Symbolic Logic. Alice E. Fischer. CSCI 1166 Discrete Mathematics for Computing February 5 6,

Size: px
Start display at page:

Download "Symbolic Logic. Alice E. Fischer. CSCI 1166 Discrete Mathematics for Computing February 5 6,"

Transcription

1 Symbolic Logic Alice E. Fischer CSCI 1166 Discrete Mathematics for Computing February 5 6, Alice E. Fischer Symbolic Logic... 1/19

2 1 2 3 Alice E. Fischer Symbolic Logic... 2/19

3 Symbolic Logic Outline This lecture begins the study of symbolic logic and clear reasoning. Symbolic and valid reasoning are the philosophical bases of clear reasoning. A first-year law student studies logic. Digital circuits and program-control statements evolved from philosophical logic. The laws of logic are closely related to the laws of set algebra. Sets and logic are the foundations of databases. Alice E. Fischer Symbolic Logic... 3/19

4 Alice E. Fischer Symbolic Logic... 4/19

5 Symbolic Logic Outline Algebra is a field of mathematics where operators are used to do computations on numbers. In set theory, set relations and operators are used to do computations on sets. In symbolic logic, boolean functions are applied to statements (not number or sets). All three fields have their own sets of symbols and follow a set of laws and relationships. The laws and relationships of all three areas are much the same and often have the same names, although the symbols used make them look different. Alice E. Fischer Symbolic Logic... 5/19

6 Statements Outline In logic, a statement is: A meaningful declarative sentence that is either true or false. or some other formulation that expresses the same meaning. Examples of Statements: Bridges can fall down. Dams last forever. Computer systems malfunction eventually. Some aircraft crash, some disappear, and some land safely. The word proposition is a synonym for logical statement. Alice E. Fischer Symbolic Logic... 6/19

7 What is NOT a Statement? It is not a statement (proposition) if it is... Neither true nor false: The King of France is wise. Truth depends on a pronoun: She is ready now. Opinion, not fact: Broccoli tastes good. Not declarative: Are you ready? Not declarative: Run! Meaningless: Greenness perambulates. Meaningless: I had one grunch but the eggplant over there. Not a sentence: This sentence no verb. Alice E. Fischer Symbolic Logic... 7/19

8 Ambiguity can Confound Reasoning. When you use logic, you must be very careful that your initial propositions have one meaning and only one meaning in your universe. Consider the word bipedal. X is bipedal can have three meanings: 1 X is a member of an animal species that has two feet. 2 X has two feet. 3 X walks on two feet. Suppose X refers to me. Then all three interpretations are true. Now suppose X is a 1-legged man in a wheelchair. Number 1 is true but numbers 2 and 3 are false. Alice E. Fischer Symbolic Logic... 8/19

9 Ambiguity Leads to Trouble. Suppose you are writing a business contract or specifying safety requirements or proving a theorem. If you use the same word twice, it must have the same meaning. Failure to ensure a consistent meaning has led to disaster. This incident earned a Darwin Award: There was a group of terrorists in Palestine. They planned to set off a car bomb in Jerusalem at 10:00 am. The bomb maker set the bomb s timer to explode at 10:00, Jerusalem time. The car drivers planned to get there just before 10:00. But they used Palestine time, which was 1 hour different. All three terrorists died when the bomb went off a half-hour outside Jerusalem. Alice E. Fischer Symbolic Logic... 9/19

10 Talking about Statements and Propositions In order to talk about propositions, we frequently define a symbol to be a name for a proposition. Example: Let A be the proposition: The World Trade Center collapsed. Then we can say that A terrorist attack caused A. Using symbols makes it much easier for us to talk and reason about logical issues, just as using variable names in a program makes it easier for us to talk about calculations. The Boolean functions have names in symbolic logic that are older than the names used for logic gates. Alice E. Fischer Symbolic Logic... 10/19

11 Alice E. Fischer Symbolic Logic... 11/19

12 Negation and Conjunction:, and, && is Logic s symbol for the not or! operator. Let S be the proposition I am skinny. Then this is a proposition: S means I am not skinny. is Logic s symbol for and or && operator. Let C be the proposition Aircraft C53 crashed Let S be the proposition The alien spaceship exploded Then this is a proposition: C S Aircraft C53 crashed the alien spaceship exploded. Alice E. Fischer Symbolic Logic... 12/19

