bozen-Bolzano Faculty of Computer Science http://www.inf.unibz.it/ artale Descrete Mathematics and Logic BSc course Thanks to Prof.
Section 1.2 Propositional Equivalences. A tautology is a proposition which is always true. A contradiction is a proposition which is always false.
Section 1.2 Propositional Equivalences A tautology is a proposition which is always true. Classic Example: P P A contradiction is a proposition which is always false. Classic Example: P P A contingency
Packet #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
3. 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,
Today. 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
Chapter 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)
Propositional logic (revision) & semantic entailment. p. 1/34
Propositional logic (revision) & semantic entailment p. 1/34 Reading The background reading for propositional logic is Chapter 1 of Huth/Ryan. (This will cover approximately the first three lectures.)
Announcements. 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
Department 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
Recitation 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
Review. 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,
2. The Logic of Compound Statements Summary. Aaron Tan August 2017
2. The Logic of Compound Statements Summary Aaron Tan 21 25 August 2017 1 2. The Logic of Compound Statements 2.1 Logical Form and Logical Equivalence Statements; Compound Statements; Statement Form (Propositional
Learning Goals of CS245 Logic and Computation
Learning Goals of CS245 Logic and Computation Alice Gao April 27, 2018 Contents 1 Propositional Logic 2 2 Predicate Logic 4 3 Program Verification 6 4 Undecidability 7 1 1 Propositional Logic Introduction
Announcement. Homework 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
Description Logics. Foundations of Propositional Logic. franconi. Enrico Franconi
(1/27) Description Logics Foundations of Propositional Logic Enrico Franconi franconi@cs.man.ac.uk http://www.cs.man.ac.uk/ franconi Department of Computer Science, University of Manchester (2/27) Knowledge
Propositional Logic and Semantics
Propositional Logic and Semantics English is naturally ambiguous. For example, consider the following employee (non)recommendations and their ambiguity in the English language: I can assure you that no
Chapter 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
CSE 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
Proposition/Statement. Boolean Logic. Boolean variables. Logical operators: And. Logical operators: Not 9/3/13. Introduction to Logical Operators
Proposition/Statement Boolean Logic CS 231 Dianna Xu A proposition is either true or false but not both he sky is blue Lisa is a Math major x == y Not propositions: Are you Bob? x := 7 1 2 Boolean variables
Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014
Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014 1. Translate each of the following English sentences into formal statements using the logical operators (,,,,, and ). You may also use mathematical
1.1 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,
Introduction 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)
Announcements 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
COMP 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
Discrete Mathematics & Mathematical Reasoning Predicates, Quantifiers and Proof Techniques
Discrete Mathematics & Mathematical Reasoning Predicates, Quantifiers and Proof Techniques Colin Stirling Informatics Some slides based on ones by Myrto Arapinis Colin Stirling (Informatics) Discrete Mathematics
2-4: The Use of Quantifiers
2-4: The Use of Quantifiers The number x + 2 is an even integer is not a statement. When x is replaced by 1, 3 or 5 the resulting statement is false. However, when x is replaced by 2, 4 or 6 the resulting
HW1 graded review form? HW2 released CSE 20 DISCRETE MATH. Fall
CSE 20 HW1 graded review form? HW2 released DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Translate sentences from English to propositional logic using appropriate
Negations 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
Boolean 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
software 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
MAT 243 Test 1 SOLUTIONS, FORM A
t MAT 243 Test 1 SOLUTIONS, FORM A 1. [10 points] Rewrite the statement below in positive form (i.e., so that all negation symbols immediately precede a predicate). ( x IR)( y IR)((T (x, y) Q(x, y)) R(x,
Lecture 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
Conjunction: p q is true if both p, q are true, and false if at least one of p, q is false. The truth table for conjunction is as follows.
Chapter 1 Logic 1.1 Introduction and Definitions Definitions. A sentence (statement, proposition) is an utterance (that is, a string of characters) which is either true (T) or false (F). A predicate is
CSC165. Larry Zhang, September 23, 2014
CSC165 Larry Zhang, September 23, 2014 Tutorial classrooms T0101, Tuesday 9:10am~10:30am: BA3102 A-F (Jason/Jason) BA3116 G-L (Eleni/Eleni) BA2185 M-T (Madina/Madina) BA2175 V-Z (Siamak/Siamak) T0201:
Packet #1: Logic & Proofs. Applied Discrete Mathematics
Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 3-13 Course Objectives At the conclusion of this course, you should
Logic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.
![Logic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.](/thumbs/87/97228443.jpg "Logic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.")
Math 0413 Appendix A.0 Logic Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another. This type of logic is called propositional.
More Propositional Logic Algebra: Expressive Completeness and Completeness of Equivalences. Computability and Logic
More Propositional Logic Algebra: Expressive Completeness and Completeness of Equivalences Computability and Logic Equivalences Involving Conditionals Some Important Equivalences Involving Conditionals
Example. 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.
