The document discusses propositional functions and quantifiers in predicate logic. It defines propositional functions as functions that take arguments and evaluate to true or false, expressing a predicate. Quantifiers like the universal quantifier ∀ and existential quantifier ∃ are used to make predicates into propositions by specifying whether the predicate is true for all or some values in the universe of discourse. Examples demonstrate writing English statements in logical notation using propositional functions and quantifiers.
This document provides an introduction to predicate logic and quantifiers. It begins with terminology like propositional functions, arguments, and universe of discourse. It then defines and provides examples of quantifiers like universal and existential quantifiers. It discusses how to mix quantifiers and their truth values. It also covers binding variables, scope, and negation of quantified statements. Finally, it provides a brief introduction to Prolog, a logic programming language based on predicate logic.
This document provides an overview of predicate logic, including:
- The basic components of predicate logic like variables, predicates, quantifiers, and propositional functions
- Explanations of the universal and existential quantifiers
- How to negate quantified expressions using De Morgan's laws
- Examples of translating statements between English and predicate logic
This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can express properties and relationships for objects. Laws of quantifier equivalence are also presented.
This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can represent concepts like "all parking spaces are full" or "some parking space is full." Laws of quantifier equivalence and negation rules with quantifiers are also presented.
The document defines logical quantifiers such as existence and uniqueness quantifiers. It discusses how quantifiers can be used to restrict domains and bind variables. It provides examples of translating English statements to logical expressions using quantifiers and discusses precedence, logical equivalences, and negating quantifier expressions.
The document provides an overview of predicate logic, including:
- Predicates and quantifiers are introduced as the building blocks of predicate logic. Predicates allow representing properties and relations, while quantifiers like "for all" and "there exists" are used to make statements about predicates.
- Examples demonstrate how predicates and quantifiers can be used to represent concepts in logic and translate statements between English and logical expressions.
- Key concepts like universal and existential quantification, propositional functions, logical equivalences for quantifiers, and translating between English and logical expressions are defined and illustrated with examples.
- The document also discusses domains of discourse, precedence of quantifiers, thinking of quantifiers as conjunction
The document provides an overview of predicate logic, including:
- Predicates and quantifiers are introduced as the building blocks of predicate logic. Predicates allow representing properties and relations, while quantifiers like "for all" and "there exists" are used to make statements about predicates.
- Examples demonstrate how predicates and quantifiers can be used to represent concepts in logic and translate statements between English and logical expressions.
- Key concepts like universal and existential quantification, propositional functions, logical equivalences for quantifiers, and translating between English and logical expressions are defined and illustrated with examples.
- The document also discusses domains of discourse, precedence of quantifiers, thinking of quantifiers as conjunction
The document provides an introduction to discrete structures and mathematical reasoning. It discusses key concepts like propositions, logical operators, quantification, and proof techniques. Propositions can be combined using logical operators like negation, conjunction, disjunction, etc. Quantifiers like universal and existential are used to represent statements about all or some elements. Mathematical reasoning involves using axioms, rules of inference, and deductive proofs to establish theorems from given conditions.
This document provides an introduction to predicate logic and quantifiers. It begins with terminology like propositional functions, arguments, and universe of discourse. It then defines and provides examples of quantifiers like universal and existential quantifiers. It discusses how to mix quantifiers and their truth values. It also covers binding variables, scope, and negation of quantified statements. Finally, it provides a brief introduction to Prolog, a logic programming language based on predicate logic.
This document provides an overview of predicate logic, including:
- The basic components of predicate logic like variables, predicates, quantifiers, and propositional functions
- Explanations of the universal and existential quantifiers
- How to negate quantified expressions using De Morgan's laws
- Examples of translating statements between English and predicate logic
This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can express properties and relationships for objects. Laws of quantifier equivalence are also presented.
This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can represent concepts like "all parking spaces are full" or "some parking space is full." Laws of quantifier equivalence and negation rules with quantifiers are also presented.
The document defines logical quantifiers such as existence and uniqueness quantifiers. It discusses how quantifiers can be used to restrict domains and bind variables. It provides examples of translating English statements to logical expressions using quantifiers and discusses precedence, logical equivalences, and negating quantifier expressions.
The document provides an overview of predicate logic, including:
- Predicates and quantifiers are introduced as the building blocks of predicate logic. Predicates allow representing properties and relations, while quantifiers like "for all" and "there exists" are used to make statements about predicates.
- Examples demonstrate how predicates and quantifiers can be used to represent concepts in logic and translate statements between English and logical expressions.
- Key concepts like universal and existential quantification, propositional functions, logical equivalences for quantifiers, and translating between English and logical expressions are defined and illustrated with examples.
- The document also discusses domains of discourse, precedence of quantifiers, thinking of quantifiers as conjunction
The document provides an overview of predicate logic, including:
- Predicates and quantifiers are introduced as the building blocks of predicate logic. Predicates allow representing properties and relations, while quantifiers like "for all" and "there exists" are used to make statements about predicates.
- Examples demonstrate how predicates and quantifiers can be used to represent concepts in logic and translate statements between English and logical expressions.
- Key concepts like universal and existential quantification, propositional functions, logical equivalences for quantifiers, and translating between English and logical expressions are defined and illustrated with examples.
- The document also discusses domains of discourse, precedence of quantifiers, thinking of quantifiers as conjunction
The document provides an introduction to discrete structures and mathematical reasoning. It discusses key concepts like propositions, logical operators, quantification, and proof techniques. Propositions can be combined using logical operators like negation, conjunction, disjunction, etc. Quantifiers like universal and existential are used to represent statements about all or some elements. Mathematical reasoning involves using axioms, rules of inference, and deductive proofs to establish theorems from given conditions.
The document summarizes key concepts from a lecture on discrete structures including:
1) Predicates are statements with variables that become true or false when values are substituted. The truth set of a predicate contains values that make the statement true.
2) Universal statements are true if the predicate is true for all values, while existential statements are true if the predicate is true for at least one value.
3) Statements can be translated between formal logic notation using quantifiers and informal English. Negations of universal statements are existential, and vice versa.
The document summarizes key concepts from a lecture on discrete structures, including:
1) It defines predicates as sentences containing variables that become statements when values are substituted, and introduces truth sets as the set of elements making a predicate true.
2) It discusses universal and existential statements, where a universal statement is true if a predicate is true for all variables, and an existential is true if true for at least one variable.
3) It explains translating between formal quantified statements and informal English statements, and shows several examples of translating in both directions.
The document discusses propositional logic and first-order logic. It states that propositional logic has limited expressive power and cannot represent certain statements involving relationships between objects. First-order logic extends propositional logic by adding predicates and quantifiers, allowing it to more concisely represent natural language statements and relationships between objects. The key characteristics of first-order logic in AI are that it allows logical inference, more accurately represents facts about the real world, and provides a better theoretical foundation for program design.