13 Disjunction:, or, is Logic s symbol for the or, Let P be the proposition I will buy the pink dress. Let L be the proposition I will buy the lavender hat. Then P L is a proposition. Or has two common meanings in English, inclusive and exclusive. The logical symbol stands for Inclusive-Or. Inclusive: The P L uses inclusive-or. It means that I will buy something. It might be a dress. It might be a hat. It might be both. Exclusive: Consider the proposition that Lee will walk the dog or grade papers. This uses the Exclusive-Or. He certainly cannot do both at the same time. Alice E. Fischer Symbolic Logic... 13/19

14 Operator Precedence If you see an expression with multiple operators in it like S P Q, how do you know which operation to do first? In formal logic Negation has the highest precedence and should be done first. Conjunction and Disjunction have the same precedence. Therefore, in formal logic, if you have a sequence of conjunctions and disjunctions you must include parentheses to designate the proper order to remove ambiguity. Alice E. Fischer Symbolic Logic... 14/19

15 Translating from English to Symbolic Logic-1 Presume we start with a sentence or paragraph, in English. For example, There was a meeting on Friday and I was not sick, so I went to work. First, identify and name the statements: Let M = There was a meeting on Friday. Let S = I was sick. Let W = I went to work. Then restate the sentence(s) using symbols: M S so W We will soon see that so translates as in logic, giving: M S W Alice E. Fischer Symbolic Logic... 15/19

16 Translating from English to Symbolic Logic 2 Sunday is a weekend day but I went to work. Use wedge (and) to translate BUT. Let E be Sunday is a weekend day. Let W be I went to work. E W Use (not) with (or) to symbolize neither... nor. Today is a neither a weekend day nor is it a work day for me. (E W ) Alice E. Fischer Symbolic Logic... 16/19

17 Alice E. Fischer Symbolic Logic... 17/19

18 Using a Truth Table John is designing the logic for an elevator. The safety specifications require the doors to be closed (that is, not open). when the elevator is moving. Also, the doors must open when the elevator stops. Let O be The elevator doors are open. Let M be The elevator is moving. Then the requirements are: M and not O or not M and O We can symbolize the requirements as: ( O M) (O M) Alice E. Fischer Symbolic Logic... 18/19

19 A Truth Table for the Elevator The requirements are: ( O M) (O M) Make a truth table to evaluate the proposition. ( O M) O M O M O M O M (O M) T T F F F F F T F F T F T T F T T F T F T F F T T F F F The elevator controller must keep the system in a safe state. We see safe states on the second and third line of the table. So the controller must keep the system in one of these states. Alice E. Fischer Symbolic Logic... 19/19

A statement is a sentence that is definitely either true or false but not both.

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

More information

DISCRETE STRUCTURES WEEK5 LECTURE1

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

More information

Section 1.1: Logical Form and Logical Equivalence

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,

More information

Discrete Mathematical Structures. Chapter 1 The Foundation: Logic

Discrete 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 information

Overview. 1. Introduction to Propositional Logic. 2. Operations on Propositions. 3. Truth Tables. 4. Translating Sentences into Logical Expressions

Overview. 1. Introduction to Propositional Logic. 2. Operations on Propositions. 3. Truth Tables. 4. Translating Sentences into Logical Expressions Note 01 Propositional Logic 1 / 10-1 Overview 1. Introduction to Propositional Logic 2. Operations on Propositions 3. Truth Tables 4. Translating Sentences into Logical Expressions 5. Preview: Propositional

More information

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) 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 information

Equivalence and Implication

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

More information

An 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 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 information

HW1 graded review form? HW2 released CSE 20 DISCRETE MATH. Fall

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

More information

3.2: Compound Statements and Connective Notes

3.2: Compound Statements and Connective Notes 3.2: Compound Statements and Connective Notes 1. Express compound statements in symbolic form. _Simple_ statements convey one idea with no connecting words. _Compound_ statements combine two or more simple

More information

CSE 20 DISCRETE MATH. Fall

CSE 20 DISCRETE MATH. Fall CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Describe and use algorithms for integer operations based on their expansions Relate algorithms for integer

More information

CSC Discrete Math I, Spring Propositional Logic

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,...)

