PHI Propositional Logic Lecture 2. Truth Tables
|
|
|
- Ernest Barton
- 6 years ago
- Views:
Transcription
1 PHI Propositional Logic Lecture 2 ruth ables
2 ruth ables Part 1 - ruth unctions for Logical Operators ruth unction - the truth-value of any compound proposition determined solely by the truth-value of its components. Statement Variable - a variable that represents any proposition (by convention we use lower-case letters p, q, r, s, etc.). ruth able - a calculation matrix used to demonstrate all logically possible truth-values of a given proposition. Let any statement be represented by p: p p
3 ruth ables Part 1 - ruth unctions for Logical Operators ruth unction - the truth-value of any compound proposition determined solely by the truth-value of its components. Statement Variable - any variable that represents a proposition (by convention we use lower-case letters p, q, r, s, etc.). ruth able - a calculation matrix used to demonstrate all logically possible truth-values of a given proposition. his is the negation of any proposition p. Negation ~ opposite truth values p ~ p
4 ruth ables Part 1 - ruth unctions for Logical Operators Conjunction p q p q
5 ruth ables Part 1 - ruth unctions for Logical Operators Conjunction p q p q number of lines equals 2 to the? power of the number of propositions L=2 n 2 2 = 4
6 ruth ables Part 1 - ruth unctions for Logical Operators Conjunction p q p q both are true
7 ruth ables Part 1 - ruth unctions for Logical Operators Disjunction p q p q at least one is true
8 ruth ables Part 1 - ruth unctions for Logical Operators Material Equivalence p q p q equivalent truth values
9 ruth ables Part 1 - ruth unctions for Logical Operators Material Implication p q p q in all cases except when the antecedent is true and the consequent is false
10 ruth ables Part 1 - ruth unctions for Logical Operators Material Implication in all cases except when the antecedent is true and the consequent is false If you get an A on the final exam, then you will pass Logic. (Getting an A on the final is sufficient to pass Logic.) I got an A on the final! Why did you fail me? If you get an A on the final exam, then you will pass Logic.
11 ruth ables Part 2 - ruth ables for Compound Propositions Method 1.a - if you already know the truth-values of the components (the long method) suppose A, B, and C are true, and D, E, and are false (A D) ( )
12 ruth ables Part 2 - ruth ables for Compound Propositions Method 1.b - if you already know the truth-values of the components (the short method) suppose A, B, and C are true, and D, E, and are false (A D)
13 ruth ables Part 3 - ruth ables for Compound Propositions Method 2 - if you do not know the actual truth value of the components, solve for all possible worlds A D (A D) line 1 line 2 line 3 line 4 L=2 3 line 5 line 6 line 7 line 8
14 Logical Status of Propositions I. hree types of Logical statements: A. Logically rue (tautologies) - always true B. Logically alse (self-contradiction) - always false C. Logically Contingent (truth-value dependent) - sometimes true/false
15 Logical Status of Propositions autology (logically true) p p ~ p every case is true
16 Logical Status of Propositions Self- Contradiction (logically false) p p ~ p every case is false
17 Logical Status of Propositions Contingent (logically dependent) p q p q variable truth-value
18 II. Logical Relations between two statements: A. Contradictory - opposite truth-values p q (p q) (p ~ q) Propositional Logic Logical Relationships Between Propositions
19 II. Logical Relations between statements: A. Contradictory - opposite truth-values B. Equivalent - identical truth-values p q (p q) :: (~q ~ p) Propositional Logic Logical Relationships Between Propositions
20 Logical Relationships Between Propositions II. Logical Relations between statements: A. Contradictory - opposite truth-values B. Equivalent - identical truth-values C. Consistent - at least one common (true) truth-value p q (p q) (p q)
21 Logical Relationships Between Propositions II. Logical Relations between statements: A. Contradictory - opposite truth-values B. Equivalent - identical truth-values C. Consistent - at least one common (true) truth-value D. Inconsistent - no common (true) truth-value p q (p q) (p ~ q)
