Announcements. CS311H: Discrete Mathematics. Propositional Logic II. Inverse of an Implication. Converse of a Implication
|
|
- Clifford Little
- 6 years ago
- Views:
Transcription
1 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 lecture starts! Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 1/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 2/36 Converse of a Implication Inverse of an Implication Recall implication p q when does it evaluate to false? The converse of an implication p q is q p. What is the converse of If I am a CS major, then I can program? What is the converse of If I get an A in CS311, then I am smart? Note: It is possible for a implication to be true, but its converse to be false, e.g., F T is true, but converse false The inverse of an implication p q is p q. What is the inverse of If I am a CS major, then I can program? What is the inverse of If I get an A in CS311, then I am smart? Note: It is possible for a implication to be true, but its inverse to be false. F T is true, but inverse is false Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 3/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 4/36 Contrapositive of Implication Conditional and its Contrapositive The contrapositive of an implication p q is q p. What is the contrapositive of If I am a CS major, then I can program? What is the contrapositive of If I get an A in CS311, then I am smart? Question: Is it possible for an implication to be true, but its contrapositive to be false? A conditional p q and its contrapositive q p always have the same truth value. Proof: We consider all four possible cases: p = T, q = T : Both T T and F F are true p = T, q = F : Both T F and T F are false p = F, q = T : Both F T and F T are true p = F, q = F : Both F F and T T are true Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 5/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 6/36 1
2 Question Summary Conditional is of the form p q Consider a conditional p q Is it possible that its converse is true, but inverse is false? Converse: q p Inverse: p q Contrapositive: q p Conditional and contrapositive have same truth value Inverse and converse always have same truth value Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 7/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 8/36 Biconditionals Operator Precedence A biconditional p q is the proposition p if and only if q. The biconditional p q is true if p and q have same truth value, and false otherwise. Exercise: Construct a truth table for p q Question: How can we express p q using the other boolean connectives? Given a formula p q r, do we parse this as (p q) r or p (q r)? Without settling on a convention, formulas without explicit paranthesization are ambiguous. To avoid ambiguity, we will specify precedence for logical connectives. Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 9/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 10/36 Operator Precedence, cont. Operator Precedence Example Negation ( ) has higher precedence than all other connectives. Question: Does p q mean (i) (p q) or (ii) ( p) q? Conjunction ( ) has next highest predence. Question: Does p q q mean (i) (p q) r or (ii) p (q r)? Disjunction ( ) has third highest precedence. Next highest is precedence is, and lowest precedence is Which is the correct interpretation of the formula p q r q r (A) ((p (q r)) q) ( r) (B) ((p q) r) q) ( r) (C) (p (q r)) (q ( r)) (D) (p ((q r) q)) ( r) Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 11/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 12/36 2
3 Validity, Unsatisfiability Interpretations The truth value of a propositional formula depends on truth assignments to variables Example: p evaluates to true under under the assignment p = F and to false under p = T Some formulas evaluate to true for every assignment, e.g., p p Such formulas are called tautologies or valid formulas Some formulas evaluate to false for every assignment, e.g., p p To make satisfability/validity precise, we ll define interpretation of formula An interpretation I for a formula F is a mapping from each propositional variables in F to exactly one truth value I : {p true, q false, } Each interpretation corresponds to one row in the truth table, so 2 n possible interpretations Such formulas are called unsatisfiable formulas or contradictions Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 13/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 14/36 Entailment Examples Consider the formula F : p q p q Under an interpretation, every propositional formula evaluates to T or F Formula F + Interpretation I = Truth value We write I = F if F evaluates to true under I Similarly, I = F if F evaluates to false under I. Theorem: I = F if and only if I = F Let I 1 be the interpretation such that [p true, q false] What does F evaluate to under I 1? Thus, I 1 = F Let I 2 be the interpretation such that [p true, q true] What does F evaluate to under I 2? Thus, I 2 = F Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 15/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 16/36 Another Example Satisfiability, Validity Let F 1 and F 2 be two propositional formulas Suppose F 1 evaluates to true under interpretation I What does F 2 F 1 evaluate to under I? F is satisfiable iff there exists interpretation I s.t. I = F F is valid iff for all interpretations I, I = F F is unsatisfiable iff for all interpretations I, I = F F is contingent if it is satisfiable, but not valid. Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 17/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 18/36 3