More information

TRUTH TABLES LOGIC (CONTINUED) Philosophical Methods

TRUTH TABLES LOGIC (CONTINUED) Philosophical Methods TRUTH TABLES LOGIC (CONTINUED) Philosophical Methods Here s some Vocabulary we will be talking about in this PowerPoint. Atomic Sentences: Statements which express one proposition Connectives: These are

More information

Propositional logic ( ): Review from Mat 1348

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

More information

1.1 Statements and Compound Statements

1.1 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 information

THE LOGIC OF COMPOUND STATEMENTS

THE LOGIC OF COMPOUND STATEMENTS CHAPTER 2 THE LOGIC OF COMPOUND STATEMENTS Copyright Cengage Learning. All rights reserved. SECTION 2.1 Logical Form and Logical Equivalence Copyright Cengage Learning. All rights reserved. Logical Form

More information

Chapter 1, Section 1.1 Propositional Logic

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

More information

AMTH140 Lecture 8. Symbolic Logic

AMTH140 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 information

Discrete Mathematics and Applications COT3100

Discrete 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 information

Chapter Summary. Propositional Logic. Predicate Logic. Proofs. The Language of Propositions (1.1) Applications (1.2) Logical Equivalences (1.

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)

More information

THE LOGIC OF COMPOUND STATEMENTS

THE LOGIC OF COMPOUND STATEMENTS THE LOGIC OF COMPOUND STATEMENTS All dogs have four legs. All tables have four legs. Therefore, all dogs are tables LOGIC Logic is a science of the necessary laws of thought, without which no employment

More information

Propositions. Frequently, we will use the word statement instead of proposition.

Propositions. Frequently, we will use the word statement instead of proposition. Propositional Logic Propositions A proposition is a declaration of fact that is either true or false, but not both. Examples and non-examples: One plus two equals four (proposition) Mozart is the greatest

More information

Logic and Propositional Calculus

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

More information

AI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic and Predicate Logic

AI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic and Predicate Logic AI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic and Predicate Logic Propositional logic Logical connectives Rules for wffs Truth tables for the connectives Using Truth Tables to evaluate

More information

Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014

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

More information

ARTIFICIAL INTELLIGENCE

ARTIFICIAL INTELLIGENCE ARTIFICIAL INTELLIGENCE LECTURE # 03 Artificial Intelligence 2012 Lecture 03 Delivered By Zahid Iqbal 1 Review of Last Lecture Artificial Intelligence 2012 Lecture 03 Delivered By Zahid Iqbal 2 Today s

More information

ICS141: Discrete Mathematics for Computer Science I

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

More information

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

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

More information

CS100: DISCRETE STRUCTURES. Lecture 5: Logic (Ch1)

CS100: 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 information

Discrete Mathematics

Discrete Mathematics Discrete Mathematics Discrete mathematics is devoted to the study of discrete or distinct unconnected objects. Classical mathematics deals with functions on real numbers. Real numbers form a continuous

More information

Mathematical Logic Part One

Mathematical Logic Part One Mathematical Logic Part One Question: How do we formalize the definitions and reasoning we use in our proofs? Where We're Going Propositional Logic (Today) Basic logical connectives. Truth tables. Logical

More information

PROPOSITIONAL CALCULUS

PROPOSITIONAL 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 information

Introduction to Sets and Logic (MATH 1190)

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

More information

Symbolic Logic Outline

Symbolic Logic Outline Symbolic Logic Outline 1. Symbolic Logic Outline 2. What is Logic? 3. How Do We Use Logic? 4. Logical Inferences #1 5. Logical Inferences #2 6. Symbolic Logic #1 7. Symbolic Logic #2 8. What If a Premise

More information

Compound Propositions

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

More information

8. Reductio ad absurdum

8. Reductio ad absurdum 8. Reductio ad absurdum 8.1 A historical example In his book, The Two New Sciences, 10 Galileo Galilea (1564-1642) gives several arguments meant to demonstrate that there can be no such thing as actual

More information

Boolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table.

Boolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table. The Laws of Boolean Boolean algebra As well as the logic symbols 0 and 1 being used to represent a digital input or output, we can also use them as constants for a permanently Open or Closed circuit or

More information

Unit 1. Propositional Logic Reading do all quick-checks Propositional Logic: Ch. 2.intro, 2.2, 2.3, 2.4. Review 2.9

Unit 1. Propositional Logic Reading do all quick-checks Propositional Logic: Ch. 2.intro, 2.2, 2.3, 2.4. Review 2.9 Unit 1. Propositional Logic Reading do all quick-checks Propositional Logic: Ch. 2.intro, 2.2, 2.3, 2.4. Review 2.9 Typeset September 23, 2005 1 Statements or propositions Defn: A statement is an assertion

More information

Proposition/Statement. Boolean Logic. Boolean variables. Logical operators: And. Logical operators: Not 9/3/13. Introduction to Logical Operators