22 Logical Relationships Between Propositions I. hree types of Logical statements: A. Logically rue (tautologies) - always true B. Logically alse (self-contradiction) - always false C. Logically Contingent (truth-value dependent) - sometimes true/false II. Logical Relations between statements: A. Contradictory - opposite truth-values B. Equivalent - identical truth-values C. Consistent - at least one common (true) truth-value D. Inconsistent - no common (true) truth-value
23 III. esting Arguments for Validity: A. Symbolize the argument Propositional Logic ruth ables or Arguments B. Separate each proposition (premises and conclusion) with slashes on a single one C. ill in truth values (just as you would for a compound proposition) D. Look for a line with true premises and false conclusion
24 ruth ables or Arguments III. esting Arguments for Validity: A. Symbolize B. Separate C. ill in D. Look If minors who commit murder are equally responsible for their crimes as adults, they should receive the death penalty. But they re not. So, minors should not receive the death penalty. M D M D / ~M // ~D Invalid (denying the antecedent)
25 III. esting Arguments for Validity: A.Symbolize B. Separate C. ill in D.Look Propositional Logic ruth ables or Arguments If minors who commit murder are equally responsible for their crimes as adults, they should receive the death penalty. And, they are. So, minors should receive the death penalty. M D M D / M // D Valid (modus ponens)
26 III. esting Arguments for Validity: A.Symbolize B. Separate C. ill in D.Look Propositional Logic ruth ables or Arguments If minors who commit murder are equally responsible for their crimes as adults, they should receive the death penalty. But it s not the case that they deserve the death penalty. So, minors are not equally responsible for their crimes. M D M D / ~ D // ~ M Valid (modus tollens)
27 III. esting Arguments for Validity: A.Symbolize B. Separate C. ill in D.Look Propositional Logic ruth ables or Arguments Either minors who commit murder are equally responsible for their crimes as adults, or they should receive the death penalty. But it s not the case that they are equally responsible. So, they should receive the death penalty. M D M D / ~ M // D Valid (disjunctive syllogism)
28 M D W M D / D W // M W If minors who commit murder are equally responsible for their crimes as adults, then they should receive the death penalty. If they should receive the death penalty, then we should feel sorry for them. So, if minors who commit murder are equally responsible, then we should feel sorry for them. Valid (hypothetical syllogism) Propositional Logic ruth ables or Arguments
2. The Logic of Compound Statements Summary. Aaron Tan August 2017
2. The Logic of Compound Statements Summary Aaron Tan 21 25 August 2017 1 2. The Logic of Compound Statements 2.1 Logical Form and Logical Equivalence Statements; Compound Statements; Statement Form (Propositional
Lecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook)
Lecture 2 Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits Reading (Epp s textbook) 2.1-2.4 1 Logic Logic is a system based on statements. A statement (or
13.3 Truth Tables for the Conditional and the Biconditional
ruthablesconditionalbiconditional.nb 1 13.3 ruth ables for the Conditional and the Biconditional Conditional Earlier, we mentioned that the statement preceding the conditional symbol is called the antecedent
Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic.
Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms)
PROPOSITIONAL CALCULUS
PROPOSITIONAL CALCULUS A proposition is a complete declarative sentence that is either TRUE (truth value T or 1) or FALSE (truth value F or 0), but not both. These are not propositions! Connectives and
CSC Discrete Math I, Spring Propositional Logic
CSC 125 - Discrete Math I, Spring 2017 Propositional Logic Propositions A proposition is a declarative sentence that is either true or false Propositional Variables A propositional variable (p, q, r, s,...)
Example. Logic. Logical Statements. Outline of logic topics. Logical Connectives. Logical Connectives
Logic Logic is study of abstract reasoning, specifically, concerned with whether reasoning is correct. Logic focuses on relationship among statements as opposed to the content of any particular statement.
