3 Rules of Inferential Logic
|
|
|
- Martin Cook
- 5 years ago
- Views:
Transcription
1 24 FUNDAMENTALS OF MATHEMATICAL LOGIC 3 Rules of Inferential Logic The main concern of logic is how the truth of some roositions is connected with the truth of another. Thus, we will usually consider a grou of related roositions. An argument is a set of two or more roositions related to each other in such a way that all but one of them (the remises) are suosed to rovide suort for the remaining one (the conclusion). The transition from remises to conclusion is the inference uon which the argument relies. Examle 3.1 Show that the roositions \The star is made of milk, and strawberries are red. My dog has eas." do not form an argument. Indeed, the truth or falsity of each of the roositions has no bearing on that of the others Examle 3.2 Show that the roositions: \Mark is a lawyer. So Mark went to law school since all lawyers have gone to law school" form an argument. This is an argument. The truth of the conclusion, \Mark went to law school," is inferred or deduced from its remises, \Mark is a lawyer" and \all lawyers have gone to law school." The above argument can be reresented as follows: Let : Mark is a lawyer. q: All lawyers have gone to law school. r: Mark went to law school. Then ^ q :: r The symbol :: is to indicate the inferrenced conclusion. Now, suose that the remises of an argument are all true. Then the
2 3 RULES OF INFERENTIAL LOGIC 25 conclusion may be either true or false. When the conclusion is true then the argument is said to be valid. When the conclusion is false then the argument is said to be invalid. To test an argument for validity one roceeds as follows: (1) Identify the remises and the conclusion of the argument. (2) Construct a truth table including the remises and the conclusion. (3) Find rows in which all remises are true. (4) In each row of Ste (3), if the conclusion is true then the argument is valid; otherwise the argument is invalid. Examle 3.3 is invalid q! _ q We construct the truth table as follows. q q! _ q T T T F F T T F T T F T F F T T F From the last row we see that the remises are true but the conclusion is false. The argument is then invalid Examle 3.4 (Modus Ponens or the method of a rming) a. b. _ q! r _ q :: r
3 26 FUNDAMENTALS OF MATHEMATICAL LOGIC a. q T F F F T T F F T The rst row shows that the argument b. Follows from (a) by relacing with _ q and q with r Examle 3.5 is invalid. q q T F F F T T F F T Because of the third row the argument is invalid. An argument of this form is referred to as converse error because the conclusion of the argument would follows from the remises if is relaced by its converse q! Examle 3.6 (Modus Tollens or the method of denial) q
4 3 RULES OF INFERENTIAL LOGIC 27 q q F F T F F T F F T T F T F F The last row shows that the argument is valid Examle 3.7 is invalid. q q F F T F F T F F T T F T F F The third row shows that the argument is invalid. This is known as inverse error because the conclusion of the argument would follow from the remises if is relaced by the inverse Examle 3.8 (Disjunctive Addition) a. _ q b. q _ q
5 28 FUNDAMENTALS OF MATHEMATICAL LOGIC a. q _ q T F T F T T F F F The rst and second rows show that the argument b. The rst and third rows show that the argument is valid Examle 3.9 (Conjunctive addition) Show that ; q ^ q q ^ q T F F F T F F F F The rst row shows that the argument is valid Examle 3.10 (Conjunctive Simli cation) a. ^ q b. ^ q
6 3 RULES OF INFERENTIAL LOGIC 29 a. q ^ q T F F F T F F F F The rst row shows that the argument b. The rst row shows that the argument is valid Examle 3.11 (Disjunctive Syllogism) a. _ q q b. _ q a. q q _ q T T F F T T F F T T F T T F T F F T T F The second row shows that the argument b. The third row shows that the argument is valid Examle 3.12 (Hyothetical Syllogism) q! r! r
7 30 FUNDAMENTALS OF MATHEMATICAL LOGIC q r q! r! r T T F T F F T F T F T T T F F F T F F T T F T F T F T F F T F F F The rst, fth, seventh, and eighth rows show that the argument is valid Examle 3.13 (Rule of contradiction) Show that if c is a contradiction then the following argument is valid for any.! c Constructing the truth table we nd c! c F T T F F F The rst row shows that the argument is valid
8 3 RULES OF INFERENTIAL LOGIC 31 Review Problems Problem 3.1 Use modus onens or modus tollens to ll in the blanks in the argument below so as to roduce valid inferences. If is rational, then 2 2 = a for some integers a and b: b It is not true that 2 = a for some integers a and b: b :: Problem 3.2 Use modus onens or modus tollens to ll in the blanks in the argument below so as to roduce valid inferences. If logic is easy, then I am a monkey's uncle. I am not a monkey's uncle. :: Problem 3.3 Use a truth table to determine whether the argument below q! _ q Problem 3.4 Use a truth table to determine whether the argument below q _ r :: r Problem 3.5 Use symbols to write the logical form of the given argument and then use a truth table to test the argument for validity. If Tom is not on team A, then Hua is on team B. If Hua is not on team B, then Tom is on team A. :: Tom is not on team A or Hua is not on team B.
