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

https://vu5.sfc.keio.ac.jp/slide/

1 FUNDAMENTALS OF LOGIC NO.7 PREDICATE LOGIC Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 So Far Propositional Logic Logical Connectives (,,, ) Truth Table Tautology

1 FUNDAMENTALS OF LOGIC NO.10 HERBRAND THEOREM Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 So Far Propositional Logic Logical connectives (,,, ) Truth table Tautology

1 FUNDAMENTALS OF LOGIC NO.8 SEMANTICS OF PREDICATE LOGIC Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 So Far Propositional Logic Logical connectives (,,, ) Truth

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

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

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,...)

Fundamentals of Logic

Fundamentals of Logic No.5 Soundness and Completeness Tatsuya Hagino Faculty of Environment and Information Studies Keio University 2015/5/18 Tatsuya Hagino (Faculty of Environment and InformationFundamentals

Part Two: The Basic Components of the SOFL Specification Language

Part Two: The Basic Components of the SOFL Specification Language SOFL logic Module Condition Data Flow Diagrams Process specification Function definition and specification Process decomposition Other

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

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.

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

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

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?

2 Truth Tables, Equivalences and the Contrapositive

2 Truth Tables, Equivalences and the Contrapositive 12 2 Truth Tables, Equivalences and the Contrapositive 2.1 Truth Tables In a mathematical system, true and false statements are the statements of the

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

Propositional Logic. Yimei Xiang 11 February format strictly follow the laws and never skip any step.

Propositional Logic Yimei Xiang yxiang@fas.harvard.edu 11 February 2014 1 Review Recursive definition Set up the basis Generate new members with rules Exclude the rest Subsets vs. proper subsets Sets 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

Solutions 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

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

1.3 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

Foundation of proofs. Jim Hefferon.

Foundation of proofs Jim Hefferon http://joshua.smcvt.edu/proofs The need to prove In Mathematics we prove things To a person with a mathematical turn of mind, the base angles of an isoceles triangle are

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

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

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

Discrete 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

LING 106. Knowledge of Meaning Lecture 3-1 Yimei Xiang Feb 6, Propositional logic

LING 106. Knowledge of Meaning Lecture 3-1 Yimei Xiang Feb 6, 2016 Propositional logic 1 Vocabulary of propositional logic Vocabulary (1) a. Propositional letters: p, q, r, s, t, p 1, q 1,..., p 2, q 2,...

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

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,

Logic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies

Logic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies Valentin Stockholm University September 2016 Propositions Proposition:

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:

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

Introduction to Theoretical Computer Science

Introduction to Theoretical Computer Science Zdeněk Sawa Department of Computer Science, FEI, Technical University of Ostrava 17. listopadu 15, Ostrava-Poruba 708 33 Czech republic February 11, 2018 Z.

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

1 FUNDAMENTALS OF LOGIC NO.1 WHAT IS LOGIC Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 Course Summary What is the correct deduction? Since A, therefore B. It is

Equational Logic: Part 2. Roland Backhouse March 6, 2001

1 Equational Logic: Part 2 Roland Backhouse March 6, 2001 Outline 2 We continue the axiomatisation of the propositional connectives. The axioms for disjunction (the logical or of two statements) are added

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

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

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

Propositional 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

~ p is always false. Based on the basic truth table for disjunction, if q is true then p ~

MAT 101 Solutions Exam 2 (Logic, Part I) Multiple-Choice Questions 1. D Because this sentence contains exactly ten words, it is stating that it is false. But if it is taken to be false, then it has to

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

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

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

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

Chapter 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

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

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

15414/614 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,

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,

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

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

CS206 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

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

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

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)

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

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

Numbers that are divisible by 2 are even. The above statement could also be written in other logically equivalent ways, such as:

3.4 THE CONDITIONAL & BICONDITIONAL Definition. Any statement that can be put in the form If p, then q, where p and q are basic statements, is called a conditional statement and is written symbolically

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

PHI Propositional Logic Lecture 2. Truth Tables

PHI 103 - Propositional Logic Lecture 2 ruth ables ruth ables Part 1 - ruth unctions for Logical Operators ruth unction - the truth-value of any compound proposition determined solely by the truth-value

Math 13, Spring 2013, Lecture B: Midterm

Math 13, Spring 2013, Lecture B: Midterm Name Signature UCI ID # E-mail address Each numbered problem is worth 12 points, for a total of 84 points. Present your work, especially proofs, as clearly as possible.