Today s Lecture 2/25/10. Truth Tables Continued Introduction to Proofs (the implicational rules of inference)
Today s Lecture 2/25/10 Truth Tables Continued Introduction to Proofs (the implicational rules of inference) Announcements Homework: -- Ex 7.3 pg. 320 Part B (2-20 Even). --Read chapter 8.1 pgs. 345-361.
CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS
CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS 1.1 - Logical Form and Logical Equivalence Definition. A statement or proposition is a sentence that is either true or false, but not both. ex. 1 + 2 = 3 IS a statement
Introduction Logic Inference. Discrete Mathematics Andrei Bulatov
Introduction Logic Inference Discrete Mathematics Andrei Bulatov Discrete Mathematics - Logic Inference 6-2 Previous Lecture Laws of logic Expressions for implication, biconditional, exclusive or Valid
DISCRETE MATH: LECTURE 3
DISCRETE MATH: LECTURE 3 DR. DANIEL FREEMAN 1. Chapter 2.2 Conditional Statements If p and q are statement variables, the conditional of q by p is If p then q or p implies q and is denoted p q. It is false
Chapter 1, Logic and Proofs (3) 1.6. Rules of Inference
CSI 2350, Discrete Structures Chapter 1, Logic and Proofs (3) Young-Rae Cho Associate Professor Department of Computer Science Baylor University 1.6. Rules of Inference Basic Terminology Axiom: a statement
The Logic of Compound Statements cont.
The Logic of Compound Statements cont. CSE 215, Computer Science 1, Fall 2011 Stony Brook University http://www.cs.stonybrook.edu/~cse215 Refresh from last time: Logical Equivalences Commutativity of :
Analyzing Arguments with Truth Tables
Analyzing Arguments with Truth Tables MATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Fall 2014 Introduction Euler diagrams are useful for checking the validity of simple
MACM 101 Discrete Mathematics I. Exercises on Propositional Logic. Due: Tuesday, September 29th (at the beginning of the class)
MACM 101 Discrete Mathematics I Exercises on Propositional Logic. Due: Tuesday, September 29th (at the beginning of the class) SOLUTIONS 1. Construct a truth table for the following compound proposition:
Truth Tables for Arguments
ruth ables for Arguments 1. Comparing Statements: We ve looked at SINGLE propositions and assessed the truth values listed under their main operators to determine whether they were tautologous, self-contradictory,
A Quick Lesson on Negation
A Quick Lesson on Negation Several of the argument forms we have looked at (modus tollens and disjunctive syllogism, for valid forms; denying the antecedent for invalid) involve a type of statement which
THE LOGIC OF COMPOUND STATEMENTS
THE LOGIC OF COMPOUND STATEMENTS All dogs have four legs. All tables have four legs. Therefore, all dogs are tables LOGIC Logic is a science of the necessary laws of thought, without which no employment
Compound Propositions
Discrete Structures Compound Propositions Producing new propositions from existing propositions. Logical Operators or Connectives 1. Not 2. And 3. Or 4. Exclusive or 5. Implication 6. Biconditional Truth
ANALYSIS EXERCISE 1 SOLUTIONS
ANALYSIS EXERCISE 1 SOLUTIONS 1. (a) Let B The main course will be beef. F The main course will be fish. P The vegetable will be peas. C The vegetable will be corn. The logical form of the argument is
Functions. Lecture 4: Truth functions, evaluating compound statements. Arithmetic Functions. x y x+y
Lecture 4: ruth functions, evaluating compound statements 1. unctions, arithmetic functions, and truth functions 2. Definitions of truth functions unctions A function is something that takes inputs and
Discrete Structures of Computer Science Propositional Logic III Rules of Inference
Discrete Structures of Computer Science Propositional Logic III Rules of Inference Gazihan Alankuş (Based on original slides by Brahim Hnich) July 30, 2012 1 Previous Lecture 2 Summary of Laws of Logic
CS100: DISCRETE STRUCTURES. Lecture 5: Logic (Ch1)
CS100: DISCREE SRUCURES Lecture 5: Logic (Ch1) Lecture Overview 2 Statement Logical Connectives Conjunction Disjunction Propositions Conditional Bio-conditional Converse Inverse Contrapositive Laws of
Logic. Logic is a discipline that studies the principles and methods used in correct reasoning. It includes:
Logic Logic is a discipline that studies the principles and methods used in correct reasoning It includes: A formal language for expressing statements. An inference mechanism (a collection of rules) to
Propositional Logic. Argument Forms. Ioan Despi. University of New England. July 19, 2013
Propositional Logic Argument Forms Ioan Despi despi@turing.une.edu.au University of New England July 19, 2013 Outline Ioan Despi Discrete Mathematics 2 of 1 Order of Precedence Ioan Despi Discrete Mathematics
Section 1.2: Propositional Logic
Section 1.2: Propositional Logic January 17, 2017 Abstract Now we re going to use the tools of formal logic to reach logical conclusions ( prove theorems ) based on wffs formed by some given statements.