The document discusses different methods of representing knowledge in artificial intelligence systems, including formal logic, production rules, and structured objects like semantic networks and frames. It provides examples of representing statements in propositional and predicate calculus, and how logic-based languages like Prolog can be used for knowledge representation and reasoning. Semantic networks are introduced as a way to organize knowledge representation in a graph-like structure similar to how human memory works.
This document provides an overview of a session on proof as quantified statements and nested quantifiers. The session aims to help students understand proof as a quantified statement and nested quantifiers. It will cover predicates and quantifiers, solving problems with nested quantified statements, and understanding statements with nested quantifiers. Examples are provided to illustrate nested quantifiers and how to translate statements into logical expressions. The document also discusses thinking of quantification as loops to help understand nested quantifiers. Self-assessment questions and tutorial problems are included to help students practice working with nested quantifiers.
Lecture 2 predicates quantifiers and rules of inferenceasimnawaz54
1) Predicates become propositions when variables are quantified by assigning values or using quantifiers. Quantifiers like ∀ and ∃ are used to make statements true or false for all or some values.
2) ∀ (universal quantifier) means "for all" and makes a statement true for all values of a variable. ∃ (existential quantifier) means "there exists" and makes a statement true if it is true for at least one value.
3) Predicates with unbound variables are neither true nor false. Binding variables by assigning values or using quantifiers turns predicates into propositions that can be evaluated as true or false.
This document provides an overview of automated theorem proving. It discusses:
1) The history and background of automated theorem proving, from Hobbes and Leibniz proposing algorithmic logic to modern computer-based approaches.
2) The theoretical limitations of automated reasoning due to results like Godel's incompleteness theorems, but also practical applications like verifying mathematics and computer systems.
3) How automated reasoning involves expressing statements formally and then manipulating those expressions algorithmically, as anticipated by Leibniz centuries ago.
This document provides an introduction to logic and set theory. It begins by defining key logic concepts such as propositions, truth values, and logical operators. It then explains how logical operators can combine propositions using truth tables. The document also discusses tautologies and contradictions. It introduces quantification and propositional functions. Finally, it provides examples of sets and set operations before transitioning to a discussion of set theory.
The document discusses logic agents and logical reasoning. It provides background on logic, including syntax, semantics, models, and inference rules. It then discusses how logic can be used to represent knowledge in knowledge-based agents and systems. The agents use a knowledge base and inference engine, where the inference engine derives new knowledge by applying inference rules to the knowledge base.
Introduction to Logic Spring 2007 Introduction to Discrete Structures.pptyaniarianneyani
This document provides an introduction to propositional logic and logical connectives. It begins by defining propositional logic and explaining that propositions can be either true or false. It then introduces common logical connectives like negation, conjunction, disjunction, implication and biconditional. Truth tables are presented as a way to define the semantics of logical connectives. Examples of each connective are provided. The document also discusses the usefulness of logic in areas like theoretical computer science, hardware/software specification, and programming. It provides an overview of propositional logic sentences and the satisfiability (SAT) problem. Finally, it gives an example of how logical equivalences can be used to modify a programming conditional statement.
A Probabilistic Attack On NP-Complete ProblemsBrittany Allen
This document discusses reformulating NP-complete problems in terms of continuous mathematics using probability theory. Specifically, it considers the 3-SAT NP-complete problem and introduces new probability variables to represent bit assignments. A cost function is constructed as a sum of clause satisfaction probabilities. Key properties of the cost function are that it is harmonic over subsets of variables and its Hessian has zero diagonal entries. The cost function is always positive inside the problem's domain and achieves its min/max on the boundary. The spectrum of cost function values on vertices corresponds to number of unsatisfied clauses. Overall, the approach reformulates 3-SAT in terms of a harmonic cost function to manipulate solutions without examining them individually.
This document discusses predicates and quantifiers in predicate logic. Predicate logic can express statements about objects and their properties, while propositional logic cannot. Predicates assign properties to variables, and quantifiers specify whether a predicate applies to all or some variables in a domain. There are two types of quantifiers: universal quantification with ∀ and existential quantification with ∃. Quantified statements involve predicates, variables ranging over a domain, and quantifiers to specify the scope of the predicate.
This document provides an outline and introduction to propositional logic. It discusses the history and development of logic from philosophical logic to its use in computer science. It covers propositional logic syntax using symbols and truth tables, semantics using the satisfaction relation, and the classification of formulas as valid, satisfiable, or unsatisfiable. It also introduces the decision problem of determining if a formula is satisfiable.
The document discusses propositional functions and quantifiers in logic. Some key points:
- A propositional function P(x) with domain of discourse D yields a class of propositions, one for each element in D, where each proposition is either true or false.
- Universal quantifiers of the form "for all x, P(x)" are true if P(x) is true for all x in D. Existential quantifiers of the form "there exists x such that P(x)" are true if P(x) is true for at least one x in D.
- Generalized De Morgan's laws state that the truth values of "there exists x, P(x
Last time we talked about propositional logic, a logic on simple statements.
This time we will talk about first order logic, a logic on quantified statements.
First order logic is much more expressive than propositional logic.
The topics on first order logic are:
1-Quantifiers
2-Negation
3-Multiple quantifiers
4-Arguments of quantified statements
I am Simon M. I am a Stochastic Processes Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Stochastic Processes, from Texas, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
This document discusses logic and propositional logic. It covers the following topics:
- The history and applications of logic.
- Different types of statements and their grammar.
- Propositional logic including symbols, connectives, truth tables, and semantics.
- Quantifiers, universal and existential quantification, and properties of quantifiers.
- Normal forms such as disjunctive normal form and conjunctive normal form.
- Inference rules and the principle of mathematical induction, illustrated with examples.
This document provides an outline for a lecture on discrete mathematics. It introduces topics like logic, set theory, mathematical reasoning/proof techniques, propositional/predicate calculus, Boolean algebra, induction, algorithms, recursion, counting/probability, and graph theory. The lecture begins with an introduction to logic, including statements, propositions, truth values, and logical operators like negation, conjunction, disjunction, implication, and biconditional. It provides examples of combining propositions using logical operators and truth tables. It also discusses propositional functions, quantification, and counterexamples.
The document summarizes key concepts from a lecture on discrete structures including:
1) Predicates are statements with variables that become true or false when values are substituted. The truth set of a predicate contains values that make the statement true.
2) Universal statements are true if the predicate is true for all values, while existential statements are true if the predicate is true for at least one value.
3) Statements can be translated between formal logic notation using quantifiers and informal English. Negations of universal statements are existential, and vice versa.
The document summarizes key concepts from a lecture on discrete structures, including:
1) It defines predicates as sentences containing variables that become statements when values are substituted, and introduces truth sets as the set of elements making a predicate true.
2) It discusses universal and existential statements, where a universal statement is true if a predicate is true for all variables, and an existential is true if true for at least one variable.
3) It explains translating between formal quantified statements and informal English statements, and shows several examples of translating in both directions.