Math 3336: Discrete Mathematics Practice Problems for Exam I
Math 3336: Discrete Mathematics Practice Problems for Exam I The upcoming exam on Tuesday, February 26, will cover the material in Chapter 1 and Chapter 2*. You will be provided with a sheet containing
EECS 1028 M: Discrete Mathematics for Engineers
EECS 1028 M: Discrete Mathematics for Engineers Suprakash Datta Office: LAS 3043 Course page: http://www.eecs.yorku.ca/course/1028 Also on Moodle S. Datta (York Univ.) EECS 1028 W 18 1 / 26 Why Study Logic?
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
Logic and Discrete Mathematics. Section 3.5 Propositional logical equivalence Negation of propositional formulae
Logic and Discrete Mathematics Section 3.5 Propositional logical equivalence Negation of propositional formulae Slides version: January 2015 Logical equivalence of propositional formulae Propositional
Discrete Structures of Computer Science Propositional Logic III Rules of Inference
Discrete Structures of Computer Science Propositional Logic III Rules of Inference Gazihan Alankuş (Based on original slides by Brahim Hnich) July 30, 2012 1 Previous Lecture 2 Summary of Laws of Logic
Logic and Truth Tables
Logic and Truth Tables What is a Truth Table? A truth table is a tool that helps you analyze statements or arguments in order to verify whether or not they are logical, or true. There are five basic operations
Mathematical Logic Part Two
Mathematical Logic Part Two Outline for Today Recap from Last Time The Contrapositive Using Propositional Logic First-Order Logic First-Order Translations Recap from Last Time Recap So Far A propositional
UNIT-I: Propositional Logic
1. Introduction to Logic: UNIT-I: Propositional Logic Logic: logic comprises a (formal) language for making statements about objects and reasoning about properties of these objects. Statements in a logical
2/2/2018. 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
Propositional logic ( ): Review from Mat 1348
CSI 2101 / Winter 2008: Discrete Structures. Propositional logic ( 1.1-1.2): Review from Mat 1348 Dr. Nejib Zaguia - Winter 2008 1 Propositional logic: Review Mathematical Logic is a tool for working with
Propositional Logic. Spring Propositional Logic Spring / 32
Propositional Logic Spring 2016 Propositional Logic Spring 2016 1 / 32 Introduction Learning Outcomes for this Presentation Learning Outcomes... At the conclusion of this session, we will Define the elements
Logic, Sets, and Proofs
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Operators. A logical statement is a mathematical statement that can be assigned a value either true or false.
https://vu5.sfc.keio.ac.jp/slide/
1 FUNDAMENTALS OF LOGIC NO.3 NORMAL FORMS Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 So Far What is Logic? mathematical logic symbolic logic Proposition A statement
Introduction Logic Inference. Discrete Mathematics Andrei Bulatov
Introduction Logic Inference Discrete Mathematics Andrei Bulatov Discrete Mathematics - Logic Inference 6-2 Previous Lecture Laws of logic Expressions for implication, biconditional, exclusive or Valid
ICS141: 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
Section 1.1: Logical Form and Logical Equivalence
Section 1.1: Logical Form and Logical Equivalence An argument is a sequence of statements aimed at demonstrating the truth of an assertion. The assertion at the end of an argument is called the conclusion,
Equivalence and Implication
Equivalence and Alice E. Fischer CSCI 1166 Discrete Mathematics for Computing February 7 8, 2018 Alice E. Fischer Laws of Logic... 1/33 1 Logical Equivalence Contradictions and Tautologies 2 3 4 Necessary
Tools for reasoning: Logic. Ch. 1: Introduction to Propositional Logic Truth values, truth tables Boolean logic: Implications:
Tools for reasoning: Logic Ch. 1: Introduction to Propositional Logic Truth values, truth tables Boolean logic: Implications: 1 Why study propositional logic? A formal mathematical language for precise
CISC-102 Winter 2016 Lecture 17
CISC-102 Winter 2016 Lecture 17 Logic and Propositional Calculus Propositional logic was eventually refined using symbolic logic. The 17th/18th century philosopher Gottfried Leibniz (an inventor of calculus)
Today s Topic: Propositional Logic
Today s Topic: Propositional Logic What is a proposition? Logical connectives and truth tables Translating between English and propositional logic Logic is the basis of all mathematical and analytical
1.1 Language and Logic
c Oksana Shatalov, Fall 2017 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,
PHIL12A Section answers, 16 February 2011
PHIL12A Section answers, 16 February 2011 Julian Jonker 1 How much do you know? 1. Show that the following sentences are equivalent. (a) (Ex 4.16) A B A and A B A B (A B) A A B T T T T T T T T T T T F
Direct Proof and Proof by Contrapositive
Dr. Nahid Sultana October 14, 2012 Consider an implication: p q. Then p q p q T T T T F F F T T F F T Consider an implication: p q. Then p q p q T T T T F F F T T F F T Consider x D, p(x) q(x). It can
Definition 2. Conjunction of p and q
Proposition Propositional Logic CPSC 2070 Discrete Structures Rosen (6 th Ed.) 1.1, 1.2 A proposition is a statement that is either true or false, but not both. Clemson will defeat Georgia in football
Logic 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