9 32 FUNDAMENTALS OF MATHEMATICAL LOGIC Problem 3.6 Use symbols to write the logical form of the given argument. If the argument is valid, identify the rule of inference that guarantees its validity. Otherwise state whether the converse or the inverse error is made. If Jules solved this roblem correctly, then Jules obtained the answer 2. Jules obtained the answer 2. :: Jules solved this roblem correctly. Problem 3.7 Use symbols to write the logical form of the given argument. If the argument is valid, identify the rule of inference that guarantees its validity. Otherwise state whether the converse or the inverse error is made. If this number is larger than 2, then its square is larger than 4. This number is not larger than 2. :: The square of this number is not larger than 4. Problem 3.8 Use the valid argument forms of this section to deduce the conclusion from the remises. _ q! r s_ q t! t ^ r! s Problem 3.9 Use the valid argument forms of this section to deduce the conclusion from the remises.! r^ s t! s u! w u _ w :: t _ w
[Ch 3, 4] Logic and Proofs (2) 1. Valid and Invalid Arguments ( 2.3, 3.4) 400 lecture note #2. 1) Basics
![[Ch 3, 4] Logic and Proofs (2) 1. Valid and Invalid Arguments ( 2.3, 3.4) 400 lecture note #2. 1) Basics](/thumbs/76/74441698.jpg "[Ch 3, 4] Logic and Proofs (2) 1. Valid and Invalid Arguments ( 2.3, 3.4) 400 lecture note #2. 1) Basics")
400 lecture note #2 [Ch 3, 4] Logic and Proofs (2) 1. Valid and Invalid Arguments ( 2.3, 3.4) 1) Basics An argument is a sequence of statements ( s1, s2,, sn). All statements in an argument, excet for
Resolution (7A) Young Won Lim 4/21/18
(7A) Coyright (c) 215 218 Young W. Lim. Permission is granted to coy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version ublished
Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics. Introductory Notes in Discrete Mathematics Solution Guide
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics Introductory Notes in Discrete Mathematics Solution Guide Marcel B. Finan c All Rights Reserved 2015 Edition Contents
The Logic of Compound Statements. CSE 2353 Discrete Computational Structures Spring 2018
CSE 2353 Discrete Comutational Structures Sring 2018 The Logic of Comound Statements (Chater 2, E) Note: some course slides adoted from ublisher-rovided material Outline 2.1 Logical Form and Logical Equivalence
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
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.
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.