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

More information

Propositional Logic Not Enough

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

More information

Discrete Mathematics and Its Applications

Discrete 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 information

Packet #1: Logic & Proofs. Applied Discrete Mathematics

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

More information

INTRODUCTION TO LOGIC. Propositional Logic. Examples of syntactic claims

INTRODUCTION TO LOGIC. Propositional Logic. Examples of syntactic claims Introduction INTRODUCTION TO LOGIC 2 Syntax and Semantics of Propositional Logic Volker Halbach In what follows I look at some formal languages that are much simpler than English and define validity of

More information

Mathematical Logic Part One

Mathematical Logic Part One Mathematical Logic Part One Question: How do we formalize the definitions and reasoning we use in our proofs? Where We're Going Propositional Logic (Today) Basic logical connectives. Truth tables. Logical

More information

1 Propositional Logic

1 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 information

A Little Deductive Logic

A Little Deductive Logic A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that

More information

Chapter 1: Formal Logic

Chapter 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 information

Ella failed to drop the class. Ella dropped the class.

Ella failed to drop the class. Ella dropped the class. Propositional logic In many cases, a sentence is built up from one or more simpler sentences. To see what follows from such a complicated sentence, it is helpful to distinguish the simpler sentences from

More information

Propositional Logic Basics Propositional Equivalences Normal forms Boolean functions and digital circuits. Propositional Logic.

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?

More information

2/13/2012. Logic: Truth Tables. CS160 Rosen Chapter 1. Logic?

2/13/2012. 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 information

Section L.1- Introduction to Logic

Section L.1- Introduction to Logic Section L.1- Introduction to Logic Definition: A statement, or proposition, is a declarative sentence that can be classified as either true or false, but not both. Example 1: Which of the following are

More information

Chapter 1 Elementary Logic

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

More information

Lecture 1: Propositional Logic (I)

Lecture 1: Propositional Logic (I) Lecture 1: Propositional Logic (I) Fadoua Ghourabi Ochanomizu University ghourabi.fadoua@ocha.cc.ac.jp Octobre 12, 2016 1 / 36 Contents 1 Declarative sentences 2 Constructing propositions 3 Quizz 4 Natural

More information

Boolean Logic. CS 231 Dianna Xu

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

More information

A Little Deductive Logic

A Little Deductive Logic A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that

More information

Lesson 10: True and False Equations

Lesson 10: True and False Equations Classwork Exercise 1 a. Consider the statement: The President of the United States is a United States citizen. Is the statement a grammatically correct sentence? What is the subject of the sentence? What

More information

1 The Foundations. 1.1 Logic. A proposition is a declarative sentence that is either true or false, but not both.

1 The Foundations. 1.1 Logic. A proposition is a declarative sentence that is either true or false, but not both. he oundations. Logic Propositions are building blocks of logic. A proposition is a declarative sentence that is either true or false, but not both. Example. Declarative sentences.. Ottawa is the capital

More information

10 LOGIC Introduction. Objectives

10 LOGIC Introduction. Objectives 10 LOGIC Chapter 10 Logic Objectives After studying this chapter you should understand the nature of propositional logic; understand the connectives NOT, OR, AND; understand implication and equivalence;

More information

Computer Science 280 Spring 2002 Homework 2 Solutions by Omar Nayeem

Computer Science 280 Spring 2002 Homework 2 Solutions by Omar Nayeem Computer Science 280 Spring 2002 Homework 2 Solutions by Omar Nayeem Part A 1. (a) Some dog does not have his day. (b) Some action has no equal and opposite reaction. (c) Some golfer will never be eated

More information

Sec$on Summary. Propositions Connectives. Truth Tables. Negation Conjunction Disjunction Implication; contrapositive, inverse, converse Biconditional

Sec$on Summary. Propositions Connectives. Truth Tables. Negation Conjunction Disjunction Implication; contrapositive, inverse, converse Biconditional Section 1.1 Sec$on Summary Propositions Connectives Negation Conjunction Disjunction Implication; contrapositive, inverse, converse Biconditional ruth ables 2 Proposi$ons A proposition is a declarative

More information

EECS 1028 M: Discrete Mathematics for Engineers

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?