4 True/False Questions Duality Between Validity and Unsatisfiability Are the following statements true or false? If a formula is valid, then it is also satisfiable. F is valid if and only if F is unsatisfiable If a formula is satisfiable, then its negation is unsatisfiable. If F 1 and F 2 are satisfiable, then F 1 F 2 is also satisfiable. Proof: If F 1 and F 2 are satisfiable, then F 1 F 2 is also satisfiable. Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 19/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 20/36 Proving Validity Proving Satisfiability, Unsatisfiability, Contingency Question: How can we prove that a propositional formula is a tautology? Similarly, can prove satisfiability, unsatisfiability, contingency using truth tables: Exercise: Which formulas are tautologies? Prove your answer. 1. (p q) ( q p) Satisfiable: There exists a row where formula evaluates to true Unsatisfiable: In all rows, formula evaluates to false 2. (p q) p Contingent: Exists a row where formula evaluates to true, and another row where it evaluates to false Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 21/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 22/36 Exercise Implication Is (p q) (q p) valid, unsatisfiable, or contingent? Prove your answer. Formula F 1 implies F 2 (written F 1 F 2 ) iff for all interpretations I, I = F 1 F 2 F 1 F 2 iff F 1 F 2 is valid Caveat: F 1 F 2 is not a propositional logic formula; is not part of PL syntax! Instead, F 1 F 2 is a semantic judgment, like satisfiability! Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 23/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 24/36 4
5 Example Equivalence Does p q imply p? Prove your answer. Two formulas F 1 and F 2 are equivalent if they have same truth value for every interpretation, e.g., p p and p More precisely, formulas F 1 and F 2 are equivalent, written F 1 F 2 or F 1 F 2, iff: F 1 F 2 iff F 1 F 2 is valid, not part of PL syntax; they are semantic judgments! Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 25/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 26/36 Example Important Equivalences Some important equivalences are useful to know! Prove that p q and p q are equivalent Law of double negation: p p Identity Laws: p T p p F p Domination Laws: p T T p F F Idempotent Laws: p p p p p p Negation Laws: p p F p p T Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 27/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 28/36 Commutativity and Distributivity Laws De Morgan s Laws Commutative Laws: p q q p p q q p Distributivity Law #1: (p (q r)) (p q) (p r) Distributivity Law #2: (p (q r)) (p q) (p r) Associativity Laws: p (q r) (p q) r p (q r) (p q) r Let cs311 be the proposition John took CS311 and cs312 be the proposition John took CS312 In simple English what does (cs311 cs312) mean? DeMorgan s law expresses exactly this equivalence! De Morgan s Law #1: (p q) ( p q) De Morgan s Law #2: (p q) ( p q) When you push negations in, becomes and vice versa Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 29/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 30/36 5
6 Why are These Equivalences Useful? Formalizing English Arguments in Logic We can use logic to prove correctness of English arguments. Use known equivalences to prove that two formulas are equivalent i.e., rewrite one formula into another using known equivalences Examples: Prove following formulas are equivalent: 1. (p ( p q)) and p q 2. (p q) and p q For example, consider the argument: If Joe drives fast, he gets a speeding ticket. Joe did not get a ticket. Therefore, Joe did not drive fast. Let f be the proposition Joe drives fast, and t be the proposition Joe gets a ticket How do we encode this argument as a logical formula? Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 31/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 32/36 Example, cont Another Example Can also use to logic to prove an argument is not valid. If Joe drives fast, he gets a speeding ticket. Joe did not get a ticket. Therefore, he did not drive fast. : ((f t) t) f How can we prove this argument is valid? Can do this in two ways: 1. Use truth table to show formula is tautology 2. Use known equivalences to rewrite formula to true Suppose your friend George make the following argument: If Jill carries an umbrella, it is raining. Jill is not carrying an umbrella. Therefore it is not raining. Let s use logic to prove George s argument doesn t hold water. Let u = Jill is carrying an umbrella, and r = It is raining How do we encode this argument in logic? Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 33/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 34/36 Example, cont. Summary A formula is valid if it is true for all interpretations. If Jill carries an umbrella, it is raining. Jill is not carrying an umbrella. Therefore it is not raining. : ((u r) u) r How can we prove George s argument is invalid? A formula is satisfiable if it is true for at least one interpretation. A formula is unsatisfiable if it is false for all interpretations. A formula is contingent if it is true in at least one interpretation, and false in at least one interpretation. Two formulas F 1 and F 2 are equivalent, written F 1 F 2, if F 1 F 2 is valid Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 35/36 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 36/36 6
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
More informationMath.3336: Discrete Mathematics. Propositional Equivalences
Math.3336: Discrete Mathematics Propositional Equivalences Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu Fall
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 informationCourse Staff. Textbook
Course Staff CS311H: Discrete Mathematics Intro and Propositional Logic Instructor: Işıl Dillig Instructor: Prof. Işıl Dillig TAs: Jacob Van Geffen, Varun Adiga, Akshay Gupta Class meets every Monday,
More informationCompound 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
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 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 informationMath.3336: Discrete Mathematics. Applications of Propositional Logic
Math.3336: Discrete Mathematics Applications of Propositional Logic Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu
More informationAnnouncements. CS311H: Discrete Mathematics. Introduction to First-Order Logic. A Motivating Example. Why First-Order Logic?