Why Proofs? Proof Techniques. Theorems. Other True Things. Proper Proof Technique. How To Construct A Proof. By Chuck Cusack
Proof Techniques By Chuck Cusack Why Proofs? Writing roofs is not most student s favorite activity. To make matters worse, most students do not understand why it is imortant to rove things. Here are just
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
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:
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
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
8.7 Some Common Argument Forms
M08_COPI1396_13_SE_C08.QXD 10/16/07 9:19 PM Page 349 8.7 Some Common Argument Forms 349 columns as the table was built, it is ossible for the remises and the conclusion to aear in any order at the to of
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
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)
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
DISCRETE MATH: FINAL REVIEW
DISCRETE MATH: FINAL REVIEW DR. DANIEL FREEMAN 1) a. Does 3 = {3}? b. Is 3 {3}? c. Is 3 {3}? c. Is {3} {3}? c. Is {3} {3}? d. Does {3} = {3, 3, 3, 3}? e. Is {x Z x > 0} {x R x > 0}? 1. Chapter 1 review
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
Review. p q ~p v q Contrapositive: ~q ~p Inverse: ~p ~q Converse: q p
~ v Contraositive: ~ ~ Inverse: ~ ~ Converse: Å Review ( Ú ) Ù ~( Ù ) ~( Ù ) ~ Ú ~ ~( Ú ) ~ Ù ~ is sufficient for is necessary for ~ ~ 1 Valid and Invalid Arguments CS 231 Dianna Xu 2 Associative Law:
PHI Propositional Logic Lecture 2. Truth Tables
PHI 103 - Propositional Logic Lecture 2 ruth ables ruth ables Part 1 - ruth unctions for Logical Operators ruth unction - the truth-value of any compound proposition determined solely by the truth-value
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
Section 1.3: Valid and Invalid Arguments
Section 1.3: Valid and Invalid Arguments Now we have developed the basic language of logic, we shall start to consider how logic can be used to determine whether or not a given argument is valid. In order
Proposition logic and argument. CISC2100, Spring 2017 X.Zhang
Proposition logic and argument CISC2100, Spring 2017 X.Zhang 1 Where are my glasses? I know the following statements are true. 1. If I was reading the newspaper in the kitchen, then my glasses are on the
Where are my glasses?
Proposition logic and argument CISC2100, Spring 2017 X.Zhang 1 Where are my glasses? I know the following statements are true. 1. If I was reading the newspaper in the kitchen, then my glasses are on the
PSU MATH RELAYS LOGIC & SET THEORY 2017
PSU MATH RELAYS LOGIC & SET THEORY 2017 MULTIPLE CHOICE. There are 40 questions. Select the letter of the most appropriate answer and SHADE in the corresponding region of the answer sheet. If the correct
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
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
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 :
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
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
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
CSE 20 DISCRETE MATH. Winter
CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Today's learning goals Distinguish between a theorem, an axiom, lemma, a corollary, and a conjecture. Recognize direct proofs
PHIL12A Section answers, 28 Feb 2011
PHIL12A Section answers, 28 Feb 2011 Julian Jonker 1 How much do you know? Give formal proofs for the following arguments. 1. (Ex 6.18) 1 A B 2 A B 1 A B 2 A 3 A B Elim: 2 4 B 5 B 6 Intro: 4,5 7 B Intro:
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
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
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.
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
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
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
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 MATH. Fall
CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Distinguish between a theorem, an axiom, lemma, a corollary, and a conjecture. Recognize direct proofs
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
Deductive and Inductive Logic
Deductive Logic Overview (1) Distinguishing Deductive and Inductive Logic (2) Validity and Soundness (3) A Few Practice Deductive Arguments (4) Testing for Invalidity (5) Practice Exercises Deductive and
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
DISCRETE MATH: LECTURE 6
DISCRETE MATH: LECTURE 6 DR. DANIEL FREEMAN 1) a. Does 3 = {3}? b. Is 3 {3}? c. Is 3 {3}? d. Does {3} = {3, 3, 3, 3}? e. Is {x Z x > 0} {x R x > 0}? 1. Chapter 1 review 2) a. When does (a, b) = (c, d)?
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
Steinhardt School of Culture, Education, and Human Development Department of Teaching and Learning. Mathematical Proof and Proving (MPP)
Steinhardt School of Culture, Education, and Human Development Department of Teaching and Learning Terminology, Notations, Definitions, & Principles: Mathematical Proof and Proving (MPP) 1. A statement
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
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
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
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
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
What is the decimal (base 10) representation of the binary number ? Show your work and place your final answer in the box.
Question 1. [10 marks] Part (a) [2 marks] What is the decimal (base 10) representation of the binary number 110101? Show your work and place your final answer in the box. 2 0 + 2 2 + 2 4 + 2 5 = 1 + 4
Natural deduction for truth-functional logic
Natural deduction for truth-functional logic Phil 160 - Boston University Why natural deduction? After all, we just found this nice method of truth-tables, which can be used to determine the validity or
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
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
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
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
Proof Worksheet 2, Math 187 Fall 2017 (with solutions)
Proof Worksheet 2, Math 187 Fall 2017 (with solutions) Dr. Holmes October 17, 2017 The instructions are the same as on the first worksheet, except you can use all the rules in the strategies handout. We
Propositional Logic: Part II - Syntax & Proofs 0-0
Propositional Logic: Part II - Syntax & Proofs 0-0 Outline Syntax of Propositional Formulas Motivating Proofs Syntactic Entailment and Proofs Proof Rules for Natural Deduction Axioms, theories and theorems
Maryam Al-Towailb (KSU) Discrete Mathematics and Its Applications Math. Rules Math. of1101 Inference 1 / 13
Maryam Al-Towailb (KSU) Discrete Mathematics and Its Applications Math. Rules 151 - Math. of1101 Inference 1 / 13 Maryam Al-Towailb (KSU) Discrete Mathematics and Its Applications Math. Rules 151 - Math.
