CSE 311: Foundations of Computing. Lecture 2: More Logic, Equivalence & Digital Circuits
|
|
|
- Nathaniel Ford
- 5 years ago
- Views:
Transcription
1 CSE 311: Foundations of Computing Lecture 2: More Logic, Equivalence & Digital Circuits
2 Last class: Some Connectives & Truth Tables Negation (not) p p T F F T Disjunction (or) p q p q T T T T F T F T T F F F Conjunction (and) p q p q T T T T F F F T F F F F Exclusive Or p q p q T T F T F T F T T F F F
3 Last Class: p q Implication: p implies q whenever p is true q must be true if p then q q if p p is sufficient for q p only if q q is necessary for p p q p q T T T T F F F T T F F T
4 Biconditional: p q p iff q p is equivalent to q p implies q and q implies p p is necessary and sufficient for q p q p q
5 Biconditional: p q p iff q p is equivalent to q p implies q and q implies p p is necessary and sufficient for q p q p q T T T T F F F T F F F T
6 Last class: Using Logical Connectives Measles: You can get measles Mumps: You can get mumps MMR: You had the MMR vaccine You can get measles and mumps if you didn t have the MMR vaccine, but if you had the MMR vaccine then you can t get either. ((Measles and Mumps) if not MMR) and (if MMR then not (Measles or Mumps)) ((Measles Mumps) if MMR) (if MMR then (Measles Mumps))
7 Understanding the Vaccine Sentence You can get measles and mumps if you didn t have the MMR vaccine, but if you had the MMR vaccine you can t get either. ((Measles Mumps) if MMR) (if MMR then (Measles Mumps)) ( MMR (Measles Mumps)) (MMR (Measles Mumps))
8 Understanding the Vaccine Sentence You can get measles and mumps if you didn t have the MMR vaccine, but if you had the MMR vaccine you can t get either. ((Measles Mumps) if MMR) (if MMR then (Measles Mumps)) ( MMR (Measles Mumps)) (MMR (Measles Mumps)) Define shorthand p : MMR q : Measles r : Mumps ( p q r ) (p q r )
9 Analyzing the Vaccine Sentence with a Truth Table p q r p q r p (q r) q r (q r) p (q r) p q r (p (q r)) T T T T T F T F T T F F F T T F T F F F T F F F
10 Analyzing the Vaccine Sentence with a Truth Table p q r p q r p (q r) q r (q r) p (q r) p q r (p (q r)) T T T F T T T F F F T T F F F T T F F F T F T F F T T F F F T F F F F T F T T T F T T T T T T F T T F T F T F F T F T F F F T T F F T F T F F F F T F F F T T F
11 Converse, Contrapositive Implication: p q Converse: q p Contrapositive: q p Inverse: p q Consider p: x is divisible by 2 q: x is divisible by 4 p q q p q p p q
12 Converse, Contrapositive Implication: p q Converse: q p Contrapositive: q p Inverse: p q Consider p: x is divisible by 2 q: x is divisible by 4 p q q p q p p q Divisible By 4 Not Divisible By 4 Divisible By 2 Not Divisible By 2
13 Converse, Contrapositive Implication: p q Converse: q p Contrapositive: q p Inverse: p q Consider p: x is divisible by 2 q: x is divisible by 4 Divisible By 2 Not Divisible By 2 p q q p q p p q Divisible By 4 4,8,12,... Impossible Not Divisible By 4 2,6,10,... 1,3,5,...
14 Converse, Contrapositive Implication: p q Converse: q p Contrapositive: q p Inverse: p q How do these relate to each other? p q p q q p p q p q q p T T F F T F T F
15 Converse, Contrapositive Implication: p q Converse: q p Contrapositive: q p Inverse: p q An implication and it s contrapositive have the same truth value! p q p q q p p q p q q p T T T T F F T T T F F T F T T F F T T F T F F T F F T T T T T T
16 Tautologies! Terminology: A compound proposition is a Tautology if it is always true Contradiction if it is always false Contingency if it can be either true or false p p p p (p q) p
17 Tautologies! Terminology: A compound proposition is a Tautology if it is always true Contradiction if it is always false Contingency if it can be either true or false p p This is a tautology. It s called the law of the excluded middle. If p is true, then p p is true. If p is false, then p p is true. p p This is a contradiction. It s always false no matter what truth value p takes on. (p q) p This is a contingency. When p=t, q=t, (T T) T is true. When p=t, q=f, (T F) T is false.