The document discusses propositional logic and first-order logic. It states that propositional logic has limited expressive power and cannot represent certain statements involving relationships between objects. First-order logic extends propositional logic by adding predicates and quantifiers, allowing it to more concisely represent natural language statements and relationships between objects. The key characteristics of first-order logic in AI are that it allows logical inference, more accurately represents facts about the real world, and provides a better theoretical foundation for program design.
The document discusses different methods of representing knowledge in artificial intelligence systems, including formal logic, production rules, and structured objects like semantic networks and frames. It provides examples of representing statements in propositional and predicate calculus, and how logic-based languages like Prolog can be used for knowledge representation and reasoning. Semantic networks are introduced as a way to organize knowledge representation in a graph-like structure similar to how human memory works.
This document provides an overview of a session on proof as quantified statements and nested quantifiers. The session aims to help students understand proof as a quantified statement and nested quantifiers. It will cover predicates and quantifiers, solving problems with nested quantified statements, and understanding statements with nested quantifiers. Examples are provided to illustrate nested quantifiers and how to translate statements into logical expressions. The document also discusses thinking of quantification as loops to help understand nested quantifiers. Self-assessment questions and tutorial problems are included to help students practice working with nested quantifiers.
Lecture 2 predicates quantifiers and rules of inferenceasimnawaz54
1) Predicates become propositions when variables are quantified by assigning values or using quantifiers. Quantifiers like ∀ and ∃ are used to make statements true or false for all or some values.
2) ∀ (universal quantifier) means "for all" and makes a statement true for all values of a variable. ∃ (existential quantifier) means "there exists" and makes a statement true if it is true for at least one value.
3) Predicates with unbound variables are neither true nor false. Binding variables by assigning values or using quantifiers turns predicates into propositions that can be evaluated as true or false.
This document provides an overview of automated theorem proving. It discusses:
1) The history and background of automated theorem proving, from Hobbes and Leibniz proposing algorithmic logic to modern computer-based approaches.
2) The theoretical limitations of automated reasoning due to results like Godel's incompleteness theorems, but also practical applications like verifying mathematics and computer systems.
3) How automated reasoning involves expressing statements formally and then manipulating those expressions algorithmically, as anticipated by Leibniz centuries ago.
This document provides an introduction to logic and set theory. It begins by defining key logic concepts such as propositions, truth values, and logical operators. It then explains how logical operators can combine propositions using truth tables. The document also discusses tautologies and contradictions. It introduces quantification and propositional functions. Finally, it provides examples of sets and set operations before transitioning to a discussion of set theory.
The document discusses logic agents and logical reasoning. It provides background on logic, including syntax, semantics, models, and inference rules. It then discusses how logic can be used to represent knowledge in knowledge-based agents and systems. The agents use a knowledge base and inference engine, where the inference engine derives new knowledge by applying inference rules to the knowledge base.
Introduction to Logic Spring 2007 Introduction to Discrete Structures.pptyaniarianneyani
This document provides an introduction to propositional logic and logical connectives. It begins by defining propositional logic and explaining that propositions can be either true or false. It then introduces common logical connectives like negation, conjunction, disjunction, implication and biconditional. Truth tables are presented as a way to define the semantics of logical connectives. Examples of each connective are provided. The document also discusses the usefulness of logic in areas like theoretical computer science, hardware/software specification, and programming. It provides an overview of propositional logic sentences and the satisfiability (SAT) problem. Finally, it gives an example of how logical equivalences can be used to modify a programming conditional statement.
A Probabilistic Attack On NP-Complete ProblemsBrittany Allen
This document discusses reformulating NP-complete problems in terms of continuous mathematics using probability theory. Specifically, it considers the 3-SAT NP-complete problem and introduces new probability variables to represent bit assignments. A cost function is constructed as a sum of clause satisfaction probabilities. Key properties of the cost function are that it is harmonic over subsets of variables and its Hessian has zero diagonal entries. The cost function is always positive inside the problem's domain and achieves its min/max on the boundary. The spectrum of cost function values on vertices corresponds to number of unsatisfied clauses. Overall, the approach reformulates 3-SAT in terms of a harmonic cost function to manipulate solutions without examining them individually.
This document discusses predicates and quantifiers in predicate logic. Predicate logic can express statements about objects and their properties, while propositional logic cannot. Predicates assign properties to variables, and quantifiers specify whether a predicate applies to all or some variables in a domain. There are two types of quantifiers: universal quantification with ∀ and existential quantification with ∃. Quantified statements involve predicates, variables ranging over a domain, and quantifiers to specify the scope of the predicate.
This document provides an outline and introduction to propositional logic. It discusses the history and development of logic from philosophical logic to its use in computer science. It covers propositional logic syntax using symbols and truth tables, semantics using the satisfaction relation, and the classification of formulas as valid, satisfiable, or unsatisfiable. It also introduces the decision problem of determining if a formula is satisfiable.
The document discusses propositional functions and quantifiers in logic. Some key points:
- A propositional function P(x) with domain of discourse D yields a class of propositions, one for each element in D, where each proposition is either true or false.
- Universal quantifiers of the form "for all x, P(x)" are true if P(x) is true for all x in D. Existential quantifiers of the form "there exists x such that P(x)" are true if P(x) is true for at least one x in D.
- Generalized De Morgan's laws state that the truth values of "there exists x, P(x
Last time we talked about propositional logic, a logic on simple statements.
This time we will talk about first order logic, a logic on quantified statements.
First order logic is much more expressive than propositional logic.
The topics on first order logic are:
1-Quantifiers
2-Negation
3-Multiple quantifiers
4-Arguments of quantified statements
I am Simon M. I am a Stochastic Processes Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Stochastic Processes, from Texas, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
This document discusses logic and propositional logic. It covers the following topics:
- The history and applications of logic.
- Different types of statements and their grammar.
- Propositional logic including symbols, connectives, truth tables, and semantics.
- Quantifiers, universal and existential quantification, and properties of quantifiers.
- Normal forms such as disjunctive normal form and conjunctive normal form.
- Inference rules and the principle of mathematical induction, illustrated with examples.
This document provides an outline for a lecture on discrete mathematics. It introduces topics like logic, set theory, mathematical reasoning/proof techniques, propositional/predicate calculus, Boolean algebra, induction, algorithms, recursion, counting/probability, and graph theory. The lecture begins with an introduction to logic, including statements, propositions, truth values, and logical operators like negation, conjunction, disjunction, implication, and biconditional. It provides examples of combining propositions using logical operators and truth tables. It also discusses propositional functions, quantification, and counterexamples.