Announcements CS311H: Discrete Mathematics Introduction to First-Order Logic Instructor: Işıl Dillig Homework due at the beginning of next lecture Please bring a hard copy of solutions to class! Instructor:
More information2. 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
More informationTruth-Functional Logic
Truth-Functional Logic Syntax Every atomic sentence (A, B, C, ) is a sentence and are sentences With ϕ a sentence, the negation ϕ is a sentence With ϕ and ψ sentences, the conjunction ϕ ψ is a sentence
More informationPropositional 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:
More information2/13/2012. Logic: Truth Tables. CS160 Rosen Chapter 1. Logic?
Logic: Truth Tables CS160 Rosen Chapter 1 Logic? 1 What is logic? Logic is a truth-preserving system of inference Truth-preserving: If the initial statements are true, the inferred statements will be true
More informationPropositional 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
More informationThe 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 :
More informationPropositional 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
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 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 information10/5/2012. 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
More informationSection 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
More informationLanguage of Propositional Logic
Logic A logic has: 1. An alphabet that contains all the symbols of the language of the logic. 2. A syntax giving the rules that define the well formed expressions of the language of the logic (often called
More informationLogic: 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.
More informationWhat 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
More informationPropositional 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
More informationTautologies, 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
More informationPropositional 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?
More informationLearning 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
More informationUNIT-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
More informationProposition/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
More informationANALYSIS EXERCISE 1 SOLUTIONS
ANALYSIS EXERCISE 1 SOLUTIONS 1. (a) Let B The main course will be beef. F The main course will be fish. P The vegetable will be peas. C The vegetable will be corn. The logical form of the argument is
More informationPropositional 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.)
More informationCISC-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)
More informationOverview, cont. Overview, cont. Logistics. Optional Reference #1. Optional Reference #2. Workload and Grading
Course staff CS389L: Automated Logical Reasoning Lecture 1: ntroduction and Review of Basics şıl Dillig nstructor: şil Dillig E-mail: isil@cs.utexas.edu Office hours: Thursday after class until 6:30 pm
More informationChapter 1: Formal Logic
Chapter 1: Formal Logic Dr. Fang (Daisy) Tang ftang@cpp.edu www.cpp.edu/~ftang/ CS 130 Discrete Structures Logic: The Foundation of Reasoning Definition: the foundation for the organized, careful method
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 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
More informationDescription 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
More informationExample ( x.(p(x) Q(x))) ( x.p(x) x.q(x)) premise. 2. ( x.(p(x) Q(x))) -elim, 1 3. ( x.p(x) x.q(x)) -elim, x. P(x) x.
