Packet #1: Logic & Proofs. Applied Discrete Mathematics

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Packet #1: Logic & Proofs. Applied Discrete Mathematics"

Transcription

1 Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 3-13

2 Course Objectives At the conclusion of this course, you should be able to 1. Represent logical statements in propositional and predicate calculus, and use truth tables and formal proofs to determine their truth values. 2. Create a truth table for a logical expression. Derive a logical expression from a given truth table. Design a circuit to perform a simple task. 3. Construct a circuit from a logical expression using AND, OR, and NOT gates. Simplify logical expressions. Derive a logical expression from a given circuit. 4. Describe set notations using predicate calculus. Determine the power of a set. Use predicate calculus to prove set theoretic propositions. 5. Describe and use the first, second, and general principles of proof by induction. Derive closed form representations for recursively defined sequences; prove their correctness by induction. Derive recursive sequences from closed form functions and prove their equivalence by induction. 6. Describe asymptotic growth of functions, compare functions using big-oh notation. Compare asymptotic growth and prove inequalities by induction. Determine and solve recurrences arising from algorithms. Determine big-oh running times for algorithms. 7. Define binary relations and their properties using predicate calculus. Represent binary relations as ordered pairs, matrices, predicates, or graphs. Combine binary relations by union, intersection, and composition using matrix operations. Find the reflexive, symmetric, and transitive closures of a binary relation. 8. Describe and calculate permutations and combinations with and without replacement and with and without distinguishable objects. Describe and apply the pigeonhole principle. 9. Describe and determine the existence of Euler circuits and paths and Hamilton circuits and paths in graphs. Determine the minimum spanning tree of a graph. Construct and analyze Hasse diagrams for partially ordered sets. 2

3 Propositional Calculus Outline I. Proposition (Sections 1.1 and 1.2) A. Definition B. Compound 1. Connectives:,,,,,,, 2. Variables II. Truth Tables (Sections 1.1 and 1.2) A. Table of compound results using all combinations of input variables B. Definitions C. Tautology and Contradiction - using truth tables for short proofs D. Logical Implications E. Logical Equivalences III. Formal Proofs using statements and reasons (Section 3.1) A. Axioms of Logical Implications B. Axioms of Logical Equivalences C. Two substitution rules IV. Types of formal proofs (Section 3.1) A. Direct B. Indirect 1. Contradiction 2. Other V. Hints for Attacking a propositional calculus proof 3

4 VI. Logic Functions (Section 9.1) A. Number of cases B. Disjunctive Normal Form C. Generation of logic statements from truth table VII. Logic Circuits (Section 9.3) A. Finding logic circuit for given logic function B. Finding logic function for given logic circuit C. Manipulating the logic function for specific characteristics and designing new logic circuit D. Creating logic function from input-output specifications 4

5 Propositional Calculus A. Proposition - a sentence or statement which is either true or false but NOT BOTH. It cannot be unspecified (ambiguous) or vacuous. 1. n 2 > n for all integers n>1 2. n 2 = n (ambiguous) 3. How are you? (vacuous) B. Compound propositions 1. lower case letters stand for propositions -- q, p, r, etc. 2. connectives between propositions not or negative and or (inclusive) implies if and only if or (exclusive) Examples: Converse: p q (p q) (s t ) At this time we will consider only a finite number of connectives. p q q p is converse Contrapositive: p q q p is contrapositive Truth Tables: 1. Simple propositions are input (independent) variables that are either true or false. 2. Compound proposition (output) is either true or false = true; 0 = false 4. Must consider all combinations of input variables. 5

6 Example 1.1: p q; that is p q q p (Equivalence rule) p q p q q p If p q is true (=1) then p = q Tautology A proposition that is always true independent of input values (e.g. p p). Contradiction A proposition that is always false independent of input values (e.g. p p). Tautology Contradiction p p p p p p