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMSIJNSA Journal
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...IJECEIAES
Medical image analysis has witnessed significant advancements with deep learning techniques. In the domain of brain tumor segmentation, the ability to
precisely delineate tumor boundaries from magnetic resonance imaging (MRI)
scans holds profound implications for diagnosis. This study presents an ensemble convolutional neural network (CNN) with transfer learning, integrating
the state-of-the-art Deeplabv3+ architecture with the ResNet18 backbone. The
model is rigorously trained and evaluated, exhibiting remarkable performance
metrics, including an impressive global accuracy of 99.286%, a high-class accuracy of 82.191%, a mean intersection over union (IoU) of 79.900%, a weighted
IoU of 98.620%, and a Boundary F1 (BF) score of 83.303%. Notably, a detailed comparative analysis with existing methods showcases the superiority of
our proposed model. These findings underscore the model’s competence in precise brain tumor localization, underscoring its potential to revolutionize medical
image analysis and enhance healthcare outcomes. This research paves the way
for future exploration and optimization of advanced CNN models in medical
imaging, emphasizing addressing false positives and resource efficiency.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
1. 1/ 33
Introduction
Propositional Functions
Slides by Christopher M. Bourke
Propositional
Functions Instructor: Berthe Y. Choueiry
Quantifiers
Logic
Programming
Transcribing
English into
Logic Fall 2007
Predicate Logic and Quantifiers
2. 2/ 33
Further Computer Science & Engineering 235
Examples &
Exercises Introduction to Discrete Mathematics
Sections 1.3–1.4 of Rosen cse235@cse.unl.edu
Introduction
Predicate
Logic and
Quantifiers
CSE235
Consider the following statements:
x > 3, x = y + 3, x + y = z
4. 4/ 33
Terminology:
affirmed =
holds = is true;
denied = does not hold = is not true.
|{z} | {z }
Propositional
subject predicate
Functions
Predicate
Logic and
Quantifiers
CSE235
Introduction
To write in predicate logic:
“ x is greater than 3”
Propositional
Functions
Quantifiers
Logic
Programming
Transcribing
English into
We introduce a (functional) symbol for the predicate, and put
the subject as an argument (to the functional symbol): P(x)
Examples:
6. Examples &
Exercises
6/ 33
Propositional Functions
Predicate
Logic and
Quantifiers Definition
CSE235
A statement of the form P(x1,x2,...,xn) is the value of the Introduction
propositional function P. Here, (x1,x2,...,xn) is an n-tuple Propositional and
P is a predicate.
Functions
Propositional
Universe of
Discourse
Quantifiers
You can think of a propositional function as a function that
7. Examples &
Exercises
7/ 33
Propositional Functions
Functions
Evaluates to true or false.
Logic Programming
Takes one or more arguments.
Transcribing
intoExpresses a predicate involving the argument(s).
English
Logic
Becomes a proposition when values are assigned to the
Further
arguments.
8. Examples &
Exercises
8/ 33
Propositional Functions
Example
Predicate
Logic and
Quantifiers Example
CSE235
Let Q(x,y,z) denote the statement “x2 + y2 = z2”. What is
Introduction the truth value of Q(3,4,5)? What is the truth value of
Propositional
Functions Q(2,2,3)? How many values of (x,y,z) make the predicate
Propositional true?
9. Examples &
Exercises
9/ 33
Propositional Functions
Functions
Universe of
Discourse
Quantifiers
Logic
Programming
Transcribing
English into Logic
Further
Example
Predicate
Logic and
11. Examples &
Exercises
11/ 33
Propositional Functions
CSE235
Let Q(x,y,z) denote the statement “x2 + y2 = z2”. What is
Introduction the truth value of Q(3,4,5)? What is the truth value of
Propositional
Quantifiers
Logic
Programming
Transcribing
English into Logic
Further
Since 32 + 42 = 25 = 52, Q(3,4,5) is true.
Since 22 + 22 = 8 6= 32 = 9, Q(2,2,3) is false.
There are infinitely many values for (x,y,z) that make this
propositional function true—how many right triangles are
there?
14. Examples &
Exercises
6/ 33
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Consider the previous example. Does it make sense to assign to
x the value “blue”?
Intuitively, the universe of discourse is the set of all things we
16. Transcribing
English into Logic
Further8/ 33
Examples &
Transcribing
English into
Logic
What would be the universe of discourse for the propositional
function P(x) = “The test will be on x the 23rd” be?
17. Examples &
Exercises
9/ 33
Universe of Discourse
Universe of wish to talk about; that is, the set of all objects that we can
Discourse
Further
18. Transcribing
English into Logic
Further10/ 33
Examples &
Universe of Discourse
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Moreover, each variable in an n-tuple may have a different
universe of discourse.
Let P(r,g,b,c) = “The rgb-value of the color c is (r,g,b)”.
20. Transcribing
English into Logic
Further12/ 33
Examples &
Multivariate Functions
Universe of
Discourse
Transcribing
English into
Logic
What are the universes of discourse for (r,g,b,c)?
22. Transcribing
English into Logic
Further14/ 33
Examples &
Quantifiers
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
A predicate becomes a proposition when we assign it fixed
values. However, another way to make a predicate into a
proposition is to quantify it. That is, the predicate is true (or
false) for all possible values in the universe of discourse or for
26. Transcribing
English into Logic
Further18/ 33
Examples &
Universal Quantifier
Example I
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
Definition
P (x)
P (x) x
∀xP (x)
which can be read “for all x
28. Transcribing
English into Logic
Further20/ 33
Examples &
Universal Quantifier
Example I
Propositional Functions
Quantifiers
Universal
Quantifier
∀xP(x) ⇐⇒ P(n1) ∧ P(n2) ∧ ··· ∧ P(nk)
Mixing
Quantifiers
Binding
Variables
Negation
Logic
Programming
If the universe of discourse is finite, say {n1,n2,...,nk}, then the
universal quantifier is simply the conjunction of all elements:
29. Examples &
Universal Quantifier
Example I
Predicate
Logic and
Quantifiers
CSE235
Let P(x) be the predicate “x must take a discrete
mathematics course” and let Q(x) be the predicate “x is a
computer science student”.
IntroductionThe
universe of
discourse
for both P(x)
and Q(x) is all PropositionalUNL students.
Functions Express the statement “Every computer science student
Propositional
Functions
Quantifiers
must take a discrete mathematics course”.
30. Transcribing
English into Logic
Further22/ 33
Examples &
Universal Quantifier
Example I
Universal
Quantifier
Existential Quantifier
Mixing
Quantifiers
Binding
Variables
Negation
Express the statement “Everybody must take a discrete
mathematics course or be a computer science student”.
Logic
Programming
31. Examples &
Universal Quantifier
Example I
Predicate
Logic and
Quantifiers
CSE235
Let P(x) be the predicate “x must take a discrete
mathematics course” and let Q(x) be the predicate “x is a
computer science student”.
IntroductionThe universe of discourse for both P(x) and Q(x) is all
PropositionalUNL students.
Functions
Express the statement “Every computer science student
32. Transcribing
English into Logic
Further24/ 33
Examples &
Universal Quantifier
Example I
UniversalQuantifier ∀x(Q(x) → P(x))
Existential Quantifier
Mixing
Quantifiers
Propositional
Functions
Quantifiers
must take a discrete mathematics course”.
