CSE 1400 Applied Discrete Mathematics Definitions
|
|
- Kelly Hill
- 5 years ago
- Views:
Transcription
1 CSE 1400 Applied Discrete Mathematics Definitions Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Arithmetic 1 Alphabets, Strings, Languages, & Words 2 Number Systems 3 Machine Numbers 3 Boolean Logic 4 Relations 5 Functions 6 Predicate (First-Order) Logic 6 Abstract Arithmetic Definition 1 (Zero). 0 is a natural number. Definition 2 (Successor Function). If n is a natural number, then σ(n) = n + 1 is a natural number called the successor of n. 0 is called a constant. 0 can be thought of as a function z(x) that returns the value 0 for all input values x. Definition 3 (Addition). If n and m are natural numbers, then n + m is a natural number called the sum of n and m. Definition 4 (Multiplication). If n and m are natural numbers, then nm is a natural number called the product of n and m. Definition 5 (Exponentiation). If n and m are natural numbers, then n m is a natural number called n raised to the exponent m. Definition 6 (Positive Exponents). n 0 = 1 n m = n n m 1
2 definitions 2 Definition 7 (Negation of Exponents). n m = 1 n m Definition 8 (Fractional Exponents). n a/b = b n a Definition 9 (Multipling Powers Add Exponents). n a n b = n a+b Definition 10 (Raising Powers to Powers Multiply Exponents). Definition 11 (Logarithms). (n a ) b = n ab log n a = m where n m = a Alphabets, Strings, Languages, & Words Definition 12 (Alphabet). An alphabet Σ is a finite set of characters Example 1 (Example Alphabets). The binary alphabet B = {0, 1} = {False, True} The decimal alphabet D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} The hexadecimal alphabet H = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F} The integers mod n Z n = {0, 1, 2,..., (n 1)} The quantifier alphabet (for all and there exists) Q = {, } The delimiter alphabet L = {(, ), [, ], {, }} The dna alphabet DNA = {A, C, G, T}
3 definitions 3 The English alphabet A = {a, b, c,..., x, y, z} The Greek alphabet G = {α, β, γ, δ, ɛ, ζ, η, θ, ι, κ, λ, µ, ν, ξ, o, π, ρ, σ, τ, υ, φ, χ, ψ, ω} The Unicode character set U = {c : 0 c (10FFFF) 16 } Definition 13 (String). A string is a finite list of characters from an alphabet Σ. The string with no characters is the empty string λ. Definition 14 (Kleene Closure). The Kleene closure Σ is the set of all strings over an alphabet Σ. Definition 15 (Language). A language L is a subset of Σ. Definition 16 (Regular Language). The empty set is a regular language. The set {λ} containing only the empty string is a regular language. For every character a Σ, the set {a} is a regular language. If L 0 and L 1 are regular languages, then the following are regular languages. The union L 0 L 1 = {s : s L 0 L 1 } The concatenation L 0 L 1 = {st : s L 0 and t L 1 } Number Systems Definition 17 (Natural Numbers). The natural numbers are the values in the set N = {0, 1, 2, 3, 4, 5,...} Definition 18 (Integers). The integers are the values in the set Z = {0, ±1, ±2, ±3, ±4, ±5,...} Definition 19 (Rationals). The rationals are the values in the set { a } Q = b : a, b Z, b = 0 The natural numbers values shown are written in decimal notation. The integer values shown are written in signed-magnitude decimal notation.
4 definitions 4 Machine Numbers Definition 20 (Computer Word). A fixed-length string of bits, often written in hexadecimal notation. Example 2. 8-Bit Word (00) 16 to (FF) 16 Common computer architectures have used word lengths that are multiples of 8. A string of 8 bits is called a byte. 16-Bit Word (0000) 16 to (FFFF) Bit Word ( ) 16 to (FFFF FFFF) Bit Word Definition 21 (The Machine Naturals). The machine naturals are the values in the set N mach = {0, 1, 2, 3,..., 2 n 1} where n is the computer s word length. Definition 22 (The Machine Integers). The machine integers are the values in the set { } Z n = 2 n 1, 2 n 1 + 1,..., 1, 0, 1,..., 2 n 1 1 where n is the computer s word length. Definition 23 (The Machine Rationals (Floating Point)). The (normalized) floating point numbers are the values in the set { } Q n = ( 1) s. f 2 e b : se f = 0 is a computer word {0} The meaning of a computer word depends on context or state, often called type. n bits can name 2 n things: 0 through 2 n 1. When n is a multiple of 4, n/4 hexadecimal digits can name 16 n/4 things. Common computer architectures represent integers in two s complement notation. If m is an integer written in two s complement notation, then { (m)2 if the left-most bit of m is 0. m = m 2 n if the left-most bit of m is 1. where n is the computer s word length. Common computer architectures represent the integer exponent e in biased notation. If m is an integer written in biased notation, then where b > 0 is bias. m = (m) 2 b where s is a single bit, called the sign. e is string of bits representing an integer written in biased notation. f is string of bits representing the fractional part of the number. Boolean Logic Definition 24 (Proposition). A proposition is a statement that is either always True or always False. Definition 25 (Boolean Variable). A Boolean variable is a character, for example p, that stands for a proposition.
