This document provides an overview of a course on discrete mathematics. It discusses the goals of teaching logical and mathematical thinking. The content will include topics like mathematical reasoning, combinatorics, discrete structures, algorithms, and applications. The textbook is Discrete Mathematics and its Applications. The course will be evaluated based on midterm, final exams, and regular assessments. The instructor then provides their contact information and office hours. Finally, an outline of course topics is presented, beginning with foundations of logic and proofs.
Section 1 axiomatizes intuitionistic fuzzy logic IF and proves its consistency and strong completeness: if a sequent is valid in every intuitionistic fuzzy model, it is provable.
Section 2 presents intuitionistic fuzzy set theory ZFIF, which extends intuitionistic set theory ZFI with the axioms of dependent choice and double complement. It develops the calculus of ZFIF.
Knowledge Based Reasoning: Agents, Facets of Knowledge. Logic and Inferences: Formal Logic,
Propositional and First Order Logic, Resolution in Propositional and First Order Logic, Deductive
Retrieval, Backward Chaining, Second order Logic. Knowledge Representation: Conceptual
Dependency, Frames, Semantic nets.
This document discusses the computation of presuppositions and entailments from natural language text. It begins by defining presuppositions and entailments, and explaining how they can be computed using tree transformations on semantic representations. The paper then provides examples of elementary presuppositions and entailments. It describes a system that computes presuppositions and entailments while parsing sentences using an augmented transition network. The system applies tree transformations specified in the lexicon to the semantic representation to derive inferences. The paper concludes that presuppositions and entailments exhibit computational properties not shown by the general class of inferences, such as being tied to the semantic and syntactic structure of language.
Artificial Intelligence (AI) | Prepositional logic (PL)and first order predic...Ashish Duggal
The following are the topics in this presentation Prepositional Logic (PL) and First-order Predicate Logic (FOPL) is used for knowledge representation in artificial intelligence (AI).
There are also sub-topics in this presentation like logical connective, atomic sentence, complex sentence, and quantifiers.
This PPT is very helpful for Computer science and Computer Engineer
(B.C.A., M.C.A., B.TECH. , M.TECH.)
This document provides an introduction to First Order Predicate Logic (FOPL). It discusses the differences between propositional logic and FOPL, the parts and syntax of FOPL including terms, atomic sentences, quantifiers and rules of inference. The semantics of FOPL are also explained. Pros and cons are provided, such as FOPL's ability to represent individual entities and generalizations compared to propositional logic. Applications include using FOPL as a framework for formulating theories.
This document provides an introduction to First Order Predicate Logic (FOPL). It defines FOPL as symbolized reasoning where sentences are broken down into subjects and predicates. FOPL is more expressive than propositional logic and allows representing almost any English sentence. It discusses features of FOPL such as generalization of propositional logic and more powerful representation. Applications of FOPL include presenting arguments, determining validity, and formulating theories. The document also defines key terms in FOPL such as constants, variables, functions, and predicates.
Artificial intelligence and first order logicparsa rafiq
The document discusses knowledge representation and first order logic. It defines knowledge representation as how knowledge is encoded in artificial systems. It discusses representing objects, events, performance, meta-knowledge and facts. It also discusses types of knowledge like meta knowledge, heuristic knowledge, procedural knowledge and declarative knowledge. The document then discusses first order logic syntax including logical symbols, terms, formulas, quantifiers and predicates. It also discusses semantics and the uses and history of first order logic.
Section 1 axiomatizes intuitionistic fuzzy logic IF and proves its consistency and strong completeness: if a sequent is valid in every intuitionistic fuzzy model, it is provable.
Section 2 presents intuitionistic fuzzy set theory ZFIF, which extends intuitionistic set theory ZFI with the axioms of dependent choice and double complement. It develops the calculus of ZFIF.
Knowledge Based Reasoning: Agents, Facets of Knowledge. Logic and Inferences: Formal Logic,
Propositional and First Order Logic, Resolution in Propositional and First Order Logic, Deductive
Retrieval, Backward Chaining, Second order Logic. Knowledge Representation: Conceptual
Dependency, Frames, Semantic nets.
This document discusses the computation of presuppositions and entailments from natural language text. It begins by defining presuppositions and entailments, and explaining how they can be computed using tree transformations on semantic representations. The paper then provides examples of elementary presuppositions and entailments. It describes a system that computes presuppositions and entailments while parsing sentences using an augmented transition network. The system applies tree transformations specified in the lexicon to the semantic representation to derive inferences. The paper concludes that presuppositions and entailments exhibit computational properties not shown by the general class of inferences, such as being tied to the semantic and syntactic structure of language.
Artificial Intelligence (AI) | Prepositional logic (PL)and first order predic...Ashish Duggal
The following are the topics in this presentation Prepositional Logic (PL) and First-order Predicate Logic (FOPL) is used for knowledge representation in artificial intelligence (AI).
There are also sub-topics in this presentation like logical connective, atomic sentence, complex sentence, and quantifiers.
This PPT is very helpful for Computer science and Computer Engineer
(B.C.A., M.C.A., B.TECH. , M.TECH.)
This document provides an introduction to First Order Predicate Logic (FOPL). It discusses the differences between propositional logic and FOPL, the parts and syntax of FOPL including terms, atomic sentences, quantifiers and rules of inference. The semantics of FOPL are also explained. Pros and cons are provided, such as FOPL's ability to represent individual entities and generalizations compared to propositional logic. Applications include using FOPL as a framework for formulating theories.
This document provides an introduction to First Order Predicate Logic (FOPL). It defines FOPL as symbolized reasoning where sentences are broken down into subjects and predicates. FOPL is more expressive than propositional logic and allows representing almost any English sentence. It discusses features of FOPL such as generalization of propositional logic and more powerful representation. Applications of FOPL include presenting arguments, determining validity, and formulating theories. The document also defines key terms in FOPL such as constants, variables, functions, and predicates.