7 Truth Tables for Connectives p q: p q: p q p q p q p q p: p q: p p p q p q p q: p q: p q p q p q p q

8 Example 1.2: (p r) ( p r) (implication rule) p r (p r) ( p r) p Logical equivalences Example 1.3: (p q) (q r) (p r) (Hypothetical Syllogism) p q r (p q) (q r) (p r) ok not true Note that this rule works only from LEFT to RIGHT and not vice-versa. It is called a Logical implication, whose symbol is. P logically implies Q is writtenp Q. 8

9 Example 1.4: (p q) s p s (Implication) p q s (p q) s p s Formal Proof for (p q) s p s: Statement Reason 1. (p q) s Hypothesis 2. p p q Addition 3. p s 1,2; Hypothetical Syllogism Class Lemma: if (p q) s then p s and q s 9

10 Example 1.5: [(s g) p] (p a) a g s g p a (s g) p (p a) a g g a Formal Proof: Statements Reasons 1. (s g) p Hypothesis 2. p a Hypothesis 3. g p 1; Class Lemma 4. g a 2, 3; Hypothetical Syllogism 5. a g 4; Contrapositive 10

11 Example 1.6: Formal Proof Given: p q Prove: s r q s p r Direct Proof: Statements Reasons 1. p q Assumption 2. q s Assumption 3. p r Assumption 4. (p q) (r s) 2, 3; 26a Constructive Dilemma 5. r s 1, 4; Modus Ponens 6. s r 5; Commutative In a direct proof one needs to show that given p then q, i.e. p q. For an indirect proof one can use any logically equivalent method. A common type of indirect proof is proof by contradiction. One assumes the negation of the conclusion along with the assumptions and shows a contradiction. i.e. (p q) 0 p q p q c (p q) 0 p q p q logically equivalent 11

12 Example 1.7: Proof by Contradiction: Statements Reasons 1. p q Assumption 2. q s Assumption 3. p r Assumption 4. (s r) Negation of Conclusion 5. s r 4; DeMorgan 6. s 5; Simplification 7. q 2, 6; Modus Tollens 8. r s 5; Commutative 9. r 8; Simplification 10. p 3, 9; Modus Tollens 11. p q 7, 10; Conjunction 12. (p q) 11; DeMorgan 13. (p q) (p q) 1, 12; Conjunction 14. Contradiction 13; Rule 7b Two Substitution Rules 1. If a compound proposition P is a tautology and if all occurrences of some variable of P, say a, are replaced by the same proposition E, then the resulting compound proposition P * is also a tautology. [Does not necessarily give same proposition.] a q a q is a tautology. If for all occurrences of p we substitute (s r). (s r) q (s r) q is also a tautology. 2. If compound proposition P contains a proposition Q and if Q is replaced by a logically equivalent proposition Q *, then the resulting compound proposition P * is logically equivalent to P. p q q p is a proposition. p q p q i.e., p q is logically equivalent to p q. Therefore, p q q p is logically equivalent to p q q p. 12

13 Logic Circuits X Y and XY X Y nand (XY) X Y or X+Y X Y nor (X+Y) X Y (X+Y)(XY) XY +X Y xor X X X X X X invertor nor invertor nand invertor De Morgan's: (X + Y)' = X'Y' (XY)' = X' + Y' [P + (QR)] = (P + Q)(P + R) P(Q + R) = PQ + PR Useful reduction rules: 13

A. Propositional Logic

A. Propositional Logic CmSc 175 Discrete Mathematics A. Propositional Logic 1. Statements (Propositions ): Statements are sentences that claim certain things. Can be either true or false, but not both. Propositional logic deals

More information

Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination II (Fall 2007)

Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination II (Fall 2007) Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination II (Fall 2007) Problem 1: Specify two different predicates P (x) and

More information

WUCT121. Discrete Mathematics. Logic. Tutorial Exercises

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

More information

Lecture Notes 1 Basic Concepts of Mathematics MATH 352

Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,

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

HANDOUT AND SET THEORY. Ariyadi Wijaya

HANDOUT AND SET THEORY. Ariyadi Wijaya HANDOUT LOGIC AND SET THEORY Ariyadi Wijaya Mathematics Education Department Faculty of Mathematics and Natural Science Yogyakarta State University 2009 1 Mathematics Education Department Faculty of Mathematics