5 definitions 5 Definition 26 (Negation). p is the negation of Boolean variable p and defined by the formula False if p = True p = True if p = False Definition 27 (Negation). p is the negation of Boolean variable p and defined by the formula False if p = True p = True if p = False Definition 28 (Boolean Expressions). True and False are Boolean expressions A Boolean variable is a Boolean expressions expressions formed by applying the above operations are Boolean expressions. Definition 29 (Truth Assignment). Let Π = {p 0, p 1,..., p n 1 } be a set of Boolean variables. A truth assignment for Π is a Definition 30 (Tautology). A tautology is a Boolean expression that is always True. Definition 31 (Contradiction). A contradiction is a Boolean expression that is always False. Definition 32 (Contingency). A contingency is a Boolean expression that is sometimes True and sometimes False. And (Conjunction) p q is true only when both p and q are true. Or (Disjunction) p q is false only when both p and q are false. Conditional p q is true when (1) p is false and when (2) both p and q are true. Equivalence p q is true when both p and q are true or when both p and q are false. Relations Definition 33 (Relation). A relation r from X to Y is a subset of ordered pairs from the Cartesian product X Y. Definition 34 (Reflexive Relation). A relation r from X to X is reflexive if (x, x) r for all x X.
6 definitions 6 Definition 35 (Symmetric Relation). A relation r from X to X is symmetric if (x, y) r implies (y x) r. Definition 36 (Transitive Relation). A relation r from X to X is transitive if (x, y) r and (y, z) r implies (x, z) r. Definition 37 (Antisymmetric Relation). A relation r from X to X is antisymmetric if (x, y) r and (y, x) r implies x = y. Definition 38 (Equivalence Relation). An relation is an equivalence if it is reflexive, symmetric, and transitive Definition 39 (Partition). The subset S 0, S 0,..., S n 1 partition a set U if U = n 1 k=0 S i S k = S k for i = j Definition 40 (Partial Orders). A relation is a partial order if it is reflexive, antisymmetric, and transitive. Functions Definition 41 (Function). A function f from X to Y is a relation such that (x, y 1 ) f and (x, y 2 ) f implies y 1 = y 2. Definition 42 (Onto Function). A function f from X to Y is onto if for every y Y there is an x X such that (x, y 1 ) f. Definition 43 (One-to-One Function). A function f from X to Y is one-to-one if (x 1, y) f and (x 2, y) f implies x 1 = x 2. Predicate (First-Order) Logic First-order logic is a syntax capable of expressing detailed mathematical statements, semantics that identify a sentence with its intended mathematical application, and a generic proof system that is surprisingly comprehensive. Christos H. Papadimitriou, Computational Complexity Definition 44 (Predicates). A predicate is a statement that is true or false depending of the value or one or more (domain) variables. Definition 45 (Universal Quantification). A predicate can be prefixed with a universal quantifier meaning the predicate is true (or false) for all values of the domain variable(s).