18 Logical Equivalence A = B means A and B are identical strings : p q = p q p q q p
19 Logical Equivalence A = B means A and B are identical strings : p q = p q These are equal, because they are character-for-character identical. p q q p These are NOT equal, because they are different sequences of characters. They mean the same thing though. A B means A and B have identical truth values: p q p q p q q p p q q p
20 Logical Equivalence A = B means A and B are identical strings : p q = p q These are equal, because they are character-for-character identical. p q q p These are NOT equal, because they are different sequences of characters. They mean the same thing though. A B means A and B have identical truth values: p q p q Two formulas that are equal also are equivalent. p q q p These two formulas have the same truth table! p q q p When p=t and q=f, p q is false, but p q is true!
21 A B vs. A B A B is an assertion over all possible truth values that A and B always have the same truth values. A B is a proposition that may be true or false depending on the truth values of the variables in A and B. A B and (A B) T have the same meaning.
22 De Morgan s Laws (p q) p q (p q) p q Negate the statement: My code compiles or there is a bug. To negate the statement, ask when is the original statement false.
23 De Morgan s Laws (p q) p q (p q) p q Negate the statement: My code compiles or there is a bug. To negate the statement, ask when is the original statement false. It s false when not(my code compiles) AND not(there is a bug). Translating back into English, we get: My code doesn t compile and there is not a bug.
24 De Morgan s Laws Example: (p q) ( p q) p q p q p q p q (p q) (p q) ( p q) T T T F F T F F
25 De Morgan s Laws Example: (p q) ( p q) p q p q p q p q (p q) (p q) ( p q) T T F F F T F T T F F T T F T T F T T F T F T T F F T T T F T T
26 De Morgan s Laws (p q) p q (p q) p q if (!(front!= null && value > front.data)) front = new ListNode(value, front); else { ListNode current = front; while (current.next!= null && current.next.data < value)) current = current.next; current.next = new ListNode(value, current.next); }
27 De Morgan s Laws (p q) p q (p q) p q!(front!= null && value > front.data) front == null value <= front.data You ve been using these for a while!
28 Law of Implication p q p q p q p q p p q p q p q T T T F F T F F
29 Law of Implication p q p q p q p q p p q p q p q T T T F T T T F F F F T F T T T T T F F T T T T
30 Some Equivalences Related to Implication p q p q p q q p p q (p q) (q p) p q p q
31 Properties of Logical Connectives We will always give you this list!
32 Digital Circuits Computing With Logic T corresponds to 1 or high voltage F corresponds to 0 or low voltage Gates Take inputs and produce outputs (functions) Several kinds of gates Correspond to propositional connectives (most of them)
33 And Gate AND Connective p q p q p q T T T T F F F T F F F F vs. AND Gate p q AND OUT p q OUT p q AND OUT block looks like D of AND
34 Or Gate OR Connective p q p q p q T T T T F T F T T F F F vs. OR Gate p q OR OUT p q OUT p q OR OUT arrowhead block looks like V
35 Not Gates NOT Connective vs. NOT Gate p p NOT OUT p p p OUT Also called inverter T F 1 0 F T 0 1 p NOT OUT
36 Blobs are Okay! You may write gates using blobs instead of shapes! p q AND OUT p q OR OUT p NOT OUT
37 Combinational Logic Circuits p NOT AND OUT q NOT AND r s OR Values get sent along wires connecting gates
38 Combinational Logic Circuits p NOT AND OUT q NOT AND r s OR Values get sent along wires connecting gates
39 Combinational Logic Circuits p AND q NOT OR OUT r AND Wires can send one value to multiple gates!
40 Combinational Logic Circuits p AND q NOT OR OUT r AND Wires can send one value to multiple gates! p q ( q r)
41 Computing Equivalence Describe an algorithm for computing if two logical expressions/circuits are equivalent. What is the run time of the algorithm? Compute the entire truth table for both of them! There are 2 n entries in the column for n variables.