Artificial intelligence and first order logicparsa rafiq
The document discusses knowledge representation and first order logic. It defines knowledge representation as how knowledge is encoded in artificial systems. It discusses representing objects, events, performance, meta-knowledge and facts. It also discusses types of knowledge like meta knowledge, heuristic knowledge, procedural knowledge and declarative knowledge. The document then discusses first order logic syntax including logical symbols, terms, formulas, quantifiers and predicates. It also discusses semantics and the uses and history of first order logic.
The document discusses various concepts in predicate logic including:
1. Universal and existential quantification allow representing statements like "for all" or "there exists".
2. Syntax of first-order logic includes constants, variables, functions, predicates, and quantifiers.
3. A predicate is satisfiable if true for some values, valid if true for all values, and unsatisfiable if false for all values.
4. Negating quantifiers flips the quantifier and negates the predicate. Free variables can be substituted while bound variables cannot. Restrictions filter domains.
1) Logic and inferences are important aspects of artificial intelligence as they allow systems to think and act rationally by making decisions based on available information and drawing conclusions.
2) Inference is the process of generating conclusions from facts and evidence. Formal logic represents knowledge through logical sentences using propositional or first-order logic.
3) Propositional logic uses symbolic variables to represent propositions that can be either true or false. Compound propositions combine simpler propositions using logical connectives like "and" and "or". Truth tables define the values of logical connectives.
Reasoning is the process of deriving logical conclusions from facts or premises. There are several types of reasoning including deductive, inductive, abductive, analogical, and formal reasoning. Reasoning is a core component of artificial intelligence as AI systems must be able to reason about what they know to solve problems and draw new inferences. Formal logic provides the foundation for building reasoning systems through symbolic representations and inference rules.
This document outlines a presentation on knowledge representation. It begins with an introduction to propositional logic, including its syntax, semantics, and properties. Several inference methods for propositional logic are discussed, including truth tables, deductive systems, and resolution. Predicate logic and semantic networks are also mentioned as topics to be covered. The overall document provides an outline of the key concepts to be presented on knowledge representation using logic.
Knowledge Based Reasoning: Agents, Facets of Knowledge. Logic and Inferences: Formal Logic,
Propositional and First Order Logic, Resolution in Propositional and First Order Logic, Deductive
Retrieval, Backward Chaining, Second order Logic. Knowledge Representation: Conceptual
Dependency, Frames, Semantic nets.
The document discusses machine learning algorithms for learning first-order logic rules from examples. It introduces the FOIL algorithm, which extends propositional rule learning algorithms like sequential covering to learn more general first-order Horn clauses. FOIL uses an iterative specialization approach, starting with a general rule and greedily adding literals to better fit examples while avoiding negative examples. The document also discusses how combining inductive learning with analytical domain knowledge can improve learning, such as in the KBANN approach where a neural network is initialized based on the domain theory before training on examples.
The document summarizes key concepts in propositional logic including:
1. Propositions are declarative sentences that are either true or false. Logical operators like negation, conjunction, disjunction, implication and biconditional are used to combine propositions.
2. Truth tables are used to determine the truth value of compound propositions based on the truth values of the individual propositions.
3. De Morgan's laws provide equivalences for negating conjunctions and disjunctions.
- The document discusses whether the truth predicate Tr in Friedman-Sheard's truth theory FS can be considered a logical connective.
- It raises the problem that Tr violates the "HARMONY" between its introduction and elimination rules, as FS is ω-inconsistent based on McGee's theorem.
- The source of the problem is that deflationism allows asserting infinite conjunctions of sentences using Tr, while also insisting that Tr not involve ontological changes, as logical connectives do not.
Classical logic has a serious limitation in that it cannot cope with the issues of vagueness and uncertainty
into which fall most modes of human reasoning. In order to provide a foundation for human knowledge
representation and reasoning in the presence of vagueness, imprecision, and uncertainty, fuzzy logic
should have the ability to deal with linguistic hedges, which play a very important role in the modification
of fuzzy predicates. In this paper, we extend fuzzy logic in narrow sense with graded syntax, introduced by
Nova´k et al., with many hedge connectives. In one case, each hedge does not have any dual one. In the
other case, each hedge can have its own dual one. The resulting logics are shown to also have the Pavelkastyle
completeness.
This document discusses knowledge representation in artificial intelligence. It covers various techniques for knowledge representation including logical representation using propositional logic and first-order predicate logic, semantic network representation, frame representation, and production rules. It also discusses issues in knowledge representation such as representing important attributes, relationships, and granularity of knowledge. Propositional logic is introduced as the simplest form of logic where statements are represented by propositions that can be either true or false. The syntax and semantics of propositional logic are also covered.
This document discusses the natural language inference system TIL and how it relates to other logical systems like natural logic and fragments of first-order logic. TIL uses concepts, contexts, and roles to represent the meaning of sentences. Contexts are used to model phenomena like negation and propositional attitudes. TIL can handle certain syllogisms and fragments of language, but more work is needed to fully evaluate it against systems like those studied by Moss and Pratt-Hartmann. Further developing TIL's ability to model language constructs like relational syllogisms and sentential negation is suggested as future work.
This document discusses symbolic reasoning under uncertainty. It introduces monotonic reasoning, where conclusions remain valid even when new information is added, and non-monotonic reasoning, where conclusions can be invalidated by new information. For non-monotonic reasoning, it provides an example where concluding a bird can fly is invalidated by learning the bird is a penguin. The document is presented by Prof. Khushali B Kathiriya and outlines introduction to monotonic reasoning, introduction to non-monotonic reasoning, and an example of non-monotonic reasoning logic.
1. The document discusses three computational modes of the mind: ordinary thought (conscious classical computation and unconscious quantum computation), and meta-thought (unconscious and non-algorithmic).
2. It proposes a quantum meta-language (QML) and probabilistic identity axiom to describe aspects of human reasoning and the disintegration of the self in conditions like schizophrenia.