7 definitions 7 Definition 46 (Existential Quantification). A predicate can be prefixed with an existential quantifier meaning the predicate is true (or false) for at least one value of the domain variable(s). Definition 47 (Vocabulary). A vocabulary V is a set of functions F, a set of relations R, and variable names {x, y, z, w,...} to which the functions and relations are applied. Definition 48 (Terms over a Vocabulary). 1. Any variable is a term. 2. If f is a function and t is a term, then f (t) is a term. Definition 49 (First-Order Expressions). 1. If is a relation and t and s are terms, then t s is an (atomic first-order) expression. 2. If p and q are (first-order) expressions, then the following are (first-order) expressions. (a) p, pronouced not p. (b) p q, pronouced p and q. (c) p q, pronouced p or q. (d) ( x)(p), pronouced for all x, p. Definition 50 (Models). A mo (graph theory, set theory, number theory) The negative of for all x, p is ( x)(p) = ( x)( p), pronouced there exists an x such that not p. Definition 51 (Valid Statements). Valid statements: statements that are True in every model. Definition 52 (True Statements). True statements: statements that True are of a given model. Definition 53 (Proof). Definition 54 (Provable Statements). Proveable statements: statements that can be proven True of a given model. Definition 55 (Complete). A model is complete if for every expression p there is a proof of p or there is a proof of p. Definition 56 (Inconsistent). A model is inconsistent if it there proof of every expression p. Definition 57 (Consistent). A model is consistent if it there is no there expression p such that both p and p have a proof in the theory.
CSE 1400 Applied Discrete Mathematics Proofs
CSE 1400 Applied Discrete Mathematics Proofs Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Axioms 1 Logical Axioms 2 Models 2 Number Theory 3 Graph Theory 4 Set Theory 4
More informationDepartment 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 informationInformal Statement Calculus
FOUNDATIONS OF MATHEMATICS Branches of Logic 1. Theory of Computations (i.e. Recursion Theory). 2. Proof Theory. 3. Model Theory. 4. Set Theory. Informal Statement Calculus STATEMENTS AND CONNECTIVES Example
More informationFirst-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms
First-Order Logic 1 Syntax Domain of Discourse The domain of discourse for first order logic is FO structures or models. A FO structure contains Relations Functions Constants (functions of arity 0) FO
More informationNotation Index. gcd(a, b) (greatest common divisor) NT-16
Notation Index (for all) B A (all functions) B A = B A (all functions) SF-18 (n) k (falling factorial) SF-9 a R b (binary relation) C(n,k) = n! k! (n k)! (binomial coefficient) SF-9 n! (n factorial) SF-9
More informationAutomata Theory and Formal Grammars: Lecture 1
Automata Theory and Formal Grammars: Lecture 1 Sets, Languages, Logic Automata Theory and Formal Grammars: Lecture 1 p.1/72 Sets, Languages, Logic Today Course Overview Administrivia Sets Theory (Review?)
More informationPropositional 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 informationRecall that the expression x > 3 is not a proposition. Why?
Predicates and Quantifiers Predicates and Quantifiers 1 Recall that the expression x > 3 is not a proposition. Why? Notation: We will use the propositional function notation to denote the expression "
More informationProblem 1: Suppose A, B, C and D are finite sets such that A B = C D and C = D. Prove or disprove: A = B.
Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination III (Spring 2007) Problem 1: Suppose A, B, C and D are finite sets
More informationPropositional Logic: Models and Proofs
Propositional Logic: Models and Proofs C. R. Ramakrishnan CSE 505 1 Syntax 2 Model Theory 3 Proof Theory and Resolution Compiled at 11:51 on 2016/11/02 Computing with Logic Propositional Logic CSE 505
More informationMathematical Preliminaries. Sipser pages 1-28
Mathematical Preliminaries Sipser pages 1-28 Mathematical Preliminaries This course is about the fundamental capabilities and limitations of computers. It has 3 parts 1. Automata Models of computation
More informationAutomata Theory for Presburger Arithmetic Logic
Automata Theory for Presburger Arithmetic Logic References from Introduction to Automata Theory, Languages & Computation and Constraints in Computational Logic Theory & Application Presented by Masood
More informationD I S C R E T E M AT H E M AT I C S H O M E W O R K
D E PA R T M E N T O F C O M P U T E R S C I E N C E S C O L L E G E O F E N G I N E E R I N G F L O R I D A T E C H D I S C R E T E M AT H E M AT I C S H O M E W O R K W I L L I A M S H O A F F S P R
More information3. Only sequences that were formed by using finitely many applications of rules 1 and 2, are propositional formulas.
1 Chapter 1 Propositional Logic Mathematical logic studies correct thinking, correct deductions of statements from other statements. Let us make it more precise. A fundamental property of a statement is
More informationDiscrete Mathematics. W. Ethan Duckworth. Fall 2017, Loyola University Maryland
Discrete Mathematics W. Ethan Duckworth Fall 2017, Loyola University Maryland Contents 1 Introduction 4 1.1 Statements......................................... 4 1.2 Constructing Direct Proofs................................