DISCRETE MATH: LECTURE Chapter 3.3 Statements with Multiple Quantifiers If you want to establish the truth of a statement of the form
DISCRETE MATH: LECTURE 5 DR. DANIEL FREEMAN 1. Chapter 3.3 Statements with Multiple Quantifiers If you want to establish the truth of a statement of the form x D, y E such that P (x, y) your challenge
Review. Propositions, propositional operators, truth tables. Logical Equivalences. Tautologies & contradictions
Review Propositions, propositional operators, truth tables Logical Equivalences. Tautologies & contradictions Some common logical equivalences Predicates & quantifiers Some logical equivalences involving
CSE 20 DISCRETE MATH WINTER
CSE 20 DISCRETE MATH WINTER 2016 http://cseweb.ucsd.edu/classes/wi16/cse20-ab/ Today's learning goals Evaluate which proof technique(s) is appropriate for a given proposition Direct proof Proofs by contraposition
CSE 20 DISCRETE MATH SPRING
CSE 20 DISCRETE MATH SPRING 2016 http://cseweb.ucsd.edu/classes/sp16/cse20-ac/ Today's learning goals Evaluate which proof technique(s) is appropriate for a given proposition Direct proof Proofs by contraposition
CSC 125 :: Final Exam May 3 & 5, 2010
CSC 125 :: Final Exam May 3 & 5, 2010 Name KEY (1 5) Complete the truth tables below: p Q p q p q p q p q p q T T T T F T T T F F T T F F F T F T T T F F F F F F T T 6-15. Match the following logical equivalences
5. Use a truth table to determine whether the two statements are equivalent. Let t be a tautology and c be a contradiction.
Statements Compounds and Truth Tables. Statements, Negations, Compounds, Conjunctions, Disjunctions, Truth Tables, Logical Equivalence, De Morgan s Law, Tautology, Contradictions, Proofs with Logical Equivalent
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
Rules of Inference. Agenda. Rules of Inference Dr Patrick Chan. p r. p q. q r. Rules of Inference for Quantifiers. Hypothetical Syllogism
Discr ete Mathem atic Chater 1: Logic and Proof 1.5 Rules of Inference Dr Patrick Chan School of Comuter Science and Engineering South China University of echnology Recall John is a co. John knows first
MAT 243 Test 1 SOLUTIONS, FORM A
t MAT 243 Test 1 SOLUTIONS, FORM A 1. [10 points] Rewrite the statement below in positive form (i.e., so that all negation symbols immediately precede a predicate). ( x IR)( y IR)((T (x, y) Q(x, y)) R(x,
Discrete Mathematics
Discrete Mathematics Chih-Wei Yi Dept. of Computer Science National Chiao Tung University March 9, 2009 Overview of ( 1.5-1.7, ~2 hours) Methods of mathematical argument (i.e., proof methods) can be formalized
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
Logic and Inferences
Artificial Intelligence Logic and Inferences Readings: Chapter 7 of Russell & Norvig. Artificial Intelligence p.1/34 Components of Propositional Logic Logic constants: True (1), and False (0) Propositional
Math.3336: Discrete Mathematics. Nested Quantifiers/Rules of Inference
Math.3336: Discrete Mathematics Nested Quantifiers/Rules of Inference Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu
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
Chapter 1 Elementary Logic
2017-2018 Chapter 1 Elementary Logic The study of logic is the study of the principles and methods used in distinguishing valid arguments from those that are not valid. The aim of this chapter is to help
PHIL012. SYMBOLIC LOGIC PROPOSITIONAL LOGIC DERIVATIONS
HIL012 SYMBOLIC LOGIC ROOSITIONL LOGIC DERIVTIONS When we argue, what we want are (i) clearly specifiable rules, (ii) that apply to any particular subject matter, and (iii) that legitimate transitions
Predicate Logic. Andreas Klappenecker
Predicate Logic Andreas Klappenecker Predicates A function P from a set D to the set Prop of propositions is called a predicate. The set D is called the domain of P. Example Let D=Z be the set of integers.