10/5/2012. Logic? What is logic? Propositional Logic. Propositional Logic (Rosen, Chapter ) Logic is a truth-preserving system of inference
Logic? Propositional Logic (Rosen, Chapter 1.1 1.3) TOPICS Propositional Logic Truth Tables Implication Logical Proofs 10/1/12 CS160 Fall Semester 2012 2 What is logic? Logic is a truth-preserving system
Completeness for FOL
Completeness for FOL Overview Adding Witnessing Constants The Henkin Theory The Elimination Theorem The Henkin Construction Lemma 12 This lemma assures us that our construction of M h works for the atomic
Propositional Equivalence
Propositional Equivalence Tautologies and contradictions A compound proposition that is always true, regardless of the truth values of the individual propositions involved, is called a tautology. Example:
PLEASE HAND IN PLEASE HAND IN. Solutions Sample
PLEASE HAND IN UNIVERSITY OF TORONTO Faculty of Arts and Science Term test #1 CSC 165H1 Section L0101 Duration 60 minutes PLEASE HAND IN No aids allowed Last Name: First Name: Solutions Sample Student
Discrete Structures for Computer Science
Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #2: Propositional Logic Based on materials developed by Dr. Adam Lee Today s Topic: Propositional
Discrete 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
Tautologies, Contradictions, and Contingencies
Section 1.3 Tautologies, Contradictions, and Contingencies A tautology is a proposition which is always true. Example: p p A contradiction is a proposition which is always false. Example: p p A contingency
CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS
CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS 1.1 - Logical Form and Logical Equivalence Definition. A statement or proposition is a sentence that is either true or false, but not both. ex. 1 + 2 = 3 IS a statement
Chapter 1 Elementary Logic
2017-2018 Chapter 1 Elementary Logic The study of logic is the study of the principles and methods used in distinguishing valid arguments from those that are not valid. The aim of this chapter is to help
CSCE 222 Discrete Structures for Computing. Propositional Logic. Dr. Hyunyoung Lee. !!!!!! Based on slides by Andreas Klappenecker
CSCE 222 Discrete Structures for Computing Propositional Logic Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Propositions A proposition is a declarative sentence that is either true or false
Part I: Propositional Calculus
Logic Part I: Propositional Calculus Statements Undefined Terms True, T, #t, 1 False, F, #f, 0 Statement, Proposition Statement/Proposition -- Informal Definition Statement = anything that can meaningfully
Propositional Logic Basics Propositional Equivalences Normal forms Boolean functions and digital circuits. Propositional Logic.
Propositional Logic Winter 2012 Propositional Logic: Section 1.1 Proposition A proposition is a declarative sentence that is either true or false. Which ones of the following sentences are propositions?
Chapter 1, Section 1.1 Propositional Logic
Discrete Structures Chapter 1, Section 1.1 Propositional Logic These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 6 th ed., by Kenneth H. Rosen, published
Propositional 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
The Logic of Compound Statements cont.
The Logic of Compound Statements cont. CSE 215, Computer Science 1, Fall 2011 Stony Brook University http://www.cs.stonybrook.edu/~cse215 Refresh from last time: Logical Equivalences Commutativity of :
DISCRETE STRUCTURES WEEK5 LECTURE1
DISCRETE STRUCTURES WEEK5 LECTURE1 Let s get started with... Logic! Spring 2010 CPCS 222 - Discrete Structures 2 Logic Crucial for mathematical reasoning Important for program design Used for designing
Propositional 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
Propositional 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
Propositional 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
Logic and Propositional Calculus
CHAPTER 4 Logic and Propositional Calculus 4.1 INTRODUCTION Many algorithms and proofs use logical expressions such as: IF p THEN q or If p 1 AND p 2, THEN q 1 OR q 2 Therefore it is necessary to know
AN INTRODUCTION TO MATHEMATICAL PROOFS NOTES FOR MATH Jimmy T. Arnold
AN INTRODUCTION TO MATHEMATICAL PROOFS NOTES FOR MATH 3034 Jimmy T. Arnold i TABLE OF CONTENTS CHAPTER 1: The Structure of Mathematical Statements.............................1 1.1. Statements..................................................................
Chapter 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
Exercise Set 1 Solutions Math 2020 Due: January 30, Find the truth tables of each of the following compound statements.
1. Find the truth tables of each of the following compound statements. (a) ( (p q)) (p q), p q p q (p q) q p q ( (p q)) (p q) 0 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 (b) [p ( p q)] [( (p
22c:145 Artificial Intelligence
22c:145 Artificial Intelligence Fall 2005 Propositional Logic Cesare Tinelli The University of Iowa Copyright 2001-05 Cesare Tinelli and Hantao Zhang. a a These notes are copyrighted material and may not
A statement is a sentence that is definitely either true or false but not both.
5 Logic In this part of the course we consider logic. Logic is used in many places in computer science including digital circuit design, relational databases, automata theory and computability, and artificial
Review: 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)