33. Examples &
Universal Quantifier
Example I
Binding
Variables
Negation
Express the statement “Everybody must take a discrete
mathematics course or be a computer science student”.
Logic
Programming
Transcribing
English into Logic
Further10 /33
34. Examples &
Universal Quantifier
Example I
Predicate
Logic and
Quantifiers
CSE235
Let P(x) be the predicate “x must take a discrete
mathematics course” and let Q(x) be the predicate “x is a
computer science student”.
IntroductionThe universe of discourse for both P(x) and Q(x) is all
PropositionalUNL students.
Functions
Express the statement “Every computer science student
35. Examples &
Universal Quantifier
Example I
Propositional
Functions
Quantifiers
must take a discrete mathematics course”.
Existential Quantifier
Mixing
Quantifiers
36. Examples &
Universal Quantifier
Example II
UniversalQuantifier ∀x(Q(x) → P(x))
Transcribing
English into Logic
Further10 /33
Predicate
Logic and
Quantifiers
CSE235
Let P(x) be the predicate “x must take a discrete
mathematics course” and let Q(x) be the predicate “x is a
computer science student”.
Binding
Variables
Negation
Logic
Programming
Express the statement “Everybody must take a discrete
mathematics course or be a computer science student”.
∀x(Q(x) ∨ P(x))
37. Examples &
Universal Quantifier
Example I
IntroductionThe universe of discourse for both P(x) and Q(x) is all
PropositionalUNL students.
Functions
Propositional
Functions
Quantifiers
must take a discrete mathematics course”.
Existential Quantifier
Mixing
Quantifiers
38. Examples &
Universal Quantifier
Example II
Express the statement “Every computer science student
UniversalQuantifier ∀x(Q(x) → P(x))
Predicate
Logic and
Binding
Variables
Negation
Logic
Programming
Transcribing
English into Logic
Further10 /33
Express the statement “Everybody must take a discrete
mathematics course or be a computer science student”.
∀x(Q(x) ∨ P(x))
Are hetse statements true or false?
39. Examples &
Universal Quantifier
Example I
Quantifiers
Express the statement “for every x and for every y, x+y > 10”
CSE235
Introduction
Propositional Functions
Propositional Functions
Quantifiers
Universal
Quantifier
Existential Quantifier
Mixing
Quantifiers
Binding
Variables
Negation
42. Examples &
Universal Quantifier
Example II
Universal
Quantifier
Existential Quantifier
Mixing
Quantifiers
Binding
Variables
Negation
Logic
Programming
Transcribing
English into Logic
CSE235
Introduction
Propositional
Functions
Express the statement “for every x and for every y, x+y > 10”
Let P(x,y) be the statement x + y > 10 where the universe of
discourse for x,y is the set of integers.
44. Examples &
Universal Quantifier
Example II
Quantifiers
Universal
Quantifier
Existential Quantifier
Mixing
Quantifiers
Binding
Variables
Negation
Logic
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Express the statement “for every x and for every y, x+y > 10”
Let P(x,y) be the statement x + y > 10 where the universe of
discourse for x,y is the set of integers.
Answer:
45. Transcribing
English into Logic
Further13 /33
Examples &
Universal Quantifier
Example II
Programming
Transcribing
English into Logic
Further11 /33
Predicate
Logic and
46. Examples &
Universal Quantifier
Example II
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Quantifiers
Express the statement “for every x and for every y, x+y > 10”
Let P(x,y) be the statement x + y > 10 where the universe of
discourse for x,y is the set of integers.
Answer:
∀x∀yP(x,y)
47. Transcribing
English into Logic
Further15 /33
Examples &
Universal Quantifier
Example II
Universal
Quantifier
∀x,yP(x,y)
Logic
Programming
Existential Quantifier
Mixing
Quantifiers
Binding
Variables
Negation
Note that we can also use the shorthand
49. Transcribing
English into Logic
Further17 /33
Examples &
Existential Quantifier
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
Definition
P (x)
x
P (x) is true.” We use the notation
∃xP (x)
which can be read “there exists anx
50. Transcribing
English into Logic
Further18 /33
Examples &
Existential Quantifier
Example II
Mixing
Quantifiers Again, if the universe of discourse is finite, {n1,n2,...,nk}, then
51. Transcribing
English into Logic
Further19 /33
Examples &
Existential Quantifier
Propositional Functions
Quantifiers
Universal
Quantifier
Existential
Quantifier
∃xP(x) ⇐⇒ P(n1) ∨ P(n2) ∨ ··· ∨ P(nk)
Binding
Variables
Negation
Logic
Programming
the existential quantifier is simply the disjunction of all
elements:
52. Transcribing
English into Logic
Further20 /33
Examples &
Existential Quantifier
Example II
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Quantifiers
Universal
Quantifier
Let P(x,y) denote the statement, “x + y = 5”.
What does the expression,
∃x∃yP(x,y)
53. Transcribing
English into Logic
Further21 /33
Examples &
Existential Quantifier
Example I
Existential mean?
Quantifier
Mixing
Quantifiers
Binding What universe(s) of discourse make it true?
Variables
Negation
Logic
Programming
54. Transcribing
English into Logic
Further22 /33
Examples &
Existential Quantifier
Example II
Predicate
Logic and
Quantifiers
Propositional Functions
Propositional Functions
Quantifiers
Universal
Quantifier
Existential Quantifier
CSE235
Introduction
Express the statement “there exists a real solution to
ax2 + bx − c = 0”
56. Examples &
Existential Quantifier
Example II
Predicate
Logic and
Quantifiers
Propositional
Functions Let P(x) be the statementwhere the universe
Universal
Quantifier
Existential Quantifier
Mixing
Quantifiers
Binding
CSE235
Introduction
Express the statement “there exists a real solution to
ax2 + bx − c = 0”
Propositional
Functions
Quantifiers
of discourse for x is the set of reals. Note here that a,b,c are all
fixed constants.
57. Examples &
Existential Quantifier
Example II Continued
Variables
Negation
Logic
Programming
Transcribing
English into Logic
Further14 /33
Predicate
Logic and
Quantifiers
Propositional
CSE235
Introduction
Express the statement “there exists a real solution to
ax2 + bx − c = 0”
58. Examples &
Existential Quantifier
Example II
Functions Let P(x) be the statementwhere the universe
The statement can thus be expressed as
Existential
Quantifier
Mixing
Propositional
Functions
Quantifiers
Universal
Quantifier
of discourse for x is the set of reals. Note here that a,b,c are all
fixed constants.
59. Examples &
Existential Quantifier
Example II Continued
Quantifiers
Binding
Variables
Negation
∃xP(x)
Logic
Programming
Transcribing
English into Logic
Further14 /33
Predicate
Logic and
Quantifiers
CSE235
Introduction Question: what is the truth value of ∃xP(x)?