More informationSample 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 informationECE473 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
More informationCSE 20. Final Review. CSE 20: Final Review
CSE 20 Final Review Final Review Representation of integers in base b Logic Proof systems: Direct Proof Proof by contradiction Contraposetive Sets Theory Functions Induction Final Review Representation
More informationECE260: Fundamentals of Computer Engineering
Data Representation & 2 s Complement James Moscola Dept. of Engineering & Computer Science York College of Pennsylvania Based on Computer Organization and Design, 5th Edition by Patterson & Hennessy Data
More informationCS 455/555: Mathematical preliminaries
CS 455/555: Mathematical preliminaries Stefan D. Bruda Winter 2019 SETS AND RELATIONS Sets: Operations: intersection, union, difference, Cartesian product Big, powerset (2 A ) Partition (π 2 A, π, i j
More informationFoundations of Mathematics MATH 220 FALL 2017 Lecture Notes
Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements
More informationCSE 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 informationConjunction: 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
More informationTheory of Languages and Automata
Theory of Languages and Automata Chapter 0 - Introduction Sharif University of Technology References Main Reference M. Sipser, Introduction to the Theory of Computation, 3 nd Ed., Cengage Learning, 2013.
More informationSyntax. Notation Throughout, and when not otherwise said, we assume a vocabulary V = C F P.
First-Order Logic Syntax The alphabet of a first-order language is organised into the following categories. Logical connectives:,,,,, and. Auxiliary symbols:.,,, ( and ). Variables: we assume a countable
More informationPropositional and Predicate Logic - II
Propositional and Predicate Logic - II Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - II WS 2016/2017 1 / 16 Basic syntax Language Propositional logic
More informationComp487/587 - 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
More informationInformatics 1 - Computation & Logic: Tutorial 3
Informatics 1 - Computation & Logic: Tutorial 3 Counting Week 5: 16-20 October 2016 Please attempt the entire worksheet in advance of the tutorial, and bring all work with you. Tutorials cannot function
More informationTecniche di Verifica. Introduction to Propositional Logic
Tecniche di Verifica Introduction to Propositional Logic 1 Logic A formal logic is defined by its syntax and semantics. Syntax An alphabet is a set of symbols. A finite sequence of these symbols is called
More informationPropositional logic (revision) & semantic entailment. p. 1/34
Propositional logic (revision) & semantic entailment p. 1/34 Reading The background reading for propositional logic is Chapter 1 of Huth/Ryan. (This will cover approximately the first three lectures.)
More informationTheory of Computation
Theory of Computation Dr. Sarmad Abbasi Dr. Sarmad Abbasi () Theory of Computation / Lecture 3: Overview Decidability of Logical Theories Presburger arithmetic Decidability of Presburger Arithmetic Dr.
More informationMat 243 Exam 1 Review
OBJECTIVES (Review problems: on next page) 1.1 Distinguish between propositions and non-propositions. Know the truth tables (i.e., the definitions) of the logical operators,,,, and Write truth tables for
More informationDiscrete Mathematical Structures. Chapter 1 The Foundation: Logic
Discrete Mathematical Structures Chapter 1 he oundation: Logic 1 Lecture Overview 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Quantifiers l l l l l Statement Logical Connectives Conjunction
More informationFoundations of Mathematics
Foundations of Mathematics L. Pedro Poitevin 1. Preliminaries 1.1. Sets We will naively think of a set as a collection of mathematical objects, called its elements or members. To indicate that an object
More informationLecture 7. Logic. Section1: Statement Logic.
Ling 726: Mathematical Linguistics, Logic, Section : Statement Logic V. Borschev and B. Partee, October 5, 26 p. Lecture 7. Logic. Section: Statement Logic.. Statement Logic..... Goals..... Syntax of Statement
More informationContents Propositional Logic: Proofs from Axioms and Inference Rules
Contents 1 Propositional Logic: Proofs from Axioms and Inference Rules... 1 1.1 Introduction... 1 1.1.1 An Example Demonstrating the Use of Logic in Real Life... 2 1.2 The Pure Propositional Calculus...
More informationA statement is a sentence that is definitely either true or false but not both.