ICS141: Discrete Mathematics for Computer Science I
ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Originals slides by Dr. Baek and Dr. Still, adapted by J. Stelovsky Based on slides Dr. M. P. Frank and Dr. J.L. Gross
Logic and Truth Tables
Logic and ruth ables What is a ruth able? A truth table is a tool that helps you analyze statements or arguments in order to verify whether or not they are logical, or true. here are five basic operations
Rules Build Arguments Rules Building Arguments
Section 1.6 1 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments Rules of Inference for Quantified Statements Building Arguments for Quantified
Propositional Logic. Jason Filippou UMCP. ason Filippou UMCP) Propositional Logic / 38
Propositional Logic Jason Filippou CMSC250 @ UMCP 05-31-2016 ason Filippou (CMSC250 @ UMCP) Propositional Logic 05-31-2016 1 / 38 Outline 1 Syntax 2 Semantics Truth Tables Simplifying expressions 3 Inference
Formal Logic 2. This lecture: Standard Procedure of Inferencing Normal forms Standard Deductive Proofs in Logic using Inference Rules
ormal Logic 2 HW2 Due Now & ickup HW3 handout! Last lecture ropositional Logic ropositions, Statements, Connectives, ruth table, ormula W roperties: autology, Contradiction, Validity, Satisfiability Logical
Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014
Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014 1. Translate each of the following English sentences into formal statements using the logical operators (,,,,, and ). You may also use mathematical
A. Propositional Logic
CmSc 175 Discrete Mathematics A. Propositional Logic 1. Statements (Propositions ): Statements are sentences that claim certain things. Can be either true or false, but not both. Propositional logic deals
CSE 20: Discrete Mathematics
Spring 2018 Summary Last time: Today: Logical connectives: not, and, or, implies Using Turth Tables to define logical connectives Logical equivalences, tautologies Some applications Proofs in propositional
Propositional Logic. Spring Propositional Logic Spring / 32
Propositional Logic Spring 2016 Propositional Logic Spring 2016 1 / 32 Introduction Learning Outcomes for this Presentation Learning Outcomes... At the conclusion of this session, we will Define the elements
Packet #1: Logic & Proofs. Applied Discrete Mathematics
Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 3-13 Course Objectives At the conclusion of this course, you should
Logic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.
![Logic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.](/thumbs/87/97228443.jpg "Logic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.")
Math 0413 Appendix A.0 Logic Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another. This type of logic is called propositional.
n logical not (negation) n logical or (disjunction) n logical and (conjunction) n logical exclusive or n logical implication (conditional)
Discrete Math Review Discrete Math Review (Rosen, Chapter 1.1 1.6) TOPICS Propositional Logic Logical Operators Truth Tables Implication Logical Equivalence Inference Rules What you should know about propositional
What is Logic? Introduction to Logic. Simple Statements. Which one is statement?