More information

MAT 243 Test 1 SOLUTIONS, FORM A

MAT 243 Test 1 SOLUTIONS, FORM A t MAT 243 Test 1 SOLUTIONS, FORM A 1. [10 points] Rewrite the statement below in positive form (i.e., so that all negation symbols immediately precede a predicate). ( x IR)( y IR)((T (x, y) Q(x, y)) R(x,

More information

Chapter 1: The Logic of Compound Statements. January 7, 2008

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

More information

Boolean Algebra and Proof. Notes. Proving Propositions. Propositional Equivalences. Notes. Notes. Notes. Notes. March 5, 2012

Boolean Algebra and Proof. Notes. Proving Propositions. Propositional Equivalences. Notes. Notes. Notes. Notes. March 5, 2012 March 5, 2012 Webwork Homework. The handout on Logic is Chapter 4 from Mary Attenborough s book Mathematics for Electrical Engineering and Computing. Proving Propositions We combine basic propositions

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

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

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 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

More information

Propositional Logic, Predicates, and Equivalence

Propositional Logic, Predicates, and Equivalence Chapter 1 Propositional Logic, Predicates, and Equivalence A statement or a proposition is a sentence that is true (T) or false (F) but not both. The symbol denotes not, denotes and, and denotes or. If

More information

Unit I LOGIC AND PROOFS. B. Thilaka Applied Mathematics

Unit I LOGIC AND PROOFS. B. Thilaka Applied Mathematics Unit I LOGIC AND PROOFS B. Thilaka Applied Mathematics UNIT I LOGIC AND PROOFS Propositional Logic Propositional equivalences Predicates and Quantifiers Nested Quantifiers Rules of inference Introduction

More information

Section 1.2 Propositional Equivalences. A tautology is a proposition which is always true. A contradiction is a proposition which is always false.

Section 1.2 Propositional Equivalences. A tautology is a proposition which is always true. 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 information

5. Use a truth table to determine whether the two statements are equivalent. Let t be a tautology and c be a contradiction.

5. Use a truth table to determine whether the two statements are equivalent. Let t be a tautology and c be a contradiction. Statements Compounds and Truth Tables. Statements, Negations, Compounds, Conjunctions, Disjunctions, Truth Tables, Logical Equivalence, De Morgan s Law, Tautology, Contradictions, Proofs with Logical Equivalent

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

Propositional Logic. Fall () Propositional Logic Fall / 30

Propositional 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 information

3 The Semantics of the Propositional Calculus

3 The Semantics of the Propositional Calculus 3 The Semantics of the Propositional Calculus 1. Interpretations Formulas of the propositional calculus express statement forms. In chapter two, we gave informal descriptions of the meanings of the logical

More information

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications Chapter 1: Foundations: Sets, Logic, and Algorithms Discrete Mathematical Structures: Theory and Applications Learning Objectives Learn about sets Explore various operations on sets Become familiar with

More information

Logic and Set Notation

Logic and Set Notation Logic and Set Notation Logic Notation p, q, r: statements,,,, : logical operators p: not p p q: p and q p q: p or q p q: p implies q p q:p if and only if q We can build compound sentences using the above

More information

CHAPTER 1. MATHEMATICAL LOGIC 1.1 Fundamentals of Mathematical Logic

CHAPTER 1. MATHEMATICAL LOGIC 1.1 Fundamentals of Mathematical Logic CHAPER 1 MAHEMAICAL LOGIC 1.1 undamentals of Mathematical Logic Logic is commonly known as the science of reasoning. Some of the reasons to study logic are the following: At the hardware level the design

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

The proposition p is called the hypothesis or antecedent. The proposition q is called the conclusion or consequence.