3. The document introduces the concept of quantum coherent states of the mind, drawing an analogy to coherent states in quantum field theory which are eigenstates of the annihilation operator.
This document provides an introduction to knowledge representation in artificial intelligence. It discusses how knowledge representation and reasoning forms the basis of intelligent behavior through computational means. The key types of knowledge that need to be represented are defined, including objects, events, facts, and meta-knowledge. Different types of knowledge such as declarative, procedural, structural and heuristic knowledge are explained. The importance of knowledge representation for modeling intelligent behavior in agents is highlighted. The requirements for effective knowledge representation including representational adequacy, inferential adequacy, inferential efficiency, and acquisitional efficiency are outlined. Propositional logic is introduced as the simplest form of logic using propositions.
Knowledge representation is a field of artificial intelligence that represents information about the world in a way that a computer system can understand to perform complex tasks. It simplifies complex systems through modeling human psychology and problem-solving. Examples of knowledge representation include semantic nets, frames, rules, and ontologies. Knowledge representation allows for automated reasoning about represented knowledge and asserting new knowledge. While first-order logic provides powerful and compact representation, it lacks ease of use and practical implementation for real-world problems. Effective knowledge representation requires balancing expressive power with practical considerations like execution efficiency.
This document discusses weak slot and filler structures in artificial intelligence. It describes semantic net representation, which represents knowledge as a graphical network of nodes and arcs. It provides examples of representing statements about a cat named Jerry in a semantic net. The document also discusses frame representation, which organizes knowledge into structured records called frames that contain slots and slot values. An example frame is provided for a person named Ram. Advantages and disadvantages of both semantic nets and frames are outlined.
The document discusses constructive description logics and provides three options for constructing description logics constructively:
1) Translating description logic syntax into intuitionistic first-order logic (IFOL) to obtain the logic IALC.
2) Translating description logic syntax into intuitionistic modal logic (IK) to obtain the logic iALC.
3) Translating description logic syntax into constructive modal logic (CK) to obtain the logic cALC.
The talk outlines the translation approaches and discusses some pros and cons of the different constructive description logics, but notes that the work is preliminary and more criteria are needed to identify the best constructive system(s).
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.
Artificial intelligence and knowledge representationSajan Sahu
The document discusses artificial intelligence and knowledge representation. It describes how computers can be made intelligent through speed of computation, filtering responses, using algorithms and neural networks. It also discusses knowledge representation techniques in AI like propositional logic, semantic networks, frames, predicate logic and nonmonotonic reasoning. The document provides examples and applications of AI like pattern recognition, robotics and natural language processing. It also discusses some fundamental problems of AI.
This document discusses logical inference in recognizing textual entailment (RTE). It covers topics such as logics, formal languages, semantics, proof theories, propositional and first-order resolution, unification, and paramodulation. The document is authored by Kilian Evang and outlines these topics for applying logical inference to RTE.
The document discusses various concepts in predicate logic including:
1. Universal and existential quantification allow representing statements like "for all" or "there exists".
2. Syntax of first-order logic includes constants, variables, functions, predicates, and quantifiers.
3. A predicate is satisfiable if true for some values, valid if true for all values, and unsatisfiable if false for all values.
4. Negating quantifiers flips the quantifier and negates the predicate. Free variables can be substituted while bound variables cannot. Restrictions filter domains.
1) Logic and inferences are important aspects of artificial intelligence as they allow systems to think and act rationally by making decisions based on available information and drawing conclusions.
2) Inference is the process of generating conclusions from facts and evidence. Formal logic represents knowledge through logical sentences using propositional or first-order logic.
3) Propositional logic uses symbolic variables to represent propositions that can be either true or false. Compound propositions combine simpler propositions using logical connectives like "and" and "or". Truth tables define the values of logical connectives.
Reasoning is the process of deriving logical conclusions from facts or premises. There are several types of reasoning including deductive, inductive, abductive, analogical, and formal reasoning. Reasoning is a core component of artificial intelligence as AI systems must be able to reason about what they know to solve problems and draw new inferences. Formal logic provides the foundation for building reasoning systems through symbolic representations and inference rules.
This document outlines a presentation on knowledge representation. It begins with an introduction to propositional logic, including its syntax, semantics, and properties. Several inference methods for propositional logic are discussed, including truth tables, deductive systems, and resolution. Predicate logic and semantic networks are also mentioned as topics to be covered. The overall document provides an outline of the key concepts to be presented on knowledge representation using logic.
Knowledge Based Reasoning: Agents, Facets of Knowledge. Logic and Inferences: Formal Logic,
Propositional and First Order Logic, Resolution in Propositional and First Order Logic, Deductive
Retrieval, Backward Chaining, Second order Logic. Knowledge Representation: Conceptual
Dependency, Frames, Semantic nets.
The document discusses machine learning algorithms for learning first-order logic rules from examples. It introduces the FOIL algorithm, which extends propositional rule learning algorithms like sequential covering to learn more general first-order Horn clauses. FOIL uses an iterative specialization approach, starting with a general rule and greedily adding literals to better fit examples while avoiding negative examples. The document also discusses how combining inductive learning with analytical domain knowledge can improve learning, such as in the KBANN approach where a neural network is initialized based on the domain theory before training on examples.
The document summarizes key concepts in propositional logic including:
1. Propositions are declarative sentences that are either true or false. Logical operators like negation, conjunction, disjunction, implication and biconditional are used to combine propositions.
2. Truth tables are used to determine the truth value of compound propositions based on the truth values of the individual propositions.
3. De Morgan's laws provide equivalences for negating conjunctions and disjunctions.
- The document discusses whether the truth predicate Tr in Friedman-Sheard's truth theory FS can be considered a logical connective.
- It raises the problem that Tr violates the "HARMONY" between its introduction and elimination rules, as FS is ω-inconsistent based on McGee's theorem.