What is Logic? Introduction to Logic Peter Lo Logic is the study of reasoning It is specifically concerned with whether reasoning is correct Logic is also known as Propositional Calculus CS218 Peter Lo
2.2: Logical Equivalence: The Laws of Logic
Example (2.7) For primitive statement p and q, construct a truth table for each of the following compound statements. a) p q b) p q Here we see that the corresponding truth tables for two statement p q
Review The Conditional Logical symbols Argument forms. Logic 5: Material Implication and Argument Forms Jan. 28, 2014
Logic 5: Material Implication and Argument Forms Jan. 28, 2014 Overview I Review The Conditional Conditional statements Material implication Logical symbols Argument forms Disjunctive syllogism Disjunctive
Formal Logic. Critical Thinking
ormal Logic Critical hinking Recap: ormal Logic If I win the lottery, then I am poor. I win the lottery. Hence, I am poor. his argument has the following abstract structure or form: If P then Q. P. Hence,
COMP 182 Algorithmic Thinking. Proofs. Luay Nakhleh Computer Science Rice University
COMP 182 Algorithmic Thinking Proofs Luay Nakhleh Computer Science Rice University 1 Reading Material Chapter 1, Section 3, 6, 7, 8 Propositional Equivalences The compound propositions p and q are called
8.8 Statement Forms and Material Equivalence
M08_COPI1396_13_SE_C08.QXD 10/16/07 9:19 PM Page 357 8.8 Statement Forms and Material Equivalence 357 murdered. So either lawlessness will be rewarded or innocent hostages will be murdered. 8. If people
DEDUCTIVE REASONING Propositional Logic
7 DEDUCTIVE REASONING Propositional Logic Chapter Objectives Connectives and Truth Values You will be able to understand the purpose and uses of propositional logic. understand the meaning, symbols, and
HANDOUT AND SET THEORY. Ariyadi Wijaya
HANDOUT LOGIC AND SET THEORY Ariyadi Wijaya Mathematics Education Department Faculty of Mathematics and Natural Science Yogyakarta State University 2009 1 Mathematics Education Department Faculty of Mathematics
Methods of Proof. 1.6 Rules of Inference. Argument and inference 12/8/2015. CSE2023 Discrete Computational Structures
Methods of Proof CSE0 Discrete Computational Structures Lecture 4 When is a mathematical argument correct? What methods can be used to construct mathematical arguments? Important in many computer science
10/5/2012. Logic? What is logic? Propositional Logic. Propositional Logic (Rosen, Chapter ) Logic is a truth-preserving system of inference
Logic? Propositional Logic (Rosen, Chapter 1.1 1.3) TOPICS Propositional Logic Truth Tables Implication Logical Proofs 10/1/12 CS160 Fall Semester 2012 2 What is logic? Logic is a truth-preserving system
Artificial Intelligence Knowledge Representation I
Artificial Intelligence Knowledge Representation I Agents that reason logically knowledge-based approach implement agents that know about their world and reason about possible courses of action needs to
Symbolic Logic 3. For an inference to be deductively valid it is impossible for the conclusion to be false if the premises are true.
Symbolic Logic 3 Testing deductive validity with truth tables For an inference to be deductively valid it is impossible for the conclusion to be false if the premises are true. So, given that truth tables
Readings: Conjecture. Theorem. Rosen Section 1.5
Readings: Conjecture Theorem Lemma Lemma Step 1 Step 2 Step 3 : Step n-1 Step n a rule of inference an axiom a rule of inference Rosen Section 1.5 Provide justification of the steps used to show that a
Boolean Algebra and Proof. Notes. Proving Propositions. Propositional Equivalences. Notes. Notes. Notes. Notes. March 5, 2012
March 5, 2012 Webwork Homework. The handout on Logic is Chapter 4 from Mary Attenborough s book Mathematics for Electrical Engineering and Computing. Proving Propositions We combine basic propositions
Proofs. Example of an axiom in this system: Given two distinct points, there is exactly one line that contains them.