The proposition p is called the hypothesis or antecedent. The proposition q is called the conclusion or consequence. The Conditional (IMPLIES) Operator The conditional operation is written p q. The proposition p is called the hypothesis or antecedent. The proposition q is called the conclusion or consequence. The Conditional

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 Distinguish between a theorem, an axiom, lemma, a corollary, and a conjecture. Recognize direct proofs

More information

ECOM Discrete Mathematics

ECOM Discrete Mathematics ECOM 2311- Discrete Mathematics Chapter # 1 : The Foundations: Logic and Proofs Fall, 2013/2014 ECOM 2311- Discrete Mathematics - Ch.1 Dr. Musbah Shaat 1 / 85 Outline 1 Propositional Logic 2 Propositional

More information

Propositional Logic. Logical Expressions. Logic Minimization. CNF and DNF. Algebraic Laws for Logical Expressions CSC 173

Propositional Logic. Logical Expressions. Logic Minimization. CNF and DNF. Algebraic Laws for Logical Expressions CSC 173 Propositional Logic CSC 17 Propositional logic mathematical model (or algebra) for reasoning about the truth of logical expressions (propositions) Logical expressions propositional variables or logical

More information

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 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:

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

DISCRETE MATH: FINAL REVIEW

DISCRETE MATH: FINAL REVIEW DISCRETE MATH: FINAL REVIEW DR. DANIEL FREEMAN 1) a. Does 3 = {3}? b. Is 3 {3}? c. Is 3 {3}? c. Is {3} {3}? c. Is {3} {3}? d. Does {3} = {3, 3, 3, 3}? e. Is {x Z x > 0} {x R x > 0}? 1. Chapter 1 review

More information

MaanavaN.Com MA1256 DISCRETE MATHEMATICS. DEPARTMENT OF MATHEMATICS QUESTION BANK Subject & Code : MA1256 DISCRETE MATHEMATICS

MaanavaN.Com MA1256 DISCRETE MATHEMATICS. DEPARTMENT OF MATHEMATICS QUESTION BANK Subject & Code : MA1256 DISCRETE MATHEMATICS DEPARTMENT OF MATHEMATICS QUESTION BANK Subject & Code : UNIT I PROPOSITIONAL CALCULUS Part A ( Marks) Year / Sem : III / V. Write the negation of the following proposition. To enter into the country you

More information

Proofs: A General How To II. Rules of Inference. Rules of Inference Modus Ponens. Rules of Inference Addition. Rules of Inference Conjunction

Proofs: A General How To II. Rules of Inference. Rules of Inference Modus Ponens. Rules of Inference Addition. Rules of Inference Conjunction Introduction I Proofs Computer Science & Engineering 235 Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu A proof is a proof. What kind of a proof? It s a proof. A proof is a proof. And when

More information

Propositional Equivalence

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:

More information

Advanced Topics in LP and FP

Advanced Topics in LP and FP Lecture 1: Prolog and Summary of this lecture 1 Introduction to Prolog 2 3 Truth value evaluation 4 Prolog Logic programming language Introduction to Prolog Introduced in the 1970s Program = collection

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

Discrete Structures CRN Test 3 Version 1 CMSC 2123 Autumn 2013

Discrete Structures CRN Test 3 Version 1 CMSC 2123 Autumn 2013 . Print your name on your scantron in the space labeled NAME. 2. Print CMSC 223 in the space labeled SUBJECT. 3. Print the date 2-2-203, in the space labeled DATE. 4. Print your CRN, 786, in the space

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

Propositional Logics and their Algebraic Equivalents

Propositional Logics and their Algebraic Equivalents Propositional Logics and their Algebraic Equivalents Kyle Brooks April 18, 2012 Contents 1 Introduction 1 2 Formal Logic Systems 1 2.1 Consequence Relations......................... 2 3 Propositional Logic

More information

DISCRETE MATHEMATICS BA202

DISCRETE MATHEMATICS BA202 TOPIC 1 BASIC LOGIC This topic deals with propositional logic, logical connectives and truth tables and validity. Predicate logic, universal and existential quantification are discussed 1.1 PROPOSITION

More information

Logical Reasoning. Chapter Statements and Logical Operators

Logical Reasoning. Chapter Statements and Logical Operators Chapter 2 Logical Reasoning 2.1 Statements and Logical Operators Preview Activity 1 (Compound Statements) Mathematicians often develop ways to construct new mathematical objects from existing mathematical