Announcements CS311H: Discrete Mathematics More Logic Intro to Proof Techniques Homework due next lecture Instructor: Işıl Dillig Instructor: Işıl Dillig, CS311H: Discrete Mathematics More Logic Intro
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 informationAn Introduction to Logic 1.1 ~ 1.4 6/21/04 ~ 6/23/04
An Introduction to Logic 1.1 ~ 1.4 6/21/04 ~ 6/23/04 1 A Taste of Logic Logic puzzles (1) Knights and Knaves Knights: always tell the truth Knaves: always lie You encounter two people A and B. A says:
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 informationDiscrete Mathematics and Its Applications
Discrete Mathematics and Its Applications Lecture 1: Proposition logic MING GAO DASE @ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Sep. 12, 2017 Outline 1 Propositions 2 Connectives
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 informationChapter 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
More informationWhy Learning Logic? Logic. Propositional Logic. Compound Propositions
Logic Objectives Propositions and compound propositions Negation, conjunction, disjunction, and exclusive or Implication and biconditional Logic equivalence and satisfiability Application of propositional
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 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 informationCSCE 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
More informationIt is not the case that ϕ. p = It is not the case that it is snowing = It is not. r = It is not the case that Mary will go to the party =
Introduction to Propositional Logic Propositional Logic (PL) is a logical system that is built around the two values TRUE and FALSE, called the TRUTH VALUES. true = 1; false = 0 1. Syntax of Propositional
More information1.3 Propositional Equivalences
1 1.3 Propositional Equivalences The replacement of a statement with another statement with the same truth is an important step often used in Mathematical arguments. Due to this methods that produce propositions
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 informationHW1 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
More informationPropositional Calculus: Formula Simplification, Essential Laws, Normal Forms
P Formula Simplification, Essential Laws, Normal Forms Lila Kari University of Waterloo P Formula Simplification, Essential Laws, Normal CS245, Forms Logic and Computation 1 / 26 Propositional calculus
More informationA 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
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 information15414/614 Optional Lecture 1: Propositional Logic
15414/614 Optional Lecture 1: Propositional Logic Qinsi Wang Logic is the study of information encoded in the form of logical sentences. We use the language of Logic to state observations, to define concepts,
More informationCSE507. Course Introduction. Computer-Aided Reasoning for Software. Emina Torlak
Computer-Aided Reasoning for Software CSE507 courses.cs.washington.edu/courses/cse507/14au/ Course Introduction Emina Torlak emina@cs.washington.edu Today What is this course about? Course logistics Review
More informationCS 2740 Knowledge Representation. Lecture 4. Propositional logic. CS 2740 Knowledge Representation. Administration
Lecture 4 Propositional logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square dministration Homework assignment 1 is out Due next week on Wednesday, September 17 Problems: LISP programming a PL
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 information1 Propositional Logic
1 Propositional Logic Required reading: Foundations of Computation. Sections 1.1 and 1.2. 1. Introduction to Logic a. Logical consequences. If you know all humans are mortal, and you know that you are
More informationCS 4700: Artificial Intelligence
CS 4700: Foundations of Artificial Intelligence Fall 2017 Instructor: Prof. Haym Hirsh Lecture 12 Prelim Tuesday, March 21 8:40-9:55am Statler Auditorium Homework 2 To be posted on Piazza 4701 Projects:
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 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 informationEquivalence 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
More informationIntroduction 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
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 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 informationCHAPTER 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
More informationKnowledge base (KB) = set of sentences in a formal language Declarative approach to building an agent (or other system):
Logic Knowledge-based agents Inference engine Knowledge base Domain-independent algorithms Domain-specific content Knowledge base (KB) = set of sentences in a formal language Declarative approach to building
More informationPROPOSITIONAL CALCULUS
PROPOSITIONAL CALCULUS A proposition is a complete declarative sentence that is either TRUE (truth value T or 1) or FALSE (truth value F or 0), but not both. These are not propositions! Connectives and
More informationCSE507. Introduction. Computer-Aided Reasoning for Software. Emina Torlak courses.cs.washington.edu/courses/cse507/17wi/
Computer-Aided Reasoning for Software CSE507 courses.cs.washington.edu/courses/cse507/17wi/ Introduction Emina Torlak emina@cs.washington.edu Today What is this course about? Course logistics Review of
More informationPropositional Logic. Fall () Propositional Logic Fall / 30
Propositional Logic Fall 2013 () Propositional Logic Fall 2013 1 / 30 1 Introduction Learning Outcomes for this Presentation 2 Definitions Statements Logical connectives Interpretations, contexts,... Logically
More informationCS206 Lecture 03. Propositional Logic Proofs. Plan for Lecture 03. Axioms. Normal Forms
CS206 Lecture 03 Propositional Logic Proofs G. Sivakumar Computer Science Department IIT Bombay siva@iitb.ac.in http://www.cse.iitb.ac.in/ siva Page 1 of 12 Fri, Jan 03, 2003 Plan for Lecture 03 Axioms
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 informationPropositional Languages
Propositional Logic Propositional Languages A propositional signature is a set/sequence of primitive symbols, called proposition constants. Given a propositional signature, a propositional sentence is
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 informationNatural Deduction. Formal Methods in Verification of Computer Systems Jeremy Johnson
Natural Deduction Formal Methods in Verification of Computer Systems Jeremy Johnson Outline 1. An example 1. Validity by truth table 2. Validity by proof 2. What s a proof 1. Proof checker 3. Rules of
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 informationTecniche di Verifica. Introduction to Propositional Logic
Tecniche di Verifica Introduction to Propositional Logic 1 Logic A formal logic is defined by its syntax and semantics. Syntax An alphabet is a set of symbols. A finite sequence of these symbols is called
More informationProof Methods for Propositional Logic
Proof Methods for Propositional Logic Logical equivalence Two sentences are logically equivalent iff they are true in the same models: α ß iff α β and β α Russell and Norvig Chapter 7 CS440 Fall 2015 1
More informationLecture 7. Logic. Section1: Statement Logic.