Propositional Functions
60. Examples &
Existential Quantifier
Example II
Propositional Functions
Quantifiers
Universal
Quantifier
Existential Quantifier
Mixing
Quantifiers
Binding
Variables
Negation
Logic
Programming
Transcribing
English into Logic
Further15 /33
61.
62. Examples &
Existential Quantifier
Example II Continued
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Quantifiers
Universal
Quantifier
Question: what is the truth value of xP(x)?
Answer: it is false. For any real numbers such that b2 < 4ac,
there will only be complex solutions, for these cases no such
real number x can satisfy the predicate.
How can we make it so that it is true?
63. Transcribing
English into Logic
Further15 /33
Examples &
Existential Quantifier
Example II Continued
∃
Existential Quantifier
Mixing
Quantifiers
Binding
Variables
Negation
65. Transcribing
English into Logic
Further17 /33
Examples &
Existential Quantifier
Example II Continued
∃
Existential
Quantifier
Introduction
Propositional
Functions
Propositional
Functions
Quantifiers
Universal
Quantifier
Question: what is the truth value of xP(x)?
Answer: it is false. For any real numbers such that b2 < 4ac,
there will only be complex solutions, for these cases no such
real number x can satisfy the predicate.
How can we make it so that it is true?
66. Transcribing
English into Logic
Further18 /33
Examples &
Mixing
Quantifiers
Binding
Variables
Negation
Answer: change the universe of discourse to the complex
numbers, C.
Logic
Programming
67. Transcribing
English into Logic
Further19 /33
Examples &
Existential Quantifier
Example II Continued
Quantifiers
Truth Values
Functions
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
In general, when are quantified statements true/false?
Statement True When False When
68. Transcribing
English into Logic
Further20 /33
Examples &
Propositional
Functions
Quantifiers
Universal
Quantifier
Existential Quantifier
Mixing
Logic
Programming
∀xP(x) P(x) is true for every
x.
There is an x for which
P(x) is false.
∃xP(x) There is an x for which
P(x) is true.
P(x) is false for every
x.
Quantifiers
Binding
Variables
Negation
Table: Truth Values of Quantifiers
70. Transcribing
English into Logic
Further22 /33
Examples &
Mixing Quantifiers
I
Introduction
Propositional
Functions
Propositional
Functions
Quantifiers
Universal
Quantifier
Existential
Quantifier
Existential and universal quantifiers can be used together to
quantify a predicate statement; for example,
∀x∃yP(x,y)
is perfectly valid. However, you must be careful—it must be
read left to right.
71. Transcribing
English into Logic
Further23 /33
Examples &
Mixing Quantifiers
Mixing
Quantifiers
Logic
Programming
II
Binding
Variables
Negation
For example, ∀x∃yP(x,y) is not equivalent to ∃y∀xP(x,y).
Thus, ordering is important.
72. Transcribing
English into Logic
Further24 /33
Examples &
Mixing Quantifiers
Predicate Logic
and
Quantifiers
CSE235
For example:
Functions
Propositional
Functions
Quantifiers
Universal
Quantifier
Existential
Quantifier
Those expressions do not mean the same thing!
Note that ∃y∀xP(x,y) → ∀x∃yP(x,y), but the converse does not
hold
74. Transcribing
English into Logic
Further26 /33
Examples &
Mixing Quantifiers
Introduction∀x∃yLoves(x,y): everybody loves somebody
Propositional∃y∀xLoves(x,y): There is someone loved by everyone
Variables
Negation
Logic
Programming
equivalent to ∃y∃xP(x,y) (which is why our shorthand was
valid).
76. Transcribing
English into Logic
Further28 /33
Examples &
Mixing Quantifiers
∀x∀yP(x,y) P(x,y) is true for every
pair x,y.
There is at least one
pair, x,y for which
P(x,y) is false.
∀x∃yP(x,y) For every x, there is a
y for which P(x,y) is
true.
There is an x for which
P(x,y) is false for
every y.
77. Transcribing
English into Logic
Further29 /33
Examples &
Mixing Quantifiers
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Quantifiers
Universal
Quantifier
Existential Quantifier
∃x∀yP(x,y) There is an x for which
P(x,y) is true for every
y.
For every x, there is a
y for which P(x,y) is
false.
∃x∃yP(x,y) There is at least one
pair x,y for which
P(x,y) is true.
P(x,y) is false for every
pair x,y.
78. Transcribing
English into Logic
Further30 /33
Examples &
Mixing Quantifiers
Mixing
Quantifiers
Binding
Variables
Negation
Logic
Programming
Table: Truth Values of 2-variate Quantifiers
79. Transcribing
English into Logic
Further31 /33
Examples &
Mixing Quantifiers
Example I
Predicate
Logic and
Quantifiers
CSE235
Express, in predicate logic, the statement that there are an
Introduction infinite number of integers.
Propositional Functions
Propositional Functions
Quantifiers
Universal
Quantifier
80. Examples &
Mixing Quantifiers
Example I
Existential Quantifier
Mixing
Quantifiers
Binding
Variables
Negation
Logic
Programming
Predicate
Logic and
Quantifiers
CSE235
Quantifiers
81. Examples &
Mixing Quantifiers
Example I
Universal
Quantifier
Existential Quantifier
Mixing
Quantifiers
Binding
Variables
Negation
Logic
Programming
Introduction
Propositional
Functions
Propositional
Functions
Express, in predicate logic, the statement that there are an
infinite number of integers.
Let P(x,y) be the statement that x < y. Let the universe of
discourse be the integers, Z.
83. Examples &
Mixing Quantifiers
Example I
CSE235
Mixing ∀x∃yP(x,y)
Quantifiers
Binding
Variables
Negation
Introduction
Propositional
Functions
Propositional
Functions
Quantifiers
Universal
Quantifier
Existential
Quantifier
Express, in predicate logic, the statement that there are an
infinite number of integers.
Let P(x,y) be the statement that x < y. Let the universe of
discourse be the integers, Z.
Then the statement can be expressed by the following.
85. Transcribing
English into Logic
Further21 /33
Examples &
Mixing Quantifiers
Example II: More Mathematical Statements
Predicate
Logic and
Quantifiers
CSE235 Express the commutative law of addition for R.
Introduction
Propositional Functions
Propositional Functions
Quantifiers
Universal
Quantifier
86. Transcribing
English into Logic
Further21 /33
Examples &
Mixing Quantifiers
Example II: More Mathematical Statements
Existential Quantifier
Mixing
Quantifiers
Binding
Variables
Negation
Logic
Programming
Predicate
Logic and
87. Transcribing
English into Logic
Further21 /33
Examples &
Mixing Quantifiers
Example II: More Mathematical Statements
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Express the commutative law of addition for R.