Proofs A mathematical system consists of axioms, definitions and undefined terms. An axiom is assumed true. Definitions are used to create new concepts in terms of existing ones. Undefined terms are only
Discrete Structures & Algorithms. Propositional Logic EECE 320 // UBC
Discrete Structures & Algorithms Propositional Logic EECE 320 // UBC 1 Review of last lecture Pancake sorting A problem with many applications Bracketing (bounding a function) Proving bounds for pancake
Valid Reasoning. Alice E. Fischer. CSCI 1166 Discrete Mathematics for Computing February, Outline Truth and Validity Valid Reasoning
Alice E. Fischer CSCI 1166 Discrete Mathematics for Computing February, 2018 Alice E. Fischer Reasoning... 1/23 1 Truth is not the same as Validity 2 Alice E. Fischer Reasoning... 2/23 Truth is not the
Chapter 2: The Logic of Compound Statements
Chapter 2: he Logic of Compound Statements irst: Aristotle (Gr. 384-322 BC) Collection of rules for deductive reasoning to be used in every branch of knowledge Next: Gottfried Leibniz (German, 17th century)
Reexam in Discrete Mathematics
Reexam in Discrete Mathematics First Year at the Faculty of Engineering and Science and the Technical Faculty of IT and Design August 15th, 2017, 9.00-13.00 This exam consists of 11 numbered pages with
CS 2336 Discrete Mathematics
CS 2336 Discrete Mathematics Lecture 3 Logic: Rules of Inference 1 Outline Mathematical Argument Rules of Inference 2 Argument In mathematics, an argument is a sequence of propositions (called premises)
Practice Exam 2. You MUST show all your work, and indicate which value is the main operator value.
Practice Exam 2 You MUS show all your work, and indicate which value is the main operator value. I. ranslate the following statements into symbolic form, using A, B, C, and D. hen, using your knowledge
3 The Semantics of the Propositional Calculus
3 The Semantics of the Propositional Calculus 1. Interpretations Formulas of the propositional calculus express statement forms. In chapter two, we gave informal descriptions of the meanings of the logical
Logic Overview, I. and T T T T F F F T F F F F
Logic Overview, I DEFINITIONS A statement (proposition) is a declarative sentence that can be assigned a truth value T or F, but not both. Statements are denoted by letters p, q, r, s,... The 5 basic logical
(ÀB Ä (A Â C)) (A Ä ÀC) Á B. This is our sample argument. Formal Proofs
(ÀB Ä (A Â C)) (A Ä ÀC) Á B This is our sample argument. Formal Proofs From now on, formal proofs will be our main way to test arguments. We ll begin with easier proofs. Our initial strategy for constructing
Logic and Proof. Aiichiro Nakano
Logic and Proof Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Department of Computer Science Department of Physics & Astronomy Department of Chemical Engineering & Materials Science
CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter p. 1/33
CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter 4.1-4.8 p. 1/33 CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer
Proposition/Statement. Boolean Logic. Boolean variables. Logical operators: And. Logical operators: Not 9/3/13. Introduction to Logical Operators
Proposition/Statement Boolean Logic CS 231 Dianna Xu A proposition is either true or false but not both he sky is blue Lisa is a Math major x == y Not propositions: Are you Bob? x := 7 1 2 Boolean variables
1.3 Propositional Equivalences
1 1.3 Propositional Equivalences The replacement of a statement with another statement with the same truth is an important step often used in Mathematical arguments. Due to this methods that produce propositions
Outline. Rules of Inferences Discrete Mathematics I MATH/COSC 1056E. Example: Existence of Superman. Outline
Outline s Discrete Mathematics I MATH/COSC 1056E Julien Dompierre Department of Mathematics and Computer Science Laurentian University Sudbury, August 6, 2008 Using to Build Arguments and Quantifiers Outline
AMTH140 Lecture 8. Symbolic Logic
AMTH140 Lecture 8 Slide 1 Symbolic Logic March 10, 2006 Reading: Lecture Notes 6.2, 6.3; Epp 1.1, 1.2 Logical Connectives Let p and q denote propositions, then: 1. p q is conjunction of p and q, meaning