More information

DEDUCTIVE REASONING Propositional Logic

DEDUCTIVE REASONING Propositional Logic 7 DEDUCTIVE REASONING Propositional Logic Chapter Objectives Connectives and Truth Values You will be able to understand the purpose and uses of propositional logic. understand the meaning, symbols, and

More information

FUNDAMENTALS OF MATHEMATICS HANDOUT 1.3 DR. MCLOUGHLIN

FUNDAMENTALS OF MATHEMATICS HANDOUT 1.3 DR. MCLOUGHLIN 021 McLoughlin Handout 1.3, page 1 of 6 FUNDAMENTALS OF MATHEMATICS HANDOUT 1.3 DR. MCLOUGHLIN Truth Table for Not K K T F F T Truth Table for And B M B M T F F F T F F F F Truth Table for Or R S R S T

More information

3/29/2017. Logic. Propositions and logical operations. Main concepts: propositions truth values propositional variables logical operations

3/29/2017. 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 information

Mathematical Logic Part One

Mathematical Logic Part One Mathematical Logic Part One Question: How do we formalize the definitions and reasoning we use in our proofs? Where We're Going Propositional Logic (oday) Basic logical connectives. ruth tables. Logical

More information

Proofs Propositions and Calculuses

Proofs Propositions and Calculuses Lecture 2 CS 1813 Discrete Mathematics Proofs Propositions and Calculuses 1 City of Königsberg (Kaliningrad) 2 Bridges of Königsberg Problem Find a route that crosses each bridge exactly once Must the

More information

Lecture 02: Propositional Logic

Lecture 02: Propositional Logic Lecture 02: Propositional Logic CSCI 358 Discrete Mathematics, Spring 2016 Hua Wang, Ph.D. Department of Electrical Engineering and Computer Science January 19, 2015 Propositional logic Propositional logic

More information

CS 220: Discrete Structures and their Applications. Propositional Logic Sections in zybooks

CS 220: Discrete Structures and their Applications. Propositional Logic Sections in zybooks CS 220: Discrete Structures and their Applications Propositional Logic Sections 1.1-1.2 in zybooks Logic in computer science Used in many areas of computer science: ü Booleans and Boolean expressions in

More information

. T SHREE MAHAPRABHU PUBLIC SCHOOL & COLLEGE NOTES FOR BOARD EXAMINATION SUBJECT COMPUTER SCIENCE (Code: 083) Boolean Algebra

. T SHREE MAHAPRABHU PUBLIC SCHOOL & COLLEGE NOTES FOR BOARD EXAMINATION SUBJECT COMPUTER SCIENCE (Code: 083) Boolean Algebra . T SHREE MAHAPRABHU PUBLIC SCHOOL & COLLEGE NOTES FOR BOARD EXAMINATION 2016-17 SUBJECT COMPUTER SCIENCE (Code: 083) Boolean Algebra Introduction to Boolean Algebra Boolean algebra which deals with two-valued

More information

What is Logic? Introduction to Logic. Simple Statements. Which one is statement?

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

More information

UNIT-I: Propositional Logic

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

More information

Manual of Logical Style

Manual of Logical Style Manual of Logical Style Dr. Holmes January 9, 2015 Contents 1 Introduction 2 2 Conjunction 3 2.1 Proving a conjunction...................... 3 2.2 Using a conjunction........................ 3 3 Implication

More information

Propositional Logic and Semantics

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

More information

2-1. Inductive Reasoning and Conjecture. Lesson 2-1. What You ll Learn. Active Vocabulary

2-1. Inductive Reasoning and Conjecture. Lesson 2-1. What You ll Learn. Active Vocabulary 2-1 Inductive Reasoning and Conjecture What You ll Learn Scan Lesson 2-1. List two headings you would use to make an outline of this lesson. 1. Active Vocabulary 2. New Vocabulary Fill in each blank with

More information

Chapter 3: Logic. Diana Pell. A statement is a declarative sentence that is either true or false, but not both.

Chapter 3: Logic. Diana Pell. A statement is a declarative sentence that is either true or false, but not both. Chapter 3: Logic Diana Pell Section 3.1: Statements and Quantifiers A statement is a declarative sentence that is either true or false, but not both. Exercise 1. Decide which of the following are statements

More information

Today s Topic: Propositional Logic

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

More information

Logic. Def. A Proposition is a statement that is either true or false.