More information

Boolean Algebra & Logic Gates. By : Ali Mustafa

Boolean Algebra & Logic Gates. By : Ali Mustafa Boolean Algebra & Logic Gates By : Ali Mustafa Digital Logic Gates There are three fundamental logical operations, from which all other functions, no matter how complex, can be derived. These Basic functions

More information

Logic. Propositional Logic: Syntax

Logic. Propositional Logic: Syntax Logic Propositional Logic: Syntax Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about

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

ANS: If you are in Kwangju then you are in South Korea but not in Seoul.

ANS: If you are in Kwangju then you are in South Korea but not in Seoul. Math 15 - Spring 2017 - Homework 1.1 and 1.2 Solutions 1. (1.1#1) Let the following statements be given. p = There is water in the cylinders. q = The head gasket is blown. r = The car will start. (a) Translate

More information

Announcements. CS311H: Discrete Mathematics. Propositional Logic II. Inverse of an Implication. Converse of a Implication

Announcements. CS311H: Discrete Mathematics. Propositional Logic II. Inverse of an Implication. Converse of a Implication Announcements CS311H: Discrete Mathematics Propositional Logic II Instructor: Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Wed 09/13 Remember: Due before

More information

1. Propositions: Contrapositives and Converses

1. Propositions: Contrapositives and Converses Preliminaries 1 1. Propositions: Contrapositives and Converses Given two propositions P and Q, the statement If P, then Q is interpreted as the statement that if the proposition P is true, then the statement

More information

CITS2211 Discrete Structures Proofs

CITS2211 Discrete Structures Proofs CITS2211 Discrete Structures Proofs Unit coordinator: Rachel Cardell-Oliver August 13, 2017 Highlights 1 Arguments vs Proofs. 2 Proof strategies 3 Famous proofs Reading Chapter 1: What is a proof? Mathematics

More information

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, 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,...

More information

Propositional Logic Language

Propositional Logic Language Propositional Logic Language A logic consists of: an alphabet A, a language L, i.e., a set of formulas, and a binary relation = between a set of formulas and a formula. An alphabet A consists of a finite

More information

Normal Forms Note: all ppts about normal forms are skipped.

Normal Forms Note: all ppts about normal forms are skipped. Normal Forms Note: all ppts about normal forms are skipped. Well formed formula (wff) also called formula, is a string consists of propositional variables, connectives, and parenthesis used in the proper

More information

First order Logic ( Predicate Logic) and Methods of Proof

First order Logic ( Predicate Logic) and Methods of Proof First order Logic ( Predicate Logic) and Methods of Proof 1 Outline Introduction Terminology: Propositional functions; arguments; arity; universe of discourse Quantifiers Definition; using, mixing, negating

More information

CPSC 121 Sample Final Examination December 2013

CPSC 121 Sample Final Examination December 2013 CPSC 121 Sample Final Examination December 201 [6] 1. Short answers [] a. What is wrong with the following circuit? You can not connect the outputs of two or more gates together directly; what will happen

More information

The Importance of Being Formal. Martin Henz. February 5, Propositional Logic

The Importance of Being Formal. Martin Henz. February 5, Propositional Logic The Importance of Being Formal Martin Henz February 5, 2014 Propositional Logic 1 Motivation In traditional logic, terms represent sets, and therefore, propositions are limited to stating facts on sets

More information

Version January Please send comments and corrections to

Version January Please send comments and corrections to Mathematical Logic for Computer Science Second revised edition, Springer-Verlag London, 2001 Answers to Exercises Mordechai Ben-Ari Department of Science Teaching Weizmann Institute of Science Rehovot

More information

Math 13, Spring 2013, Lecture B: Midterm

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.

More information

Chapter 2: The Logic of Compound Statements

Chapter 2: The Logic of Compound Statements Chapter 2: he Logic of Compound Statements irst: Aristotle (Gr. 384-322 BC) Collection of rules for deductive reasoning to be used in every branch of knowledge Next: Gottfried Leibniz (German, 17th century)

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

LOGIC CONNECTIVES. Students who have an ACT score of at least 30 OR a GPA of at least 3.5 can receive a college scholarship.