We want to express that for every pair of reals, x,y the
following identity holds:
x + y = y + x
88. Transcribing
English into Logic
Further21 /33
Examples &
Mixing Quantifiers
Example II: More Mathematical Statements
Quantifiers
Universal
Quantifier
Existential Quantifier
Mixing
Quantifiers
Binding
Variables
Negation
Logic
Programming
Predicate
Logic and
89. Transcribing
English into Logic
Further21 /33
Examples &
Mixing Quantifiers
Example II: More Mathematical Statements
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Quantifiers
Universal
Quantifier
Existential
Quantifier
Express the commutative law of addition for R.
We want to express that for every pair of reals, x,y the
following identity holds:
x + y = y + x
Then we have the following:
90. Transcribing
English into Logic
Further21 /33
Examples &
Mixing Quantifiers
Example II: More Mathematical Statements
Mixing
QuantifiersBinding ∀x∀y(x + y = y + x)
Variables
Negation
Logic
Programming
91. Transcribing
English into Logic
Further22 /33
Examples &
Mixing Quantifiers
Example II: More Mathematical Statements Continued
Predicate
Logic and
Quantifiers
CSE235
Express the multiplicative inverse law for (nonzero) rationals
Introduction
Q {0}.
Propositional Functions
Propositional Functions
Quantifiers
92. Transcribing
English into Logic
Further22 /33
Examples &
Mixing Quantifiers
Example II: More Mathematical Statements Continued
Universal
Quantifier
Existential Quantifier
Mixing
Quantifiers
Binding
Variables
Negation
Logic
Programming
Predicate
Logic and
Quantifiers
93. Transcribing
English into Logic
Further22 /33
Examples &
Mixing Quantifiers
Example II: More Mathematical Statements Continued
CSE235
Quantifiers
Universal
Quantifier
Existential Quantifier
Mixing
Introduction
Propositional
Functions
Propositional
Functions
Express the multiplicative inverse law for (nonzero) rationals Q
{0}.
We want to express that for every real number x, there exists a
real number y such that xy = 1.
94. Transcribing
English into Logic
Further22 /33
Examples &
Mixing Quantifiers
Example II: More Mathematical Statements Continued
Quantifiers
Binding
Variables
Negation
Logic
Programming
Predicate
Logic and
Quantifiers
95. Transcribing
English into Logic
Further22 /33
Examples &
Mixing Quantifiers
Example II: More Mathematical Statements Continued
CSE235
Mixing ∀x∃y(xy = 1)
Introduction
Propositional
Functions
Propositional
Functions
Quantifiers
Universal
Quantifier
Existential
Quantifier
Express the multiplicative inverse law for (nonzero) rationals Q
{0}.
We want to express that for every real number x, there exists a
real number y such that xy = 1.
Then we have the following:
96. Transcribing
English into Logic
Further22 /33
Examples &
Mixing Quantifiers
Example II: More Mathematical Statements Continued
Quantifiers
Binding
Variables
Negation
Logic
Programming
97. Transcribing
English into Logic
Further23 /33
Examples &
Mixing Quantifiers
Example II: False Mathematical Statements
Predicate
Logic and
Quantifiers
CSE235
Introduction Is commutativity for subtraction valid over the reals?
Propositional Functions
Propositional Functions
Quantifiers
Universal
Quantifier
Existential Quantifier
98. Transcribing
English into Logic
Further23 /33
Examples &
Mixing Quantifiers
Example II: False Mathematical Statements
Mixing
Quantifiers
Binding
Variables
Negation
Logic
Programming
Predicate
Logic and
Quantifiers
99. Transcribing
English into Logic
Further23 /33
Examples &
Mixing Quantifiers
Example II: False Mathematical Statements
CSE235
Quantifiers
Universal
Quantifier
Existential Quantifier
Mixing
Quantifiers
Binding
Introduction
Propositional
Functions
Propositional
Functions
Is commutativity for subtraction valid over the reals?
That is, for all pairs of real numbers x,y does the identity x −
y = y − x hold? Express this using quantifiers.
100. Transcribing
English into Logic
Further23 /33
Examples &
Mixing Quantifiers
Example II: False Mathematical Statements
Variables
Negation
Logic
Programming
Predicate
Logic and
Quantifiers
101. Transcribing
English into Logic
Further23 /33
Examples &
Mixing Quantifiers
Example II: False Mathematical Statements
CSE235
Mixing
Introduction
Propositional
Functions
Propositional
Functions
Quantifiers
Universal
Quantifier
Existential
Quantifier
Is commutativity for subtraction valid over the reals?
That is, for all pairs of real numbers x,y does the identity x −
y = y − x hold? Express this using quantifiers.
The expression is
∀x∀y(x − y = y − x)
102. Transcribing
English into Logic
Further23 /33
Examples &
Mixing Quantifiers
Example II: False Mathematical Statements
Quantifiers
Binding
Variables
Negation
Logic
Programming
103. Transcribing
English into Logic
Further24 /33
Examples &
Mixing Quantifiers
Example II: False Mathematical Statements Continued
Predicate
Logic and
Quantifiers
CSE235
Introduction Is there a multiplicative inverse law over the nonzero integers?
Propositional Functions
Propositional Functions
Quantifiers
Universal
Quantifier
Existential Quantifier
104. Transcribing
English into Logic
Further24 /33
Examples &
Mixing Quantifiers
Example II: False Mathematical Statements Continued
Mixing
Quantifiers
Binding
Variables
Negation
Logic
Programming
Predicate
Logic and
Quantifiers
CSE235
105. Transcribing
English into Logic
Further24 /33
Examples &
Mixing Quantifiers
Example II: False Mathematical Statements Continued
Functions
Quantifiers
Universal
Quantifier
Existential Quantifier
Mixing
Quantifiers
Binding
Introduction
Propositional
Functions
Propositional
Is there a multiplicative inverse law over the nonzero integers?
That is, for every integer x does there exists a y such that xy =
1?
106. Transcribing
English into Logic
Further24 /33
Examples &
Mixing Quantifiers
Example II: False Mathematical Statements Continued
Variables
Negation
Logic
Programming
Predicate
Logic and
Quantifiers
107. Transcribing
English into Logic
Further24 /33
Examples &
Mixing Quantifiers
Example II: False Mathematical Statements Continued
CSE235
Mixing statement held, then 5 = 1/y, but for any (nonzero) integer y,
Introduction
Propositional
Functions
Propositional
Functions
Quantifiers
Universal
Quantifier
Existential
Quantifier
Is there a multiplicative inverse law over the nonzero integers?
That is, for every integer x does there exists a y such that xy =
1?
This is false, since we can find a counter example. Take any
integer, say 5 and multiply it with another integer, y. If the
108. Transcribing
English into Logic
Further24 /33
Examples &
Mixing Quantifiers
Example II: False Mathematical Statements Continued
Quantifiers
Binding Variables
|1/y| ≤ 1.