Logic. Def. A Proposition is a statement that is either true or false. Logic Logic 1 Def. A Proposition is a statement that is either true or false. Examples: Which of the following are propositions? Statement Proposition (yes or no) If yes, then determine if it is true or

More information

cse541 LOGIC FOR COMPUTER SCIENCE

cse541 LOGIC FOR COMPUTER SCIENCE cse541 LOGIC FOR COMPUTER SCIENCE Professor Anita Wasilewska Spring 2015 LECTURE 2 Chapter 2 Introduction to Classical Propositional Logic PART 1: Classical Propositional Model Assumptions PART 2: Syntax

More information

Discrete Structures for Computer Science

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

More information

Topic 1: Propositional logic

Topic 1: Propositional logic Topic 1: Propositional logic Guy McCusker 1 1 University of Bath Logic! This lecture is about the simplest kind of mathematical logic: propositional calculus. We discuss propositions, which are statements

More information

LOGIC & PROPOSITIONAL EQUIVALENCE

LOGIC & PROPOSITIONAL EQUIVALENCE KS091201 MATEMATIKA DISKRIT (DISCRETE MATHEMATICS ) LOGIC & PROPOSITIONAL EQUIVALENCE Discrete Math Team 2 -- KS091201 MD W-02 Outline Logic Proposition Propositional Variables Logical Operators Presedence

More information

MAT 101 Exam 2 Logic (Part I) Fall Circle the correct answer on the following multiple-choice questions.

MAT 101 Exam 2 Logic (Part I) Fall Circle the correct answer on the following multiple-choice questions. Name: MA 101 Exam 2 Logic (Part I) all 2017 Multiple-Choice Questions [5 pts each] Circle the correct answer on the following multiple-choice questions. 1. Which of the following is not a statement? a)

More information

2/2/2018. CS 103 Discrete Structures. Chapter 1. Propositional Logic. Chapter 1.1. Propositional Logic

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

More information

Logic and Propositional Calculus

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

More information

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 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 information

Logical Operators. Conjunction Disjunction Negation Exclusive Or Implication Biconditional

Logical 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 information

Course Staff. Textbook

Course 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 information

Predicates, Quantifiers and Nested Quantifiers

Predicates, Quantifiers and Nested Quantifiers Predicates, Quantifiers and Nested Quantifiers Predicates Recall the example of a non-proposition in our first presentation: 2x=1. Let us call this expression P(x). P(x) is not a proposition because x

More information

Logic. 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: 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 information

Menu. Let s logically analyze a simple paragraph:

Menu. Let s logically analyze a simple paragraph: Menu Introduction to Design Look into my... 1 Introduction to Design Let s logically analyze a simple paragraph: When I forget my umbrella and it rains, I get wet. I also get wet when I stick my foot in

More information

Note: The area of logic that deals with propositions is called the propositional calculus or propositional logic.

Note: The area of logic that deals with propositions is called the propositional calculus or propositional logic. Ch. 1.1 Logic Logic 1 Def. A Proposition is a statement that is either true or false. Example 1: Which of the following are propositions? Statement Proposition (yes or no) UHD is a University 1 + 3 = 0

More information

Glossary of Logical Terms

Glossary of Logical Terms Math 304 Spring 2007 Glossary of Logical Terms The following glossary briefly describes some of the major technical logical terms used in this course. The glossary should be read through at the beginning

More information

Definition 2. Conjunction of p and q

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

More information

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

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

More information

CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS

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

More information

Chapter 5: Section 5-1 Mathematical Logic

Chapter 5: Section 5-1 Mathematical Logic Chapter 5: Section 5-1 Mathematical Logic D. S. Malik Creighton University, Omaha, NE D. S. Malik Creighton University, Omaha, NE Chapter () 5: Section 5-1 Mathematical Logic 1 / 29 Mathematical Logic

More information

Proposition logic and argument. CISC2100, Spring 2017 X.Zhang

Proposition logic and argument. CISC2100, Spring 2017 X.Zhang Proposition logic and argument CISC2100, Spring 2017 X.Zhang 1 Where are my glasses? I know the following statements are true. 1. If I was reading the newspaper in the kitchen, then my glasses are on the

More information

Where are my glasses?

Where are my glasses? Proposition logic and argument CISC2100, Spring 2017 X.Zhang 1 Where are my glasses? I know the following statements are true. 1. If I was reading the newspaper in the kitchen, then my glasses are on the

More information

cis32-ai lecture # 18 mon-3-apr-2006

cis32-ai lecture # 18 mon-3-apr-2006 cis32-ai lecture # 18 mon-3-apr-2006 today s topics: propositional logic cis32-spring2006-sklar-lec18 1 Introduction Weak (search-based) problem-solving does not scale to real problems. To succeed, problem

More information

Mat 243 Exam 1 Review

Mat 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 information