42 Some Familiar Properties of Arithmetic x + y = y + x (Commutativity) p q q p p q q p x y + z = x y + x z (Distributivity) p q r p q (p r) p q r p q (p r) x + y + z = x + (y + z) (Associativity) p q r p q r p q r p (q r)
43 Understanding Connectives Reflect basic rules of reasoning and logic Allow manipulation of logical formulas Simplification Testing for equivalence Applications Query optimization Search optimization and caching Artificial Intelligence Program verification
CSE 311: Foundations of Computing. Lecture 2: More Logic, Equivalence & Digital Circuits
CSE 311: Foundations of Computing Lecture 2: More Logic, Equivalence & Digital Circuits Last class: Some Connectives & Truth Tables Negation (not) p p T F F T Disjunction (or) p q p q T T T T F T F T T
Consider the following propositions: p: lecture has started q: If I have a banana, then I have a pear r: I finished this problem
Pre-Lecture Problem Do it! Do it now! What are you waiting for?! Consider the following roositions: : lecture has started : If I have a banana, then I have a ear r: I finished this roblem (i) Which of
CSE 311: Foundations of Computing. Lecture 3: Digital Circuits & Equivalence
CSE 311: Foundations of Computing Lecture 3: Digital Circuits & Equivalence Homework #1 You should have received An e-mail from [cse311a/cse311b] with information pointing you to look at Canvas to submit
cse 311: foundations of computing Spring 2015 Lecture 3: Logic and Boolean algebra
cse 311: foundations of computing Spring 2015 Lecture 3: Logic and Boolean algebra gradescope Homework #1 is up (and has been since Friday). It is due Friday, October 9 th at 11:59pm. You should have received
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,...)
HW1 graded review form? HW2 released CSE 20 DISCRETE MATH. Fall
CSE 20 HW1 graded review form? HW2 released DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Translate sentences from English to propositional logic using appropriate
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
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
A statement is a sentence that is definitely either true or false but not both.
5 Logic In this part of the course we consider logic. Logic is used in many places in computer science including digital circuit design, relational databases, automata theory and computability, and artificial
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
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
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
CSE 311 Lecture 02: Logic, Equivalence, and Circuits. Emina Torlak and Kevin Zatloukal
CSE 311 Lecture 02: Logic, Equivalence, and Circuits Emina Torlak and Kevin Zatloukal 1 Toics Proositional logic A brief review of Lecture 01. Classifying comound roositions Converse, contraositive, and
Tautologies, Contradictions, and Contingencies
Section 1.3 Tautologies, Contradictions, and Contingencies A tautology is a proposition which is always true. Example: p p A contradiction is a proposition which is always false. Example: p p A contingency
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
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?
Chapter Summary. Propositional Logic. Predicate Logic. Proofs. The Language of Propositions (1.1) Applications (1.2) Logical Equivalences (1.
Chapter 1 Chapter Summary Propositional Logic The Language of Propositions (1.1) Applications (1.2) Logical Equivalences (1.3) Predicate Logic The Language of Quantifiers (1.4) Logical Equivalences (1.4)
Section 1.2 Propositional Equivalences. A tautology is a proposition which is always true. A contradiction is a proposition which is always false.
Section 1.2 Propositional Equivalences A tautology is a proposition which is always true. Classic Example: P P A contradiction is a proposition which is always false. Classic Example: P P A contingency
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
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
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,
Chapter 1, Section 1.1 Propositional Logic
Discrete Structures Chapter 1, Section 1.1 Propositional Logic These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 6 th ed., by Kenneth H. Rosen, published
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
Propositional Logic: Equivalence
Propositional Logic: Equivalence Alice Gao Lecture 5 Based on work by J. Buss, L. Kari, A. Lubiw, B. Bonakdarpour, D. Maftuleac, C. Roberts, R. Trefler, and P. Van Beek 1/42 Outline Propositional Logic:
1.1 Language and Logic
c Oksana Shatalov, Spring 2018 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,
Chapter 1, Part I: Propositional Logic. With Question/Answer Animations
Chapter 1, Part I: Propositional Logic With Question/Answer Animations Chapter Summary Propositional Logic The Language of Propositions Applications Logical Equivalences Predicate Logic The Language of
LING 106. Knowledge of Meaning Lecture 3-1 Yimei Xiang Feb 6, Propositional logic
LING 106. Knowledge of Meaning Lecture 3-1 Yimei Xiang Feb 6, 2016 Propositional logic 1 Vocabulary of propositional logic Vocabulary (1) a. Propositional letters: p, q, r, s, t, p 1, q 1,..., p 2, q 2,...
Introduction to Decision Sciences Lecture 2
Introduction to Decision Sciences Lecture 2 Andrew Nobel August 24, 2017 Compound Proposition A compound proposition is a combination of propositions using the basic operations. For example (p q) ( p)
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 / 26 Why Study Logic?