Negation
Logic
Programming
109. Transcribing
English into Logic
Further25 /33
Examples &
Mixing Quantifiers
Exercise
Logic andPredicate Express the statement “there is a number x such that
Quantifiers when it is added to any number, the result is that CSE235
number, and if it is multiplied by any number, the Introduction
result is x” as a logical expression.
Propositional Functions
Propositional Solution:
Functions
111. Transcribing
English into Logic
Further25 /33
Examples &
Mixing Quantifiers
Exercise
Quantifiers
Binding
Variables
Negation
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Quantifiers
Express the statement “there is a number x such that
when it is added to any number, the result is that
number, and if it is multiplied by any number, the
result is x” as a logical expression.
Solution:
Let P(x,y) be the expression “x + y = y”.
112. Transcribing
English into Logic
Further25 /33
Examples &
Mixing Quantifiers
Exercise
Logic
Programming
QuantifiersLet P(x,y) be the expression “x + y = y”.
UniversalQuantifierLet Q(x,y) be the expression “xy = x”.
Existential Quantifier
Mixing
113. Transcribing
English into Logic
Further25 /33
Examples &
Mixing Quantifiers
Exercise
Quantifiers
Binding
Variables
Negation
Logic
Programming
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Express the statement “there is a number x such that
when it is added to any number, the result is that
number, and if it is multiplied by any number, the
result is x” as a logical expression.
Solution:
114. Transcribing
English into Logic
Further25 /33
Examples &
Mixing Quantifiers
Exercise
QuantifiersLet P(x,y) be the expression “x + y = y”.
UniversalQuantifierLet Q(x,y) be the expression “xy = x”.
ExistentialQuantifierThen the expression is
Mixing
115. Transcribing
English into Logic
Further25 /33
Examples &
Mixing Quantifiers
Exercise
Quantifiers
Binding Variables
∃x∀y (P(x,y) ∧ Q(x,y))
Negation
Logic
Programming
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Express the statement “there is a number x such that
when it is added to any number, the result is that
number, and if it is multiplied by any number, the
result is x” as a logical expression.
Solution:
117. Transcribing
English into Logic
Further25 /33
Examples &
Mixing Quantifiers
Exercise
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Express the statement “there is a number x such that
when it is added to any number, the result is that
number, and if it is multiplied by any number, the
result is x” as a logical expression.
Solution:
119. Transcribing
English into Logic
Further25 /33
Examples &
Mixing Quantifiers
Exercise
UniversalQuantifierLet Q(x,y) be the expression “xy = x”.
ExistentialQuantifierThen the expression is
Mixing
Quantifiers
Variables
Negation
Logic
Programming
Over what universe(s) of discourse does this statement
hold?
120. Transcribing
English into Logic
Further25 /33
Examples &
Mixing Quantifiers
Exercise
QuantifiersLet P(x,y) be the expression “x + y = y”.
UniversalQuantifierLet Q(x,y) be the expression “xy = x”.
ExistentialQuantifierThen the expression is
Mixing
121. Transcribing
English into Logic
Further25 /33
Examples &
Mixing Quantifiers
Exercise
Quantifiers
Binding
Variables
Negation
∃x∀y (P(x,y) ∧ Q(x,y))
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
Propositional
Functions
Express the statement “there is a number x such that
when it is added to any number, the result is that
number, and if it is multiplied by any number, the
result is x” as a logical expression.
Solution:
122. Transcribing
English into Logic
Further25 /33
Examples &
Mixing Quantifiers
Exercise
LogicOver what universe(s) of discourse does this statement
Programming
hold?
This is the additive identity law and holds for N,Z,R,Q but
does not hold for Z+.
123. Examples &
Binding Variables I
Propositional
Functions Example
Quantifiers In the expression ∃x∀yP(x,y) both x and y are bound.
Universal
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
When a quantifier is used on a variable x, we say that x is
bound. If no quantifier is used on a variable in a predicate
statement, it is called free.
124. Examples &
QuantifierExistential In the expression ∀xP(x,y), x is bound, but y is free.
Quantifier
Mixing
Quantifiers
BindingVariables A statement is called a well-formed formula, when all variables
Negation
are properly
quantified.
Logic
Programming
Transcribing
English into Logic
Further26 /33
125. Examples &
Binding Variables II
Propositional Example
Functions
Quantifiers In the expression ∃x,y∀zP(x,y,z,c) the scope of the
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
Functions
The set of all variables bound by a common quantifier is the
scope of that quantifier.
126. Examples &
UniversalQuantifier existential quantifier is {x,y}, the scope of the universal
ExistentialQuantifier quantifier is just z and c has no scope since it is free.
Mixing
Quantifiers
Binding
Variables
Negation
Logic
Programming
Transcribing
English into Logic
Further27 /33
131. Examples &
This is essentially a quantified version of De Morgan’s Law (in
Further28 /33
Negation
Truth Values
Predicate
Logic and
Quantifiers
CSE235
Introduction
Statement True When False When
133. Prolog
FunctionsProlog allows the user to express facts and rules
PropositionalFacts are proposational functions: student(juana),
Functions
enrolled(juana,cse235),
instructor(patel,cse235),
etc.
Quantifiers
Predicate
Logic and
Quantifiers
CSE235
Introduction
Propositional
Prolog (Programming in Logic) is a programming language
based on (a restricted form of) Predicate Calculus. It was
developped by the logicians of the artificial intelligence
community for symbolic reasoning.
Logic
Further
Examples &
Exercises
?enrolled(juana,cse478)
?enrolled(X,cse478)
?teaches(X,juana)
134. Rules are implications with conjunctions:
Logic Programming teaches(X,Y) :- instructor(X,Z), enrolled(Y,Z)
Transcribing
English intoProlog answers queries such as:
30 /33
by binding variables and doing theorem proving (i.e.,
applying inference rules) as we will see in Section 1.5.
137. 32 /33
Conclusion
English into Logic
everyone is fierce”
Transcribing
English into
Logic Use ∃ with ∧
Examples &
Exercises have at least one lion that drinks coffee
143. 32 /33
Conclusion
Quantifiers
Logic
“”. When is Q(y) true?
Predicate Logic
and
Quantifiers
CSE235
Examples? Exercises?
Rewrite the expression,
Introduction
Propositional Functions
Answer: Use the negated quantifiers and De Morgan’s law.
Logic
Programming
Transcribing
English into
Let P(x,y) denote “x is a factor of y” where x ∈ {1,2,3,...}
and y ∈ {2,3,4,...}. Let Q(y) denote
144. Propositional Functions
Quantifiers
Logic
“”. When is Q(y) true?
Answer: Only when y is a prime number.
Further
Examples &
Exercises
Logic
Programming
Transcribing
English into
Let P(x,y) denote “x is a factor of y” where x ∈ {1,2,3,...}
and y ∈ {2,3,4,...}. Let Q(y) denote
146. Extra Question
Further Examples &
Exercises Thus, the left-hand side is a proposition, but the right-hand side
is not. How can they be equivalent?
33 /33