5 Logic In this part of the course we consider logic. Logic is used in many places in computer science including digital circuit design, relational databases, automata theory and computability, and artificial
More informationPredicate Calculus. Formal Methods in Verification of Computer Systems Jeremy Johnson
Predicate Calculus Formal Methods in Verification of Computer Systems Jeremy Johnson Outline 1. Motivation 1. Variables, quantifiers and predicates 2. Syntax 1. Terms and formulas 2. Quantifiers, scope
More informationCSE 20. Lecture 4: Introduction to Boolean algebra. CSE 20: Lecture4
CSE 20 Lecture 4: Introduction to Boolean algebra Reminder First quiz will be on Friday (17th January) in class. It is a paper quiz. Syllabus is all that has been done till Wednesday. If you want you may
More informationCS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati
CS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati Important 1. No questions about the paper will be entertained during
More informationPropositional and Predicate Logic - VII
Propositional and Predicate Logic - VII Petr Gregor KTIML MFF UK WS 2015/2016 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - VII WS 2015/2016 1 / 11 Theory Validity in a theory A theory
More informationTopics in Logic and Proofs
Chapter 2 Topics in Logic and Proofs Some mathematical statements carry a logical value of being true or false, while some do not. For example, the statement 4 + 5 = 9 is true, whereas the statement 2
More informationAI 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 informationNumbering Systems. Contents: Binary & Decimal. Converting From: B D, D B. Arithmetic operation on Binary.
Numbering Systems Contents: Binary & Decimal. Converting From: B D, D B. Arithmetic operation on Binary. Addition & Subtraction using Octal & Hexadecimal 2 s Complement, Subtraction Using 2 s Complement.
More information1.1 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,
More informationGödel s Incompleteness Theorems
Seminar Report Gödel s Incompleteness Theorems Ahmet Aspir Mark Nardi 28.02.2018 Supervisor: Dr. Georg Moser Abstract Gödel s incompleteness theorems are very fundamental for mathematics and computational
More informationPacket #2: Set Theory & Predicate Calculus. Applied Discrete Mathematics
CSC 224/226 Notes Packet #2: Set Theory & Predicate Calculus Barnes Packet #2: Set Theory & Predicate Calculus Applied Discrete Mathematics Table of Contents Full Adder Information Page 1 Predicate Calculus
More informationREVIEW QUESTIONS. Chapter 1: Foundations: Sets, Logic, and Algorithms
REVIEW QUESTIONS Chapter 1: Foundations: Sets, Logic, and Algorithms 1. Why can t a Venn diagram be used to prove a statement about sets? 2. Suppose S is a set with n elements. Explain why the power set
More informationOn the satisfiability problem for a 4-level quantified syllogistic and some applications to modal logic
On the satisfiability problem for a 4-level quantified syllogistic and some applications to modal logic Domenico Cantone and Marianna Nicolosi Asmundo Dipartimento di Matematica e Informatica Università
More informationPreliminaries to the Theory of Computation
Preliminaries to the Theory of Computation 2 In this chapter, we explain mathematical notions, terminologies, and certain methods used in convincing logical arguments that we shall have need of throughout
More informationHandbook of Logic and Proof Techniques for Computer Science
Steven G. Krantz Handbook of Logic and Proof Techniques for Computer Science With 16 Figures BIRKHAUSER SPRINGER BOSTON * NEW YORK Preface xvii 1 Notation and First-Order Logic 1 1.1 The Use of Connectives
More informationPropositional Logic and Semantics
Propositional Logic and Semantics English is naturally ambiguous. For example, consider the following employee (non)recommendations and their ambiguity in the English language: I can assure you that no
More informationIntroduction to Sets and Logic (MATH 1190)
Introduction to Sets Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics Statistics York University Sept 18, 2014 Outline 1 2 Tautologies Definition A tautology is a compound proposition
More informationLogic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.
Math 0413 Appendix A.0 Logic Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another. This type of logic is called propositional.