LOGIC CONNECTIVES. Students who have an ACT score of at least 30 OR a GPA of at least 3.5 can receive a college scholarship. LOGIC In mathematical and everyday English language, we frequently use logic to express our thoughts verbally and in writing. We also use logic in numerous other areas such as computer coding, probability,

More information

Analysis 1. Lecture Notes 2013/2014. The original version of these Notes was written by. Vitali Liskevich

Analysis 1. Lecture Notes 2013/2014. The original version of these Notes was written by. Vitali Liskevich Analysis 1 Lecture Notes 2013/2014 The original version of these Notes was written by Vitali Liskevich followed by minor adjustments by many Successors, and presently taught by Misha Rudnev University

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. Logic and Proof

Chapter 1. Logic and Proof Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known

More information

Propositional Calculus: Formula Simplification, Essential Laws, Normal Forms

Propositional 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 information

CS Discrete Mathematics Dr. D. Manivannan (Mani)

CS Discrete Mathematics Dr. D. Manivannan (Mani) CS 275 - Discrete Mathematics Dr. D. Manivannan (Mani) Department of Computer Science University of Kentucky Lexington, KY 40506 Course Website: www.cs.uky.edu/~manivann/cs275 Notes based on Discrete Mathematics

More information

COMP219: Artificial Intelligence. Lecture 19: Logic for KR

COMP219: Artificial Intelligence. Lecture 19: Logic for KR COMP219: Artificial Intelligence Lecture 19: Logic for KR 1 Overview Last time Expert Systems and Ontologies Today Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof

More information

Math Real Analysis

Math Real Analysis 1 / 28 Math 370 - Real Analysis G.Pugh Sep 3 2013 Real Analysis 2 / 28 3 / 28 What is Real Analysis? Wikipedia: Real analysis... has its beginnings in the rigorous formulation of calculus. It is a branch

More information

Equivalents of Mingle and Positive Paradox

Equivalents of Mingle and Positive Paradox Eric Schechter Equivalents of Mingle and Positive Paradox Abstract. Relevant logic is a proper subset of classical logic. It does not include among itstheoremsanyof positive paradox A (B A) mingle A (A

More information

Combinational Logic Design Principles

Combinational Logic Design Principles Combinational Logic Design Principles Switching algebra Doru Todinca Department of Computers Politehnica University of Timisoara Outline Introduction Switching algebra Axioms of switching algebra Theorems

More information

Switches: basic element of physical implementations

Switches: basic element of physical implementations Combinational logic Switches Basic logic and truth tables Logic functions Boolean algebra Proofs by re-writing and by perfect induction Winter 200 CSE370 - II - Boolean Algebra Switches: basic element

More information

Logical forms and substitution instances. Philosophy and Logic Unit 2, Section 2.1

Logical forms and substitution instances. Philosophy and Logic Unit 2, Section 2.1 Logical forms and substitution instances Philosophy and Logic Unit 2, Section 2.1 Avoiding impossibility A valid deductive argument is an argument with a valid logical form. An argument has a valid logical

More information

2 Truth Tables, Equivalences and the Contrapositive

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

More information

Propositional Logic Logical Implication (4A) Young W. Lim 4/21/17

Propositional Logic Logical Implication (4A) Young W. Lim 4/21/17 Propositional Logic Logical Implication (4A) Young W. Lim Copyright (c) 2016-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation

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

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

Lecture : Set Theory and Logic

Lecture : Set Theory and Logic Lecture : Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014 Outline Contrapositive and Converse 1 Contrapositive and Converse 2 3 4 5 Contrapositive and Converse

More information

3.4 Set Operations Given a set A, the complement (in the Universal set U) A c is the set of all elements of U that are not in A. So A c = {x x / A}.