- The source of the problem is that deflationism allows asserting infinite conjunctions of sentences using Tr, while also insisting that Tr not involve ontological changes, as logical connectives do not.
Classical logic has a serious limitation in that it cannot cope with the issues of vagueness and uncertainty
into which fall most modes of human reasoning. In order to provide a foundation for human knowledge
representation and reasoning in the presence of vagueness, imprecision, and uncertainty, fuzzy logic
should have the ability to deal with linguistic hedges, which play a very important role in the modification
of fuzzy predicates. In this paper, we extend fuzzy logic in narrow sense with graded syntax, introduced by
Nova´k et al., with many hedge connectives. In one case, each hedge does not have any dual one. In the
other case, each hedge can have its own dual one. The resulting logics are shown to also have the Pavelkastyle
completeness.
This document discusses knowledge representation in artificial intelligence. It covers various techniques for knowledge representation including logical representation using propositional logic and first-order predicate logic, semantic network representation, frame representation, and production rules. It also discusses issues in knowledge representation such as representing important attributes, relationships, and granularity of knowledge. Propositional logic is introduced as the simplest form of logic where statements are represented by propositions that can be either true or false. The syntax and semantics of propositional logic are also covered.
This document discusses the natural language inference system TIL and how it relates to other logical systems like natural logic and fragments of first-order logic. TIL uses concepts, contexts, and roles to represent the meaning of sentences. Contexts are used to model phenomena like negation and propositional attitudes. TIL can handle certain syllogisms and fragments of language, but more work is needed to fully evaluate it against systems like those studied by Moss and Pratt-Hartmann. Further developing TIL's ability to model language constructs like relational syllogisms and sentential negation is suggested as future work.
This document discusses symbolic reasoning under uncertainty. It introduces monotonic reasoning, where conclusions remain valid even when new information is added, and non-monotonic reasoning, where conclusions can be invalidated by new information. For non-monotonic reasoning, it provides an example where concluding a bird can fly is invalidated by learning the bird is a penguin. The document is presented by Prof. Khushali B Kathiriya and outlines introduction to monotonic reasoning, introduction to non-monotonic reasoning, and an example of non-monotonic reasoning logic.
1. The document discusses three computational modes of the mind: ordinary thought (conscious classical computation and unconscious quantum computation), and meta-thought (unconscious and non-algorithmic).
2. It proposes a quantum meta-language (QML) and probabilistic identity axiom to describe aspects of human reasoning and the disintegration of the self in conditions like schizophrenia.
3. The document introduces the concept of quantum coherent states of the mind, drawing an analogy to coherent states in quantum field theory which are eigenstates of the annihilation operator.
This document provides an introduction to knowledge representation in artificial intelligence. It discusses how knowledge representation and reasoning forms the basis of intelligent behavior through computational means. The key types of knowledge that need to be represented are defined, including objects, events, facts, and meta-knowledge. Different types of knowledge such as declarative, procedural, structural and heuristic knowledge are explained. The importance of knowledge representation for modeling intelligent behavior in agents is highlighted. The requirements for effective knowledge representation including representational adequacy, inferential adequacy, inferential efficiency, and acquisitional efficiency are outlined. Propositional logic is introduced as the simplest form of logic using propositions.
Knowledge representation is a field of artificial intelligence that represents information about the world in a way that a computer system can understand to perform complex tasks. It simplifies complex systems through modeling human psychology and problem-solving. Examples of knowledge representation include semantic nets, frames, rules, and ontologies. Knowledge representation allows for automated reasoning about represented knowledge and asserting new knowledge. While first-order logic provides powerful and compact representation, it lacks ease of use and practical implementation for real-world problems. Effective knowledge representation requires balancing expressive power with practical considerations like execution efficiency.
This document discusses weak slot and filler structures in artificial intelligence. It describes semantic net representation, which represents knowledge as a graphical network of nodes and arcs. It provides examples of representing statements about a cat named Jerry in a semantic net. The document also discusses frame representation, which organizes knowledge into structured records called frames that contain slots and slot values. An example frame is provided for a person named Ram. Advantages and disadvantages of both semantic nets and frames are outlined.
The document discusses constructive description logics and provides three options for constructing description logics constructively:
1) Translating description logic syntax into intuitionistic first-order logic (IFOL) to obtain the logic IALC.
2) Translating description logic syntax into intuitionistic modal logic (IK) to obtain the logic iALC.
3) Translating description logic syntax into constructive modal logic (CK) to obtain the logic cALC.
The talk outlines the translation approaches and discusses some pros and cons of the different constructive description logics, but notes that the work is preliminary and more criteria are needed to identify the best constructive system(s).
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.
Artificial intelligence and knowledge representationSajan Sahu
The document discusses artificial intelligence and knowledge representation. It describes how computers can be made intelligent through speed of computation, filtering responses, using algorithms and neural networks. It also discusses knowledge representation techniques in AI like propositional logic, semantic networks, frames, predicate logic and nonmonotonic reasoning. The document provides examples and applications of AI like pattern recognition, robotics and natural language processing. It also discusses some fundamental problems of AI.
This document discusses logical inference in recognizing textual entailment (RTE). It covers topics such as logics, formal languages, semantics, proof theories, propositional and first-order resolution, unification, and paramodulation. The document is authored by Kilian Evang and outlines these topics for applying logical inference to RTE.
The document discusses knowledge representation and reasoning in artificial intelligence. It covers the following key points in 3 sentences:
Intelligent agents should have the capacity for perceiving, representing knowledge, reasoning about what they know, and acting. Knowledge representation involves representing an understanding of the world, while reasoning involves inferring implications of what is known. Logic provides a way to represent and reason about knowledge through specifying a logical language with syntax, semantics, and inference rules.
