Definition Applications Examples

Slides:



Advertisements
Similar presentations
The Logic of Quantified Statements
Advertisements

22 March 2009Instructor: Tasneem Darwish1 University of Palestine Faculty of Applied Engineering and Urban Planning Software Engineering Department Introduction.
Copyright © Cengage Learning. All rights reserved. CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION.
Copyright © Cengage Learning. All rights reserved.
Basic Structures: Sets, Functions, Sequences, Sums, and Matrices
CLASSICAL LOGIC and FUZZY LOGIC. CLASSICAL LOGIC In classical logic, a simple proposition P is a linguistic, or declarative, statement contained within.
Instructor: Hayk Melikya
Basic Structures: Sets, Functions, Sequences, Sums, and Matrices
– Alfred North Whitehead,
Computability and Complexity 9-1 Computability and Complexity Andrei Bulatov Logic Reminder (Cnt’d)
Analytical Methods in CS (CIS 505)
Propositional Calculus Math Foundations of Computer Science.
Discrete Math 6A Max Welling. Recap 1. Proposition: statement that is true or false. 2. Logical operators: NOT, AND, OR, XOR, ,  3. Compound proposition:
First Order Logic. This Lecture Last time we talked about propositional logic, a logic on simple statements. This time we will talk about first order.
Predicates and Quantifiers
Predicates & Quantifiers Goal: Introduce predicate logic, including existential & universal quantification Introduce translation between English sentences.
Systems Architecture I1 Propositional Calculus Objective: To provide students with the concepts and techniques from propositional calculus so that they.
A Brief Summary for Exam 1 Subject Topics Propositional Logic (sections 1.1, 1.2) –Propositions Statement, Truth value, Proposition, Propositional symbol,
Math 3121 Abstract Algebra I Section 0: Sets. The axiomatic approach to Mathematics The notion of definition - from the text: "It is impossible to define.
DECIDABILITY OF PRESBURGER ARITHMETIC USING FINITE AUTOMATA Presented by : Shubha Jain Reference : Paper by Alexandre Boudet and Hubert Comon.
Atomic Sentences Chapter 1 Language, Proof and Logic.
1st-order Predicate Logic (FOL)
Michaelmas Term 2004 Discrete Mathematics CSC 141 Discrete Mathematics Dr. Corina Sas and Ms. Nelly Bencomo
Theory and Applications
Set, Combinatorics, Probability & Number Theory Mathematical Structures for Computer Science Chapter 3 Copyright © 2006 W.H. Freeman & Co.MSCS Slides Set,
Copyright © Cengage Learning. All rights reserved. CHAPTER 3 THE LOGIC OF QUANTIFIED STATEMENTS THE LOGIC OF QUANTIFIED STATEMENTS.
Copyright © 2014 Curt Hill Sets Introduction to Set Theory.
LDK R Logics for Data and Knowledge Representation PL of Classes.
Propositional Calculus CS 270: Mathematical Foundations of Computer Science Jeremy Johnson.
PHIL012 Class Notes 1/15/2001. Outline Announcements, web page Review Homework Problems (1-7) Set Theory Review & Problem 8 (if time) Assignment for Wednesday.
CS201: Data Structures and Discrete Mathematics I
Chapter 2: Basic Structures: Sets, Functions, Sequences, and Sums (1)
Logical Systems and Knowledge Representation Fuzzy Logical Systems 1.
Sets and Sentences Open Sentences Foundations of Real Analysis.
Key Concepts Representation Inference Semantics Discourse Pragmatics Computation.
CompSci 102 Discrete Math for Computer Science
Theory and Applications
Lecture 4: Predicates and Quantifiers; Sets.
Section 2.1. Section Summary Definition of sets Describing Sets Roster Method Set-Builder Notation Some Important Sets in Mathematics Empty Set and Universal.
Naïve Set Theory. Basic Definitions Naïve set theory is the non-axiomatic treatment of set theory. In the axiomatic treatment, which we will only allude.
1 Introduction to Abstract Mathematics Sets Section 2.1 Basic Notions of Sets Section 2.2 Operations with sets Section 2.3 Indexed Sets Instructor: Hayk.
4.1 Proofs and Counterexamples. Even Odd Numbers Find a property that describes each of the following sets E={…, -4, -2, 0, 2, 4, 6, …} O={…, -3, -1,
Chapter 2 With Question/Answer Animations. Section 2.1.
Computer Science CPSC 322 Lecture 22 Logical Consequences, Proof Procedures (Ch 5.2.2)
First Order Logic Lecture 3: Sep 13 (chapter 2 of the book)
1 Propositional Logic: Fundamental Elements for Computer Scientists 0. Motivation for Computer Scientists 1. Propositions and Propositional Variables 2.
Copyright © Peter Cappello 2011 Predicates & Quantifiers.
ARTIFICIAL INTELLIGENCE [INTELLIGENT AGENTS PARADIGM] Professor Janis Grundspenkis Riga Technical University Faculty of Computer Science and Information.
Tautology. In logic, a tautology (from the Greek word ταυτολογία) is a formula that is true in every possible interpretation.logic Greek formulainterpretation.
Section 1.5 and 1.6 Predicates and Quantifiers. Vocabulary Predicate Domain Universal Quantifier Existential Quantifier Counterexample Free variable Bound.
Lecture 041 Predicate Calculus Learning outcomes Students are able to: 1. Evaluate predicate 2. Translate predicate into human language and vice versa.
Section 1.4. Propositional Functions Propositional functions become propositions (and have truth values) when their variables are each replaced by a value.
5 Lecture in math Predicates Induction Combinatorics.
1 Section 7.1 First-Order Predicate Calculus Predicate calculus studies the internal structure of sentences where subjects are applied to predicates existentially.
Chapter 2 1. Chapter Summary Sets (This Slide) The Language of Sets - Sec 2.1 – Lecture 8 Set Operations and Set Identities - Sec 2.2 – Lecture 9 Functions.
BOOLEAN INFORMATION RETRIEVAL 1Adrienn Skrop. Boolean Information Retrieval  The Boolean model of IR (BIR) is a classical IR model and, at the same time,
رياضيات متقطعة لعلوم الحاسب MATH 226. Chapter 1 Predicates and Quantifiers 1.4.
CENG 424-Logic for CS Introduction Based on the Lecture Notes of Konstantin Korovin, Valentin Goranko, Russel and Norvig, and Michael Genesereth.
Propositional Calculus: Boolean Functions and Expressions
Set, Combinatorics, Probability & Number Theory
Propositional Calculus: Boolean Functions and Expressions
Set Theory A B C.
Propositional Calculus: Boolean Algebra and Simplification
Set Operations Section 2.2.
Copyright © Cengage Learning. All rights reserved.
Logics for Data and Knowledge Representation
CHAPTER 1: LOGICS AND PROOF
A Brief Summary for Exam 1
Discrete Mathematics CMP-200 Propositional Equivalences, Predicates & Quantifiers, Negating Quantified Statements Abdul Hameed
Copyright © Cengage Learning. All rights reserved.
Presentation transcript:

Definition Applications Examples Predicate Definition Applications Examples

Predicate A predicate is commonly understood to be a Boolean-valued function P: X→ {true, false}, called the predicate on X. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory. So, for example, when a theory defines the concept of a relation, then a predicate is simply the characteristic function or the indicator function of a relation. However, not all theories have relations, or are founded on set theory, and so one must be careful with the proper definition and semantic interpretation of a predicate.

Predicate Informally, a predicate is a statement that may be true or false depending on the values of its variables. It can be thought of as an operator or function that returns a value that is either true or false. For example, predicates are sometimes used to indicate set membership: when talking about sets, it is sometimes inconvenient or impossible to describe a set by listing all of its elements. Thus, a predicate P(x) will be true or false, depending on whether x belongs to a set. Predicates are also commonly used to talk about the properties of objects, by defining the set of all objects that have some property in common. So, for example, when P is a predicate on X, one might sometimes say P is a property of X. Similarly, the notation P(x) is used to denote a sentence or statement P concerning the variable object x. The set defined by P(x) is written as {x | P(x)}, and is just a collection of all the objects for which P is true. For instance, {x | x is a positive integer less than 4} is the set {1,2,3}. If t is an element of the set {x | P(x)}, then the statement P(t) is true. Here, P(x) is referred to as the predicate, and x the subject of the proposition. Sometimes, P(x) is also called a propositional function, as each choice of x produces a proposition.

Formal definition The precise semantic interpretation of an atomic formula and an atomic sentence will vary from theory to theory. In propositional logic, atomic formulas are called propositional variables. In a sense, these are nullary (i.e. 0-arity) predicates. In first-order logic, an atomic formula consists of a predicate symbol applied to an appropriate number of terms. In set theory, predicates are understood to be characteristic functions or set indicator functions, i.e. functions from a set element to a truth value. Set-builder notation makes use of predicates to define sets. In autoepistemic logic, which rejects the law of excluded middle, predicates may be true, false, or simply unknown; i.e. a given collection of facts may be insufficient to determine the truth or falsehood of a predicate. In fuzzy logic, predicates are the characteristic functions of a probability distribution. That is, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.

Predicate

Predicate

Predicate

References and bibliography Cunningham, Daniel W. (2012). A Logical Introduction to Proof. New York: Springer. p. 29. ISBN 9781461436317. Haas, Guy M. "What If? (Predicates)". Introduction to Computer Programming. Berkeley Foundation for Opportunities in IT (BFOIT),. Retrieved 20 July 2013. Lavrov, Igor Andreevich and Larisa Maksimova (2003). Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms. New York: Springer. p. 52. ISBN 0306477122.