3.4 Set Operations Given a set A, the complement (in the Universal set U) A c is the set of all elements of U that are not in A. So A c = {x x / A}. 3.4 Set Operations Given a set, the complement (in the niversal set ) c is the set of all elements of that are not in. So c = {x x /. (This type of picture is called a Venn diagram.) Example 39 Let = {1,

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

More information

Discrete Mathematics

Discrete Mathematics Discrete Mathematics Chih-Wei Yi Dept. of Computer Science National Chiao Tung University March 9, 2009 Overview of ( 1.5-1.7, ~2 hours) Methods of mathematical argument (i.e., proof methods) can be formalized

More information

Unit i) 1)verify that [p->(q->r)] -> [(p->q)->(p->r)] is a tautology or not? 2) simplify the Boolean expression xy+(x+y) + y

Unit i) 1)verify that [p->(q->r)] -> [(p->q)->(p->r)] is a tautology or not? 2) simplify the Boolean expression xy+(x+y) + y Unit i) 1)verify that [p->(q->r)] -> [(p->q)->(p->r)] is a tautology or not? 2) simplify the Boolean expression xy+(x+y) + y 3)find a minimal sum of product representation f(w,x,y)= m(1,2,5,6) 4)simplify

More information

Intelligent Agents. First Order Logic. Ute Schmid. Cognitive Systems, Applied Computer Science, Bamberg University. last change: 19.

Intelligent Agents. First Order Logic. Ute Schmid. Cognitive Systems, Applied Computer Science, Bamberg University. last change: 19. Intelligent Agents First Order Logic Ute Schmid Cognitive Systems, Applied Computer Science, Bamberg University last change: 19. Mai 2015 U. Schmid (CogSys) Intelligent Agents last change: 19. Mai 2015

More information

Deductive Systems. Lecture - 3

Deductive Systems. Lecture - 3 Deductive Systems Lecture - 3 Axiomatic System Axiomatic System (AS) for PL AS is based on the set of only three axioms and one rule of deduction. It is minimal in structure but as powerful as the truth

More information

CS1800 Discrete Structures Fall 2016 Profs. Gold & Schnyder April 25, CS1800 Discrete Structures Final

CS1800 Discrete Structures Fall 2016 Profs. Gold & Schnyder April 25, CS1800 Discrete Structures Final CS1800 Discrete Structures Fall 2016 Profs. Gold & Schnyder April 25, 2017 CS1800 Discrete Structures Final Instructions: 1. The exam is closed book and closed notes. You may not use a calculator or any

More information

Mathematics 220 Midterm Practice problems from old exams Page 1 of 8

Mathematics 220 Midterm Practice problems from old exams Page 1 of 8 Mathematics 220 Midterm Practice problems from old exams Page 1 of 8 1. (a) Write the converse, contrapositive and negation of the following statement: For every integer n, if n is divisible by 3 then

More information

Math 55 Homework 2 solutions

Math 55 Homework 2 solutions Math 55 Homework solutions Section 1.3. 6. p q p q p q p q (p q) T T F F F T F T F F T T F T F T T F T F T F F T T T F T 8. a) Kwame will not take a job in industry and not go to graduate school. b) Yoshiko

More information

Analyzing Arguments with Truth Tables

Analyzing Arguments with Truth Tables Analyzing Arguments with Truth Tables MATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Fall 2014 Introduction Euler diagrams are useful for checking the validity of simple

More information

3 Propositional Logic

3 Propositional Logic 3 Propositional Logic 3.1 Syntax 3.2 Semantics 3.3 Equivalence and Normal Forms 3.4 Proof Procedures 3.5 Properties Propositional Logic (25th October 2007) 1 3.1 Syntax Definition 3.0 An alphabet Σ consists

More information

Basic Proof Examples

Basic Proof Examples Basic Proof Examples Lisa Oberbroeckling Loyola University Maryland Fall 2015 Note. In this document, we use the symbol as the negation symbol. Thus p means not p. There are four basic proof techniques