The document discusses knowledge representation using propositional logic and predicate logic. It begins by explaining the syntax and semantics of propositional logic for representing problems as logical theorems to prove. Predicate logic is then introduced as being more versatile than propositional logic for representing knowledge, as it allows quantifiers and relations between objects. Examples are provided to demonstrate how predicate logic can formally represent statements involving universal and existential quantification.
First-order logic (FOL) is an extension of propositional logic that allows for quantification over objects, properties, and relations. FOL uses constants, variables, predicates, functions, connectives, equality, and quantifiers as its basic elements. It can represent statements involving objects and their relationships in a concise way, overcoming limitations of propositional logic. FOL uses universal and existential quantifiers to make statements about all or some objects respectively.
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.
Introduction to logic and prolog - Part 1Sabu Francis
The document provides an introduction to logic and Prolog programming. It discusses:
1) Alan Turing's invention of the modern computer to solve complex problems like decoding encrypted messages. This established the concept of algorithms being carried out through linear instruction processing.
2) Prolog programming focuses solely on logic and removes concerns about procedural elements like instruction pointers. It allows programmers to focus only on the problem's logic.
3) Logic is a tool for reasoning that uses concepts like true, false, if-then statements, and, or, etc. It helps clarify reasoning but cannot validate conclusions on its own if premises are flawed.
Here are the proofs using mathematical induction for the two assignments:
1.3
Base Case: When n = 1, LHS = 12 = 1(1+1)(2(1)+1)/6 = RHS.
Inductive Hypothesis: Assume the formula holds for n = k.
LHS = 12 + 22 + ... + k2 = k(k+1)(2k+1)/6
Inductive Step: For n = k + 1,
LHS = 12 + 22 + ... + k2 + (k+1)2
= k(k+1)(2k+1)/6 + (k+1)2
= (k
The document discusses a proposed proof by Vinay Deolalikar that P ≠ NP, the famous open problem in computer science. The proof strategy involved showing that satisfiability problems like random k-SAT would have "simple structure" if in P, but some instances do not. However, the proof was found to have flaws, as the "simple structure" property still holds even for problems in P. Multiple objections and counterexamples were found within a week through open online discussion, suggesting the proof is likely unsalvageable. The rapid online peer review process provided both benefits and costs to rigorously evaluating the proposed proof.
This document discusses different techniques for knowledge representation in artificial intelligence, including logical representation, semantic networks, frames, and production rules. It focuses on logical representation using propositional logic and first-order logic. Propositional logic uses atomic and compound propositions connected by logical operators like negation, conjunction, disjunction, implication, and biconditional. First-order logic extends propositional logic by adding quantifiers and predicates to represent objects and relations. Knowledge representation enables AI systems to understand and utilize knowledge to solve complex problems.
This document discusses representing knowledge in predicate logic and artificial intelligence. It explains how predicate logic can represent objects, relationships, and quantification which allows more complex knowledge representation compared to propositional logic. The document provides examples of representing facts about individuals like Socrates and classes in predicate logic using predicates like man and mortal. It also discusses representing class-instance relationships and property inheritance using the predicates "instance" and "isa".
This document provides an introduction to logic and critical thinking. It defines logic as the study of correct reasoning and fallacies as incorrect arguments. It discusses the history of logic according to various philosophers like Aristotle. It also defines key logical terms like quality, quantity, and immediate inference. Immediate inferences include conversion, obversion, and contraposition. The document provides examples of each. It also discusses different types of fallacies including fallacies of relevance, presumption, and ambiguity. Fallacies of relevance aim to appear relevant but are not, such as appeals to emotion.
Logic provides tools for analyzing reasoning and arguments through symbolic representations of propositions. A proposition is a declarative sentence that is either true or false. Logical operators such as negation, conjunction, disjunction, implication, biconditional, NAND, and NOR allow building compound propositions from simpler ones. Truth tables systematically represent the relationships between the truth values of simple and compound propositions. Key logical concepts include logical equivalence, contradiction, tautology, and logical validity.
The document discusses logic and propositions. It begins by defining a proposition as a statement that is either true or false. It then provides examples of propositions and non-propositions. The document also discusses arguments and their validity. An argument is valid if the premises guarantee the conclusion. It discusses logical operators like conjunction, disjunction, negation and implication. Truth tables are used to determine the truth values of compound propositions formed using logical operators. Laws of algebra are also discussed for propositional logic.
This book is written by LOIBANGUTI, BM, it is just an online copy provided for free. No part of this book mya be republished. but can be used and stored as a softcopy book, can be shared accordingly.
Knowledge-based agents can accept new tasks in the form of explicitly described goals and adapt to changes in their environment by updating relevant knowledge. They maintain a knowledge base of facts about the environment and use an inference engine to deduce new information and determine what actions to take. The knowledge base stores sentences expressed in a knowledge representation language and the inference engine applies logical rules to deduce new facts or answer queries. Propositional logic is often used to represent knowledge, where sentences consist of proposition symbols connected by logical connectives like AND, OR, and NOT.
Lean Logic for Lean Times: Varieties of Natural LogicValeria de Paiva
This document discusses using logic to analyze natural language text. It proposes a Knowledge Inference Management Language (KIML) that represents text as concepts, roles, and contexts. KIML aims to model quantification, propositional attitudes, and inference in a way that corresponds to natural language semantics. The document also discusses using contextual constructive description logics and connexive logic to model textual entailment relationships.
Slides for a Course Based on the Text Discrete Mathematics & Its Applications (5th Edition) by Kenneth H. Rosen
Slides for a Course Based on the Text Discrete Mathematics & Its Applications (5th Edition) by Kenneth H. Rosen
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.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Section1-1
1. . Preface
目標 這門課應該教導同學如何用邏輯與數學
來思考 (how to think logically and
mathematically)。
內容 包括五個部份:mathematical
reasoning(數學推理)、combinatorial
analysis(組合分析)、discrete
structure(離散結構)、algorithmic
thinking(演算法的思考)、applications
and modeling(應用與模型)
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 1 / 41
2. . 教材
課本 Discrete Mathematics and its
Applications(sixth edition), Kenneth H.