Discrete Mathematical Structures. Chapter 1 The Foundation: Logic
Discrete Mathematical Structures Chapter 1 he oundation: Logic 1 Lecture Overview 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Quantifiers l l l l l Statement Logical Connectives Conjunction
15414/614 Optional Lecture 1: Propositional Logic
15414/614 Optional Lecture 1: Propositional Logic Qinsi Wang Logic is the study of information encoded in the form of logical sentences. We use the language of Logic to state observations, to define concepts,
software design & management Gachon University Chulyun Kim
Gachon University Chulyun Kim 2 Outline Propositional Logic Propositional Equivalences Predicates and Quantifiers Nested Quantifiers Rules of Inference Introduction to Proofs 3 1.1 Propositional Logic
Logic and Truth Tables
Logic and Truth Tables What is a Truth Table? A truth table is a tool that helps you analyze statements or arguments in order to verify whether or not they are logical, or true. There are five basic operations
n Empty Set:, or { }, subset of all sets n Cardinality: V = {a, e, i, o, u}, so V = 5 n Subset: A B, all elements in A are in B
Discrete Math Review Discrete Math Review (Rosen, Chapter 1.1 1.7, 5.5) TOPICS Sets and Functions Propositional and Predicate Logic Logical Operators and Truth Tables Logical Equivalences and Inference
Chapter 1: The Logic of Compound Statements. January 7, 2008
Chapter 1: The Logic of Compound Statements January 7, 2008 Outline 1 1.1 Logical Form and Logical Equivalence 2 1.2 Conditional Statements 3 1.3 Valid and Invalid Arguments Central notion of deductive
Advanced Topics in LP and FP
Lecture 1: Prolog and Summary of this lecture 1 Introduction to Prolog 2 3 Truth value evaluation 4 Prolog Logic programming language Introduction to Prolog Introduced in the 1970s Program = collection
1.1 Statements and Compound Statements
Chapter 1 Propositional Logic 1.1 Statements and Compound Statements A statement or proposition is an assertion which is either true or false, though you may not know which. That is, a statement is something
Definition 2. Conjunction of p and q
Proposition Propositional Logic CPSC 2070 Discrete Structures Rosen (6 th Ed.) 1.1, 1.2 A proposition is a statement that is either true or false, but not both. Clemson will defeat Georgia in football
Department of Computer Science & Software Engineering Comp232 Mathematics for Computer Science
Department of Computer Science & Software Engineering Comp232 Mathematics for Computer Science Fall 2018 Assignment 1. Solutions 1. For each of the following statements use a truth table to determine whether
PL: Truth Trees. Handout Truth Trees: The Setup
Handout 4 PL: Truth Trees Truth tables provide a mechanical method for determining whether a proposition, set of propositions, or argument has a particular logical property. For example, we can show that
Language of Propositional Logic
Logic A logic has: 1. An alphabet that contains all the symbols of the language of the logic. 2. A syntax giving the rules that define the well formed expressions of the language of the logic (often called
Math Assignment 2 Solutions - Spring Jaimos F Skriletz Provide definitions for the following:
Math 124 - Assignment 2 Solutions - Spring 2009 - Jaimos F Skriletz 1 1. Provide definitions for the following: (a) A statement is a declarative sentence that is either true or false, but not both at the
FUNDAMENTALS OF MATHEMATICS HANDOUT 1.3 DR. MCLOUGHLIN
021 McLoughlin Handout 1.3, page 1 of 6 FUNDAMENTALS OF MATHEMATICS HANDOUT 1.3 DR. MCLOUGHLIN Truth Table for Not K K T F F T Truth Table for And B M B M T F F F T F F F F Truth Table for Or R S R S T
Proof Tactics, Strategies and Derived Rules. CS 270 Math Foundations of CS Jeremy Johnson
Proof Tactics, Strategies and Derived Rules CS 270 Math Foundations of CS Jeremy Johnson Outline 1. Review Rules 2. Positive subformulas and extraction 3. Proof tactics Extraction, Conversion, Inversion,
Propositional Logic. Fall () Propositional Logic Fall / 30
Propositional Logic Fall 2013 () Propositional Logic Fall 2013 1 / 30 1 Introduction Learning Outcomes for this Presentation 2 Definitions Statements Logical connectives Interpretations, contexts,... Logically