More information

ALGEBRA. Lecture notes for MA 630/631. Rudi Weikard

ALGEBRA. Lecture notes for MA 630/631. Rudi Weikard ALGEBRA Lecture notes for MA 630/631 Rudi Weikard updated version of December 2010 Contents Chapter 1. The Language of Mathematics 1 1.1. Propositional Calculus and Laws of Inference 1 1.2. Predicate

More information

Axiomatic systems. Revisiting the rules of inference. Example: A theorem and its proof in an abstract axiomatic system:

Axiomatic systems. Revisiting the rules of inference. Example: A theorem and its proof in an abstract axiomatic system: Axiomatic systems Revisiting the rules of inference Material for this section references College Geometry: A Discovery Approach, 2/e, David C. Kay, Addison Wesley, 2001. In particular, see section 2.1,

More information

CHAPTER 10. Gentzen Style Proof Systems for Classical Logic

CHAPTER 10. Gentzen Style Proof Systems for Classical Logic CHAPTER 10 Gentzen Style Proof Systems for Classical Logic Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. By humans, not mentioning

More information

Logic for Computer Scientists

Logic for Computer Scientists Logic for Computer Scientists Pascal Hitzler http://www.pascal-hitzler.de CS 499/699 Lecture, Winter Quarter 2011 Wright State University, Dayton, OH, U.S.A. [final version: 03/10/2011] Contents 1 Propositional

More information

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 Logic and Discrete Mathematics Section 3.5 Propositional logical equivalence Negation of propositional formulae Slides version: January 2015 Logical equivalence of propositional formulae Propositional

More information

Adequate set of connectives, logic gates and circuits

Adequate set of connectives, logic gates and circuits Adequate set of connectives, logic gates and circuits Lila Kari University of Waterloo Adequate set of connectives, logic gates and circuits CS245, Logic and Computation 1 / 59 Connectives We have mentioned

More information

Ex: Boolean expression for majority function F = A'BC + AB'C + ABC ' + ABC.

Ex: Boolean expression for majority function F = A'BC + AB'C + ABC ' + ABC. Boolean Expression Forms: Sum-of-products (SOP) Write an AND term for each input combination that produces a 1 output. Write the input variable if its value is 1; write its complement otherwise. OR the

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 #6: Rules of Inference Based on materials developed by Dr. Adam Lee Today s topics n Rules of inference

More information

CS 2336 Discrete Mathematics

CS 2336 Discrete Mathematics CS 2336 Discrete Mathematics Lecture 3 Logic: Rules of Inference 1 Outline Mathematical Argument Rules of Inference 2 Argument In mathematics, an argument is a sequence of propositions (called premises)

More information

CS206 Lecture 03. Propositional Logic Proofs. Plan for Lecture 03. Axioms. Normal Forms

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

More information

Overview. Knowledge-Based Agents. Introduction. COMP219: Artificial Intelligence. Lecture 19: Logic for KR

Overview. Knowledge-Based Agents. Introduction. COMP219: Artificial Intelligence. Lecture 19: Logic for KR COMP219: Artificial Intelligence Lecture 19: Logic for KR Last time Expert Systems and Ontologies oday Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof theory Natural

More information

Chapter 2. Mathematical Reasoning. 2.1 Mathematical Models

Chapter 2. Mathematical Reasoning. 2.1 Mathematical Models Contents Mathematical Reasoning 3.1 Mathematical Models........................... 3. Mathematical Proof............................ 4..1 Structure of Proofs........................ 4.. Direct Method..........................

More information

Argument. whenever all the assumptions are true, then the conclusion is true. If today is Wednesday, then yesterday is Tuesday. Today is Wednesday.

Argument. whenever all the assumptions are true, then the conclusion is true. If today is Wednesday, then yesterday is Tuesday. Today is Wednesday. Logic and Proof Argument An argument is a sequence of statements. All statements but the first one are called assumptions or hypothesis. The final statement is called the conclusion. An argument is valid

More information

SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR. Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE) UNIT I MATHEMATICAL LOGIC

SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR. Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE) UNIT I MATHEMATICAL LOGIC SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code :DM (15A05302) Course & Branch: B.Tech - CSE Year & Sem: II- B.Tech& I-Sem

More information