Rosen
參考書籍 離散數學 (Discrete Mathematics and its
Applications 中譯本) sixth edition, 謝良
瑜陳志賢譯
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 2 / 41
3. . 自我介紹
姓名 洪春男
email spring@mail.dyu.edu.tw
電話 04-8511888 轉 2410
辦公室 工學院 H311
Homepage http://www.dyu.edu.tw/ spring
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 3 / 41
4. . 評分標準
期中考 20%
期末考 30%
平常分數 50%(點名與隨堂測驗、作業、平常考
大約各佔 1/3)
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 4 / 41
5. . Contents
1. The Foundations: Logic and Proofs(1.1-1.7)
.
2 Basic Structures: Sets, Functions, Sequences, and
Sums(2.1-2.4)
3. The Fundamentals: Algorithms, the Integers, and
Matrices(3.1-3.5, 3.8)
4. Induction and Recursion(4.1-4.3)
.
5 Counting(5.1-5.3)
6. Discrete Probability(6.1)
.
7 Advanced Counting Techniques(7.1, 7.5)
.
8 Relations(8.1, 8.3, 8.5)
9. Graphs(9.1-9.5)
.
10 Trees(10.1)
. Boolean Algebra
11
.
12 Modeling Computation
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 5 / 41
6. . 1. The Foundations: Logic and Proofs
Logic is the basis of all mathematical reasoning,
and of all automated reasoning. 邏輯是所有數
學推理與自動推理的基礎。
To understand mathematics, we must
understand what makes up a correct
mathematical argument, that is, a proof. 要了
解數學,必須了解建構正確的數學論證,也
就是證明。
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 6 / 41
7. . 1.1 Propositional Logic 命題邏輯
A proposition is a declarative sentence(that is,
a sentence that declares a fact) that is either
true or false, but not both. 命題是一個述句
(宣告事實的句子),它可能是真、也可能是
假,但不能旣真又假。
.
Example 1
.
1. Washington, D.C., is the capital of the United
States of America. 華盛頓特區是美國首都。
2. Toronto is the capital of Canada. 多倫多是加
拿大首都。
3. 1 + 1 = 2.
4. 2 + 2 = 3.
. 洪春男 . .
1. The Foundations: Logic and Proofs
. . .
March 1, 2011
.
7 / 41
8. . 1.1 Propositional Logic 命題邏輯
.
Example 2
.
下列是錯誤的 propositions
1. What time is it? 現在幾點?
2. Read this carefully. 小心閱讀。
3. x + 1 = 2.
.
. x + y = z.
4
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 8 / 41
9. . 1.1 Propositional Logic 命題邏輯
propositional variables 命題變數, p, q, r, s, · · ·
truth value: T(真)、F(假)
The area of logic that deals with propositions is
called the propositional calculus or
propositional logic. 專門處理命題的邏輯稱
為命題演算或命題邏輯,亞里斯多德
(Aristotle) 最早開始使用。
New propositions, called compound
propositions, are formed from existing
propositions using logical operators. 由已存在
的命題加上邏輯運算子形成新的命題,稱為
複合命題。 . . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 9 / 41
10. . 1.1 Propositional Logic 命題邏輯
.
Definition 1
.
Let p be a proposition. The negation of p,
denoted by ¬p(also denoted by p), is the statement
“It is not the case that p.”
The proposition ¬p is read “not p”. The truth value
of the negation of p, ¬p, is the opposite of the
truth value of p.
令 p 為一命題, p 的否定句為「p 不成立」 ,以
¬p 表示 (有時也用 p 表示)。 ¬p 讀作「非 p」 ,
其真假值與 p 的真假值剛好相反。
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 10 / 41
11. . 1.1 Propositional Logic 命題邏輯
.
Example 3
.
Find the negation of the proposition
“Today is Friday.”
and express this in simple English.
找出「今天是星期五」的否定命題,且用簡單的
英文表示。
“It is not the case that today is Friday.”
“Today is not Friday.”
“It is not Friday today.”
「今天是星期五不成立」或「今天不是星期五」
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 11 / 41
12. . 1.1 Propositional Logic 命題邏輯
.
Example 4
.
Find the negation of the proposition
“At least 10 inches of rain fell today in Miami.”
and express this in simple English.
找出「邁阿密今天至少下 10 英吋的雨」的否定
命題,且用簡單的英文表示。
“It is not the case that at least 10 inches of rain fell
today in Miami.”
“Less than 10 inches of rain fell today in Miami.”
「邁阿密今天至少下 10 英吋的雨不成立」或
「邁阿密今天下不到 10 英吋的雨」
. 。
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 12 / 41
13. . 1.1 Propositional Logic 命題邏輯
.
Definition 2
.
Let p and q be propositions. The conjunction of p
and q, denoted by p ∧ q, is the proposition “p and
q.”. The conjunction p ∧ q is true when both p and
q are true and is false otherwise.
令 p 與 q 都是命題, p 與 q 同時發生為「p 和
q」 ,記成 p ∧ q,當 p 與 q 都是真時 p ∧ q 為真,
否則為假。
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 13 / 41
14. . 1.1 Propositional Logic 命題邏輯
.
Example 5
.
Find the conjunction of the propositions p and q
where p is the proposition “Today is Friday” and q
is the proposition “It is raining today”.
令命題 p 為「今天是星期五」 ,命題 q 為「今天
下雨」 ,請找出 p 與 q 的 conjunction。
“Today is Friday and it is raining today.”
「今天是星期五且下雨」
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 14 / 41
15. . 1.1 Propositional Logic 命題邏輯
.
Definition 3
.
Let p and q be propositions. The disjunction of p
and q, denoted by p ∨ q, is the proposition “p or q.”.
The disjunction p ∨ q is false when both p and q are
false and is true otherwise.