Ling 726: Mathematical Linguistics, Logic, Section : Statement Logic V. Borschev and B. Partee, October 5, 26 p. Lecture 7. Logic. Section: Statement Logic.. Statement Logic..... Goals..... Syntax of Statement
More informationTitle: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5)
B.Y. Choueiry 1 Instructor s notes #12 Title: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5) Introduction to Artificial Intelligence CSCE 476-876, Fall 2018 URL: www.cse.unl.edu/ choueiry/f18-476-876
More informationPropositional Logic. Jason Filippou UMCP. ason Filippou UMCP) Propositional Logic / 38
Propositional Logic Jason Filippou CMSC250 @ UMCP 05-31-2016 ason Filippou (CMSC250 @ UMCP) Propositional Logic 05-31-2016 1 / 38 Outline 1 Syntax 2 Semantics Truth Tables Simplifying expressions 3 Inference
More informationCISC-102 Fall 2018 Week 11
page! 1 of! 26 CISC-102 Fall 2018 Pascal s Triangle ( ) ( ) An easy ( ) ( way ) to calculate ( ) a table of binomial coefficients was recognized centuries ago by mathematicians in India, ) ( ) China, Iran
More informationSec$on Summary. Tautologies, Contradictions, and Contingencies. Logical Equivalence. Normal Forms (optional, covered in exercises in text)
Section 1.3 1 Sec$on Summary Tautologies, Contradictions, and Contingencies. Logical Equivalence Important Logical Equivalences Showing Logical Equivalence Normal Forms (optional, covered in exercises
More informationInference in Propositional Logic
Inference in Propositional Logic Deepak Kumar November 2017 Propositional Logic A language for symbolic reasoning Proposition a statement that is either True or False. E.g. Bryn Mawr College is located
More informationLogical Structures in Natural Language: Propositional Logic II (Truth Tables and Reasoning
Logical Structures in Natural Language: Propositional Logic II (Truth Tables and Reasoning Raffaella Bernardi Università degli Studi di Trento e-mail: bernardi@disi.unitn.it Contents 1 What we have said
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 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 informationEECS 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?
More informationLogic for Computer Science - Week 4 Natural Deduction
Logic for Computer Science - Week 4 Natural Deduction 1 Introduction In the previous lecture we have discussed some important notions about the semantics of propositional logic. 1. the truth value of a
More informationChapter 4: Classical Propositional Semantics
Chapter 4: Classical Propositional Semantics Language : L {,,, }. Classical Semantics assumptions: TWO VALUES: there are only two logical values: truth (T) and false (F), and EXTENSIONALITY: the logical
More informationDiscrete Structures & Algorithms. Propositional Logic EECE 320 // UBC
Discrete Structures & Algorithms Propositional Logic EECE 320 // UBC 1 Review of last lecture Pancake sorting A problem with many applications Bracketing (bounding a function) Proving bounds for pancake
More informationAMTH140 Lecture 8. Symbolic Logic
AMTH140 Lecture 8 Slide 1 Symbolic Logic March 10, 2006 Reading: Lecture Notes 6.2, 6.3; Epp 1.1, 1.2 Logical Connectives Let p and q denote propositions, then: 1. p q is conjunction of p and q, meaning
More information1.1 Statements and Compound Statements
Chapter 1 Propositional 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
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 informationCS Module 1. Ben Harsha Apr 12, 2017
CS 50010 Module 1 Ben Harsha Apr 12, 2017 Course details Course is split into 2 modules Module 1 (this one): Covers basic data structures and algorithms, along with math review. Module 2: Probability,
More informationCITS2211 Discrete Structures. Propositional Logic
CITS2211 Discrete Structures Propositional Logic Unit coordinator: Rachel Cardell-Oliver July 31, 2017 Highlights This lecture will address the following questions: 1 What is a proposition? 2 How are propositions
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 informationMath 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
More information