Examples. Example (1) Example (2) Let x, y be two variables, and denote statements p : x = 0 and q : y = 1. Solve. x 2 + (y 1) 2 = 0.

Examples Let x, y be two variables, and denote statements p : x = 0 and q : y = 1. Example (1) Solve x 2 + (y 1) 2 = 0. The solution is x = 0 AND y = 1. [p q.] Example (2) Solve x(y 1) = 0. Examples Let

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?

The following techniques for methods of proofs are discussed in our text: - Vacuous proof - Trivial proof

Ch. 1.6 Introduction to Proofs The following techniques for methods of proofs are discussed in our text - Vacuous proof - Trivial proof - Direct proof - Indirect proof (our book calls this by contraposition)

WUCT121. Discrete Mathematics. Logic. Tutorial Exercises

WUCT11 Discrete Mathematics Logic Tutorial Exercises 1 Logic Predicate Logic 3 Proofs 4 Set Theory 5 Relations and Functions WUCT11 Logic Tutorial Exercises 1 Section 1: Logic Question1 For each of the

Section 1.1: Propositions and Connectives

Section 1.1: Propositions and Connectives Definition : Proposition: A statement that has exactly one truth value, i.e., either true (T) or false (F). Axiom (or Postulate): A statement about the primitive

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

CS 486: Lecture 2, Thursday, Jan 22, 2009

CS 486: Lecture 2, Thursday, Jan 22, 2009 Mark Bickford January 22, 2009 1 Outline Propositional formulas Interpretations and Valuations Validity and Satisfiability Truth tables and Disjunctive Normal

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)

ECE473 Lecture 15: Propositional Logic

ECE473 Lecture 15: Propositional Logic Jeffrey Mark Siskind School of Electrical and Computer Engineering Spring 2018 Siskind (Purdue ECE) ECE473 Lecture 15: Propositional Logic Spring 2018 1 / 23 What

Comp487/587 - Boolean Formulas

Comp487/587 - Boolean Formulas 1 Logic and SAT 1.1 What is a Boolean Formula Logic is a way through which we can analyze and reason about simple or complicated events. In particular, we are interested

COMP 2600: Formal Methods for Software Engineeing

COMP 2600: Formal Methods for Software Engineeing Dirk Pattinson Semester 2, 2013 What do we mean by FORMAL? Oxford Dictionary in accordance with convention or etiquette or denoting a style of writing

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

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

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

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

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,

MACM 101 Discrete Mathematics I. Exercises on Propositional Logic. Due: Tuesday, September 29th (at the beginning of the class)

MACM 101 Discrete Mathematics I Exercises on Propositional Logic. Due: Tuesday, September 29th (at the beginning of the class) SOLUTIONS 1. Construct a truth table for the following compound proposition:

CSE20: Discrete Mathematics for Computer Science. Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication

CSE20: Discrete Mathematics for Computer Science Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication Disjunctive normal form Example: Let f (x, y, z) =xy z. Write this function in DNF. Minterm

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.

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

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

n 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

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

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

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

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

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

Syntax of propositional logic. Syntax tree of a formula. Semantics of propositional logic (I) Subformulas

Syntax of propositional logic Syntax tree of a formula An atomic formula has the form A i where i =, 2, 3,.... Formulas are defined by the following inductive process: Every formula can be represented

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

CITS2211 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

Proving Things. Why prove things? Proof by Substitution, within Logic. Rules of Inference: applying Logic. Using Assumptions.

1 Proving Things Why prove things? Proof by Substitution, within Logic Rules of Inference: applying Logic Using Assumptions Proof Strategies 2 Why Proofs? Knowledge is power. Where do we get it? direct

Logic and Truth Tables

Logic and ruth ables What is a ruth able? 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. here are five basic operations

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,

Sec$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

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

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: 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

Propositional natural deduction

Propositional natural deduction COMP2600 / COMP6260 Dirk Pattinson Australian National University Semester 2, 2016 Major proof techniques 1 / 25 Three major styles of proof in logic and mathematics Model