令 p 與 q 都是命題, p 與 q 的分裂為「p 或
q」 ,記成 p ∨ q,當 p 與 q 都是假時 p ∨ q 為假,
否則為真。
.
or 有 inclusive 與 exclusive 的分別, ∨ 是
inclusive。
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 15 / 41
16. . 1.1 Propositional Logic 命題邏輯
.
Example 6
.
What is the disjunction of the propositions p and q
where p and q are the same propositions as in
Example 5.
令命題 p 為「今天是星期五」 ,命題 q 為「今天
下雨」 ,請問 p 與 q 的 disjunction。
“Today is Friday or it is raining today.”
「今天是星期五或今天下雨」
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 16 / 41
17. . 1.1 Propositional Logic 命題邏輯
.
Definition 4
.
Let p and q be propositions. The exclusive or of p
and q, denoted by p ⊕ q, is the proposition that is
true when exactly one of p and q is true and is false
otherwise.
令 p 與 q 都是命題, p 與 q 的互斥或,記成
p ⊕ q,當 p 與 q 恰為一真一假時 p ⊕ q 為真,否
則為假。
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 17 / 41
18. . 1.1 Propositional Logic 命題邏輯
The truth table 真值表
p q ¬p p ∧ q p ∨ q p ⊕ q
T T F T T F
T F F F T T
F T T F T T
F F T F F F
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 18 / 41
19. . 1.1 Propositional Logic 命題邏輯
.
Definition 5
.
Let p and q be propositions. The conditional
statement p → q is the proposition “if p, then q.”
The conditional statement p → q is false when p is
true and q is false, and true otherwise. In the
conditional statement p → q, p is called the
hypothesis(or antecedent or premise) and q is
called the conclusion(or consequence).
令 p 與 q 都是命題,條件句 p → q 代表「若 p
則 q」的命題。當 p 真 q 假時,條件句 p → q 為
假,否則為真。其中 p 稱為假設 (或前提)、而 q
稱為結論。
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 19 / 41
20. . 1.1 Propositional Logic 命題邏輯
A conditional statement is also called an
implication. 條件句有時也稱為隱涵。
下列都是「若 p 則 q」的寫法: p, then 「if
q」 「p implies q」 「if p, q」, 「p only if q」,
、 、
「p is sufficient for q」, 「a sufficient condition
for q is p」, 「q if p」, 「q whenever p」, 「q
when p」 「q is necessary for p」 「a necessary
、 、
condition for p is q」 「q follows from p」 「q
、 、
unless ¬p」
“If I am elected, then I will lower taxes.” 若我當
選就減稅。
“If you get 100% on the final, then you will get
an A.” 期末考 100 分就得 A。 . . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 20 / 41
21. . 1.1 Propositional Logic 命題邏輯
.
Example 7
.
Let p be the statement “Maria learns discrete
mathematics” and q the statement “Maria will find
a good job.” Express the statement p → q as a
statement in English.
令 p 是「瑪麗亞學離散數學」 q 為「瑪麗亞將 ,
找到好工作」 ,請用英文表達 p → q。
“If Maria learns discrete mathematics, then she will
find a good job.”,「若瑪麗亞學離散數學,她將
找到好工作」 ,“Maria will find a good job when
she learns discrete mathematics.”,“Maria will find
a good job unless she does not learn discrete
mathemathics.”
. . . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 21 / 41
22. . 1.1 Propositional Logic 命題邏輯
“If it is sunny today, then we will go to the
beach.” 若今天出太陽,我們將去海邊玩。
“If today is Friday, then 2 + 3 = 5.” 若今天是
星期五,則 2 + 3 = 5。
“If today is Friday, then 2 + 3 = 5.” 若今天是
星期五,則 2 + 3 = 6。
前題與結果未必需要有因果關係。
.
Example 8
.
if 2 + 2 = 4 then x := x + 1.
若在這個 statement 之前 x = 0 的話,執行之後 x
的值是多少?
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 22 / 41
23. . 1.1 Propositional Logic 命題邏輯
The proposition q → p is called the converse
(相反) of p → q.
The contrapositive (對換) of p → q is the
proposition ¬q → ¬p.
The proposition ¬p → ¬q is called the inverse
(相反) of p → q.
When two compound propositions always have
the same truth value we call them equivalent.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 23 / 41
24. . 1.1 Propositional Logic 命題邏輯
.
Example 9
.
What are the contrapositive, the converse, and the
inverse of the conditional statement
“The home team wins whenever it is raining.”? 每
當下雨時地主隊獲勝。
contrapositive “If the home team doesn’t win, then
it is not raining.” 若地主隊沒贏就沒有下雨。
converse “If the home team wins, then it is raining.”
若地主隊贏就下雨
inverse “If it is not raining, then the home team
doesn’t win.” 若沒下雨地主隊就沒贏。
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 24 / 41
25. . 1.1 Propositional Logic 命題邏輯
.
Definition 6
.
Let p and q be propositions. The biconditional
statement p ↔ q is the proposition “p if and only
if q.” The biconditional statement p ↔ q is true
when p and q have the same truth values, and is
false otherwise. Biconditional statements are also
called bi-implications.
令 p 與 q 都是命題,雙條件句 p ↔ q 代表「p 若
且唯若 q」的命題。當 p 與 q 有相同真假值時,
雙條件句 p ↔ q 為真,否則為假。雙條件句又稱
為雙蘊涵。
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 25 / 41
26. . 1.1 Propositional Logic 命題邏輯
當 p → q 與 q → p 都是 true 時 p ↔ q 才為
true。
「p is necessary and sufficient for q」 「if p
、
then q, and conversely」 「p iff q」都是 p ↔ q
、
的意思。iff 是 if and only if 的縮寫。
.
Example 10
.
Let p be the statement “You can take the flight”
and let q be the statement “You buy a ticket.”
Then p ↔ q is the statement
“You can take the flight if and only if you buy a
ticket.”