PS10.3 Logical implications
Warmup: Construct truth tables for these compound statements: 1) p (q r) p q r p q r p (q r) PS10.3 Logical implications Lets check it out: We will be covering Implications, logical equivalence, converse,
Packet #2: Set Theory & Predicate Calculus. Applied Discrete Mathematics
CSC 224/226 Notes Packet #2: Set Theory & Predicate Calculus Barnes Packet #2: Set Theory & Predicate Calculus Applied Discrete Mathematics Table of Contents Full Adder Information Page 1 Predicate Calculus
Rules of Inference. Arguments and Validity
Arguments and Validity A formal argument in propositional logic is a sequence of propositions, starting with a premise or set of premises, and ending in a conclusion. We say that an argument is valid if
Deductive Systems. Lecture - 3
Deductive Systems Lecture - 3 Axiomatic System Axiomatic System (AS) for PL AS is based on the set of only three axioms and one rule of deduction. It is minimal in structure but as powerful as the truth
Math 3336: Discrete Mathematics Practice Problems for Exam I
Math 3336: Discrete Mathematics Practice Problems for Exam I The upcoming exam on Tuesday, February 26, will cover the material in Chapter 1 and Chapter 2*. You will be provided with a sheet containing
COMP Intro to Logic for Computer Scientists. Lecture 6
COMP 1002 Intro to Logic for Computer Scientists Lecture 6 B 5 2 J Treasure hunt In the back of an old cupboard you discover a note signed by a pirate famous for his bizarre sense of humour and love of
Computational Intelligence Lecture 13:Fuzzy Logic
Computational Intelligence Lecture 13:Fuzzy Logic Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2011 arzaneh Abdollahi Computational Intelligence Lecture
Proofs. Joe Patten August 10, 2018
Proofs Joe Patten August 10, 2018 1 Statements and Open Sentences 1.1 Statements A statement is a declarative sentence or assertion that is either true or false. They are often labelled with a capital
Section 1.1: Logical Form and Logical Equivalence
Section 1.1: Logical Form and Logical Equivalence An argument is a sequence of statements aimed at demonstrating the truth of an assertion. The assertion at the end of an argument is called the conclusion,
The proposition p is called the hypothesis or antecedent. The proposition q is called the conclusion or consequence.
The Conditional (IMPLIES) Operator The conditional operation is written p q. The proposition p is called the hypothesis or antecedent. The proposition q is called the conclusion or consequence. The Conditional
Propositional Equivalence
Propositional Equivalence Tautologies and contradictions A compound proposition that is always true, regardless of the truth values of the individual propositions involved, is called a tautology. Example:
1.1 Language and Logic
c Oksana Shatalov, Fall 2017 1 1.1 Language and Logic Mathematical Statements DEFINITION 1. A proposition is any declarative sentence (i.e. it has both a subject and a verb) that is either true or false,
Section 3.1. Statements and Logical Connectives. Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 3.1 Statements and Logical Connectives What You Will Learn Statements, quantifiers, and compound statements Statements involving the words not, and, or, if then, and if and only if 3.1-2 HISORY
3. The Logic of Quantified Statements Summary. Aaron Tan August 2017
3. The Logic of Quantified Statements Summary Aaron Tan 28 31 August 2017 1 3. The Logic of Quantified Statements 3.1 Predicates and Quantified Statements I Predicate; domain; truth set Universal quantifier,
Logic, Sets, and Proofs
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Operators. A logical statement is a mathematical statement that can be assigned a value either true or false.
Propositional Logic Basics Propositional Equivalences Normal forms Boolean functions and digital circuits. Propositional Logic.
Propositional Logic Winter 2012 Propositional Logic: Section 1.1 Proposition A proposition is a declarative sentence that is either true or false. Which ones of the following sentences are propositions?