More informationIntroduction to Metalogic
Introduction to Metalogic Hans Halvorson September 21, 2016 Logical grammar Definition. A propositional signature Σ is a collection of items, which we call propositional constants. Sometimes these propositional
More informationLecture 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 informationINTRODUCTION 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 informationLecture Notes on DISCRETE MATHEMATICS. Eusebius Doedel
Lecture Notes on DISCRETE MATHEMATICS Eusebius Doedel c Eusebius J. Doedel, 009 Contents Logic. Introduction............................................................................... Basic logical
More information1 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
More informationIntroduction to Metalogic
Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)
More informationn Empty Set:, or { }, subset of all sets n Cardinality: V = {a, e, i, o, u}, so V = 5 n Subset: A B, all elements in A are in B
Discrete Math Review Discrete Math Review (Rosen, Chapter 1.1 1.7, 5.5) TOPICS Sets and Functions Propositional and Predicate Logic Logical Operators and Truth Tables Logical Equivalences and Inference
More informationGÖDEL S COMPLETENESS AND INCOMPLETENESS THEOREMS. Contents 1. Introduction Gödel s Completeness Theorem
GÖDEL S COMPLETENESS AND INCOMPLETENESS THEOREMS BEN CHAIKEN Abstract. This paper will discuss the completeness and incompleteness theorems of Kurt Gödel. These theorems have a profound impact on the philosophical
More informationCS 2800: Logic and Computation Fall 2010 (Lecture 13)
CS 2800: Logic and Computation Fall 2010 (Lecture 13) 13 October 2010 1 An Introduction to First-order Logic In Propositional(Boolean) Logic, we used large portions of mathematical language, namely those
More information586 Index. vertex, 369 disjoint, 236 pairwise, 272, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws
Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 465 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,
More informationIntroduction to Decision Sciences Lecture 2
Introduction to Decision Sciences Lecture 2 Andrew Nobel August 24, 2017 Compound Proposition A compound proposition is a combination of propositions using the basic operations. For example (p q) ( p)
More informationPredicate Logic: Sematics Part 1
Predicate Logic: Sematics Part 1 CS402, Spring 2018 Shin Yoo Predicate Calculus Propositional logic is also called sentential logic, i.e. a logical system that deals with whole sentences connected with
More informationCS256/Spring 2008 Lecture #11 Zohar Manna. Beyond Temporal Logics
CS256/Spring 2008 Lecture #11 Zohar Manna Beyond Temporal Logics Temporal logic expresses properties of infinite sequences of states, but there are interesting properties that cannot be expressed, e.g.,
More informationTruth-Functional Logic
Truth-Functional Logic Syntax Every atomic sentence (A, B, C, ) is a sentence and are sentences With ϕ a sentence, the negation ϕ is a sentence With ϕ and ψ sentences, the conjunction ϕ ψ is a sentence
More informationHANDOUT 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 information1. Write a program to calculate distance traveled by light
G. H. R a i s o n i C o l l e g e O f E n g i n e e r i n g D i g d o h H i l l s, H i n g n a R o a d, N a g p u r D e p a r t m e n t O f C o m p u t e r S c i e n c e & E n g g P r a c t i c a l M a
More informationChapter 0 Introduction. Fourth Academic Year/ Elective Course Electrical Engineering Department College of Engineering University of Salahaddin
Chapter 0 Introduction Fourth Academic Year/ Elective Course Electrical Engineering Department College of Engineering University of Salahaddin October 2014 Automata Theory 2 of 22 Automata theory deals
More informationA generalization of modal definability
A generalization of modal definability Tin Perkov Polytechnic of Zagreb Abstract. Known results on global definability in basic modal logic are generalized in the following sense. A class of Kripke models
More informationA Little Logic. Propositional Logic. Satisfiability Problems. Solving Sudokus. First Order Logic. Logic Programming
A Little Logic International Center for Computational Logic Technische Universität Dresden Germany Propositional Logic Satisfiability Problems Solving Sudokus First Order Logic Logic Programming A Little
More informationCSE 105 Homework 1 Due: Monday October 9, Instructions. should be on each page of the submission.
CSE 5 Homework Due: Monday October 9, 7 Instructions Upload a single file to Gradescope for each group. should be on each page of the submission. All group members names and PIDs Your assignments in this
More informationPredicate Calculus - Syntax
Predicate Calculus - Syntax Lila Kari University of Waterloo Predicate Calculus - Syntax CS245, Logic and Computation 1 / 26 The language L pred of Predicate Calculus - Syntax L pred, the formal language
More informationDiscrete Mathematics Review
CS 1813 Discrete Mathematics Discrete Mathematics Review or Yes, the Final Will Be Comprehensive 1 Truth Tables for Logical Operators P Q P Q False False False P Q False P Q False P Q True P Q True P True
More informationIntroduction to Model Theory
Introduction to Model Theory Charles Steinhorn, Vassar College Katrin Tent, University of Münster CIRM, January 8, 2018 The three lectures Introduction to basic model theory Focus on Definability More
More information5. 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 informationLogic 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 informationDefinition 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 informationPRELIMINARIES FOR GENERAL TOPOLOGY. Contents
PRELIMINARIES FOR GENERAL TOPOLOGY DAVID G.L. WANG Contents 1. Sets 2 2. Operations on sets 3 3. Maps 5 4. Countability of sets 7 5. Others a mathematician knows 8 6. Remarks 9 Date: April 26, 2018. 2
More informationSyntax and Semantics of Propositional Linear Temporal Logic
Syntax and Semantics of Propositional Linear Temporal Logic 1 Defining Logics L, M, = L - the language of the logic M - a class of models = - satisfaction relation M M, ϕ L: M = ϕ is read as M satisfies
More informationAdvanced 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 informationMaanavaN.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 informationNotes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.
Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3
More informationCSC Discrete Math I, Spring Propositional Logic
CSC 125 - Discrete Math I, Spring 2017 Propositional Logic Propositions A proposition is a declarative sentence that is either true or false Propositional Variables A propositional variable (p, q, r, s,...)
More informationDiscrete Mathematics. Benny George K. September 22, 2011
Discrete Mathematics Benny George K Department of Computer Science and Engineering Indian Institute of Technology Guwahati ben@iitg.ernet.in September 22, 2011 Set Theory Elementary Concepts Let A and
More informationCOMP1002 exam study sheet
COMP1002 exam study sheet Propositional statement: expression that has a truth value (true/false). It is a tautology if it is always true, contradiction if always false. Logic connectives: negation ( not
More informationLecturecise 22 Weak monadic second-order theory of one successor (WS1S)
Lecturecise 22 Weak monadic second-order theory of one successor (WS1S) 2013 Reachability in the Heap Many programs manipulate linked data structures (lists, trees). To express many important properties
More informationBounded-width QBF is PSPACE-complete
Bounded-width QBF is PSPACE-complete Albert Atserias Universitat Politècnica de Catalunya Barcelona, Spain atserias@lsi.upc.edu Sergi Oliva Universitat Politècnica de Catalunya Barcelona, Spain oliva@lsi.upc.edu
More informationChapter 1 : The language of mathematics.
MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :
More informationMATH 363: Discrete Mathematics
MATH 363: Discrete Mathematics Learning Objectives by topic The levels of learning for this class are classified as follows. 1. Basic Knowledge: To recall and memorize - Assess by direct questions. The
More informationn logical not (negation) n logical or (disjunction) n logical and (conjunction) n logical exclusive or n logical implication (conditional)
Discrete Math Review Discrete Math Review (Rosen, Chapter 1.1 1.6) TOPICS Propositional Logic Logical Operators Truth Tables Implication Logical Equivalence Inference Rules What you should know about propositional
More informationcse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska
cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 1 Course Web Page www3.cs.stonybrook.edu/ cse303 The webpage contains: lectures notes slides; very detailed solutions to
More informationMath 10850, fall 2017, University of Notre Dame
Math 10850, fall 2017, University of Notre Dame Notes on first exam September 22, 2017 The key facts The first midterm will be on Thursday, September 28, 6.15pm-7.45pm in Hayes-Healy 127. What you need
More informationLogic and Proofs. (A brief summary)
Logic and Proofs (A brief summary) Why Study Logic: To learn to prove claims/statements rigorously To be able to judge better the soundness and consistency of (others ) arguments To gain the foundations
More informationCourse Runtime Verification
Course Martin Leucker (ISP) Volker Stolz (Høgskolen i Bergen, NO) INF5140 / V17 Chapters of the Course Chapter 1 Recall in More Depth Chapter 2 Specification Languages on Words Chapter 3 LTL on Finite
More informationMathematical Logic. An Introduction
Mathematical Logic. An Introduction Summer 2006 by Peter Koepke Table of contents Table of contents............................................... 1 1 Introduction.................................................
More informationThe non-logical symbols determine a specific F OL language and consists of the following sets. Σ = {Σ n } n<ω
1 Preliminaries In this chapter we first give a summary of the basic notations, terminology and results which will be used in this thesis. The treatment here is reduced to a list of definitions. For the
More informationDefinition: Let S and T be sets. A binary relation on SxT is any subset of SxT. A binary relation on S is any subset of SxS.
4 Functions Before studying functions we will first quickly define a more general idea, namely the notion of a relation. A function turns out to be a special type of relation. Definition: Let S and T be
More information