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 26 / 41
27. . 1.1 Propositional Logic 命題邏輯
The truth table 真值表
p q p → q q → p ¬q → ¬p ¬p → ¬q p ↔ q
T T T T T T T
T F F T F T F
F T T F T F F
F F T T T T T
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 27 / 41
28. . 1.1 Propositional Logic 命題邏輯
.
Example 11
.
Construct the truth table of the compound
proposition
(p ∨ ¬q) → (p ∧ q).
.
p q ¬q p ∨ ¬q p ∧ q (p ∨ ¬q) → (p ∧ q)
T T F T T T
T F T T F F
F T F F F T
F F T T F F
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 28 / 41
30. . 1.1 Propositional Logic 命題邏輯
.
Example 12
.
How can this English sentence be translated into a
logical expression?
“You can access the Internet from campus only if
you are a computer science major or you are not a
freshman.”
只有當你主修電腦或不是新鮮人,才能在校園中
使用網路
a 代表 “You can access the Internet from campus.”
c 代表 “You are a computer science major.”
f 代表 “You are a freshman.”
前面的句字可翻譯為 a → (c ∨ ¬f)
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 30 / 41
31. . 1.1 Propositional Logic 命題邏輯
.
Example 13
.
How can this English sentence be translated into a
logical expression?
“You cannot ride the roller coaster if you are under
4 feet tall unless you are older than 16 years old.”
若你不到 4 英呎高就不能坐雲霄飛車,除非你超
過 16 歲。
q 代表 “You can ride the roller coaster.”
r 代表 “You are under 4 feet tall.”
s 代表 “You are older than 16 years old.”
前面的句字可翻譯為 (r ∧ ¬s) → ¬q
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 31 / 41
32. . 1.1 Propositional Logic 命題邏輯
.
Example 14
.
Express the specification “The automated reply
cannot be sent when the file system is full” using
logical connectives. 使用邏輯連詞表達下列規
定: 「當檔案系統滿了,自動回覆功能不能被送
出」 。
p 代表 “The automated reply can be sent.”
q 代表 “The file system is full.”
前面的句字可翻譯為 q → ¬p
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 32 / 41
33. . 1.1 Propositional Logic 命題邏輯
.
Example 15
.
Determine whether these system specifications are consistent:
“The diagnostic message is stored in the buffer or it is
retransmitted.”
“The diagnostic message is not stored in the buffer.”
“If the diagnostic message is stored in the buffer, then it is
retransmitted.”
p 代表 “The diagnostic message is stored in the buffer.”
q 代表 “The diagnostic message is retransmitted.”
前面三個句字為 p ∨ q, ¬p, p → q,當 p 為 F 而 q 為 T
時,三個句子都成立,因此 consistent。
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 33 / 41
34. . 1.1 Propositional Logic 命題邏輯
.
Example 16
.
Do the system specifications in Example 15 remain
consistent if the specification “The diagnostic
message is not retransmitted” is added?
p 代表 “The diagnostic message is stored in the
buffer.”
q 代表 “The diagnostic message is retransmitted.”
四個句字為 p ∨ q, ¬p, p → q, ¬q,顯然無法使四
個句子都為 true,因此不 consistent。
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 34 / 41
35. . 1.1 Propositional Logic 命題邏輯 - boolean searches
可在搜尋引擎中輸入下列關鍵字,看看結果如何
.
Example 17
.
New and Mexico and University「New Mexico
University」
(New and Mexico or Arizona) and University
(Mexico and Universities) not New 「Mexico
Universities -New」
大葉 -高島屋
大葉資訊 -資管 -會計
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 35 / 41
36. . 1.1 Propositional Logic 命題邏輯 - Logic Puzzles
.
Example 18
.
An island that has two kinds of inhabitants(居民),
knights(騎士),who always tell the truth, and their
opposites, knaves(無賴), who always lie. You
encounter two people A and B. What are A and B
if A says “B is a knight” and B says “The two of us
are opposite types”?
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 36 / 41
37. . 1.1 Propositional Logic 命題邏輯 - Logic Puzzles
答案:Both A and B are knaves.
.
Example 19
.
兩個兄妹在後院玩,兩人的前額都沾了泥巴,父
親說: 「你們兩人中至少一人前額有泥巴。 」父親
問: 「你知道你自己的前額有沒有沾泥巴呢?」
父親問兩次,請問兩個小朋友會如何回答?假設
小朋友可看到另一人的前額、看不到自己的前
額,且都說實話,兩人只能回答 Yes / No。
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 37 / 41
38. 1.1 Propositional Logic 命題邏輯 - Logic and Bit
. Operations
答案:第一次都回答 NO,第二次都是 Yes。
A bit is a symbol with two possible values,
namely, 0(zero) and 1(one). 可代表 true(1) 與
false(0)。
A variable is called a Boolean variable if its
value is either true or false.
.
Definition 7
.
A bit string is a sequence of zero or more bits. The
length of this string is the number of bits in the
string.
.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 38 / 41
39. 1.1 Propositional Logic 命題邏輯 - Logic and Bit
. Operations
.
Example 20
.
101010011 is a bit string of length nine.
.
We define the bitwise OR, bitwise AND, and
bitwise XOR of two strings of the same length to
be the strings that have as their bits the OR, AND,
and XOR of the corresponding bits in the two
strings, respectively.
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 39 / 41
40. 1.1 Propositional Logic 命題邏輯 - Logic and Bit
. Operations
.
Example 21
.
Find the bitwise OR, bitwise AND, and bitwise XOR
of the bit strings 0110110110 and 1100011101.
0110110110
1100011101
bitwise OR 1110111111
bitwise AND 0100010100
. bitwise XOR 1010101011
. . . . . .
洪春男 1. The Foundations: Logic and Proofs March 1, 2011 40 / 41
41. 謝謝大家的聆聽!
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洪春男 1. The Foundations: Logic and Proofs March 1, 2011 41 / 41