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
Announcements. Exam 1 Review
Announcements Quiz today Exam Monday! You are allowed one 8.5 x 11 in cheat sheet of handwritten notes for the exam (front and back of 8.5 x 11 in paper) Handwritten means you must write them by hand,
STUDY PROBLEMS FOR EXAM I CMSC 203 DISCRETE STRUCTURES. n (n +1)(2n +1), 6. j 2 = 1(1+1)(2 1+1) 6. k (k +1)(2k +1) 6
STUDY PROBLEMS FOR EXAM I CMSC 203 DISCRETE STRUCTURES DR. LOMONACO 1. Use the principle of mathematical induction to prove that P (n) : n (n +1)(2n +1), for all integers n 1. Answer: Proof (by weak induction):
Manual of Logical Style
Manual of Logical Style Dr. Holmes January 9, 2015 Contents 1 Introduction 2 2 Conjunction 3 2.1 Proving a conjunction...................... 3 2.2 Using a conjunction........................ 3 3 Implication
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,
Artificial Intelligence. Propositional logic
Artificial Intelligence Propositional logic Propositional Logic: Syntax Syntax of propositional logic defines allowable sentences Atomic sentences consists of a single proposition symbol Each symbol stands
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
CSE 20 DISCRETE MATH. Fall
CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Describe and use algorithms for integer operations based on their expansions Relate algorithms for integer
WUCT121. Discrete Mathematics. Logic. Tutorial Exercises
WUCT11 Discrete Mathematics Logic Tutorial Exercises 1 Logic Predicate Logic 3 Proofs 4 Set Theory 5 Relations and Functions WUCT11 Logic Tutorial Exercises 1 Section 1: Logic Question1 For each of the
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
1) Let h = John is healthy, w = John is wealthy and s = John is wise Write the following statement is symbolic form
Math 378 Exam 1 Spring 2009 Show all Work Name 1) Let h = John is healthy, w = John is wealthy and s = John is wise Write the following statement is symbolic form a) In order for John to be wealthy it
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
Sets of Real Numbers
Chater 4 Sets of Real Numbers 4. The Integers Z and their Proerties In our revious discussions about sets and functions the set of integers Z served as a key examle. Its ubiquitousness comes from the fact
(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
Logical Form 5 Famous Valid Forms. Today s Lecture 1/26/10
Logical Form 5 Famous Valid Forms Today s Lecture 1/26/10 Announcements Homework: --Read Chapter 7 pp. 277-298 (doing the problems in parts A, B, and C pp. 298-300 are recommended but not required at this
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
10/13/15. Proofs: what and why. Proposi<onal Logic Proofs. 1 st Proof Method: Truth Table. A sequence of logical arguments such that:
Proofs: what and why Rules of Inference (Rosen, Section 1.6) TOPICS Logic Proofs ² via Truth Tables ² via Algebraic Simplification ² via Inference Rules A sequence of logical arguments such that: each
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,
Propositional Logic Arguments (5A) Young W. Lim 11/30/16
Propositional Logic (5A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
Conjunction: p q is true if both p, q are true, and false if at least one of p, q is false. The truth table for conjunction is as follows.
Chapter 1 Logic 1.1 Introduction and Definitions Definitions. A sentence (statement, proposition) is an utterance (that is, a string of characters) which is either true (T) or false (F). A predicate is
EECS 1028 M: Discrete Mathematics for Engineers
EECS 1028 M: Discrete Mathematics for Engineers Suprakash Datta Office: LAS 3043 Course page: http://www.eecs.yorku.ca/course/1028 Also on Moodle S. Datta (York Univ.) EECS 1028 W 18 1 / 12 Using the laws
Discrete Mathematics Recitation Course 張玟翔
Discrete Mathematics Recitation Course 1 2013.03.07 張玟翔 Acknowledge 鄭安哲 TA 2012 About Myself English Name : Zak Chinese Name : 張玟翔 Mail:o0000032@yahoo.com.tw Lab: ED612 1-1 Propositional Logic 1-1 Ex.2