Overview. 1. Introduction to Propositional Logic. 2. Operations on Propositions. 3. Truth Tables. 4. Translating Sentences into Logical Expressions
Note 01 Propositional Logic 1 / 10-1 Overview 1. Introduction to Propositional Logic 2. Operations on Propositions 3. Truth Tables 4. Translating Sentences into Logical Expressions 5. Preview: 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
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
Logical Operators. Conjunction Disjunction Negation Exclusive Or Implication Biconditional
Logical Operators Conjunction Disjunction Negation Exclusive Or Implication Biconditional 1 Statement meaning p q p implies q if p, then q if p, q when p, q whenever p, q q if p q when p q whenever p p
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.
Introduction to Sets and Logic (MATH 1190)
Introduction to Sets Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics Statistics York University Sept 18, 2014 Outline 1 2 Tautologies Definition A tautology is a 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
Propositional Logic 1
Propositional Logic 1 Section Summary Propositions Connectives Negation Conjunction Disjunction Implication; contrapositive, inverse, converse Biconditional Truth Tables 2 Propositions A proposition is
Propositional logic ( ): Review from Mat 1348
CSI 2101 / Winter 2008: Discrete Structures. Propositional logic ( 1.1-1.2): Review from Mat 1348 Dr. Nejib Zaguia - Winter 2008 1 Propositional logic: Review Mathematical Logic is a tool for working with
Numbers that are divisible by 2 are even. The above statement could also be written in other logically equivalent ways, such as:
3.4 THE CONDITIONAL & BICONDITIONAL Definition. Any statement that can be put in the form If p, then q, where p and q are basic statements, is called a conditional statement and is written symbolically
Announcements. CS311H: Discrete Mathematics. Propositional Logic II. Inverse of an Implication. Converse of a Implication
Announcements CS311H: Discrete Mathematics Propositional Logic II Instructor: Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Wed 09/13 Remember: Due before
Chapter 1, Part I: Propositional Logic. With Question/Answer Animations
Chapter 1, Part I: Propositional Logic With Question/Answer Animations Chapter Summary! Propositional Logic! The Language of Propositions! Applications! Logical Equivalences! Predicate Logic! The Language
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
Mathematical Reasoning (Part I) 1
c Oksana Shatalov, Spring 2017 1 Mathematical Reasoning (art I) 1 Statements DEFINITION 1. A statement is any declarative sentence 2 that is either true or false, but not both. A statement cannot be neither
Topic 1: Propositional logic
Topic 1: Propositional logic Guy McCusker 1 1 University of Bath Logic! This lecture is about the simplest kind of mathematical logic: propositional calculus. We discuss propositions, which are statements
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
MAT2345 Discrete Math
Fall 2013 General Syllabus Schedule (note exam dates) Homework, Worksheets, Quizzes, and possibly Programs & Reports Academic Integrity Do Your Own Work Course Web Site: www.eiu.edu/~mathcs Course Overview
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
CISC-102 Winter 2016 Lecture 17
CISC-102 Winter 2016 Lecture 17 Logic and Propositional Calculus Propositional logic was eventually refined using symbolic logic. The 17th/18th century philosopher Gottfried Leibniz (an inventor of calculus)
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
Review. Propositional Logic. Propositions atomic and compound. Operators: negation, and, or, xor, implies, biconditional.
Review Propositional Logic Propositions atomic and compound Operators: negation, and, or, xor, implies, biconditional Truth tables A closer look at implies Translating from/ to English Converse, inverse,
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
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:
Propositional Calculus: Formula Simplification, Essential Laws, Normal Forms
P Formula Simplification, Essential Laws, Normal Forms Lila Kari University of Waterloo P Formula Simplification, Essential Laws, Normal CS245, Forms Logic and Computation 1 / 26 Propositional calculus
Logic and Proofs. (A brief summary)
Logic and Proofs (A brief summary) Why Study Logic: To learn to prove claims/statements rigorously To be able to judge better the soundness and consistency of (others ) arguments To gain the foundations
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
cse 311: foundations of computing Fall 2015 Lecture 6: Predicate Logic, Logical Inference
cse 311: foundations of computing Fall 2015 Lecture 6: Predicate Logic, Logical Inference quantifiers x P(x) P(x) is true for every x in the domain read as for all x, P of x x P x There is an x in the
Boolean Logic. CS 231 Dianna Xu
Boolean Logic CS 231 Dianna Xu 1 Proposition/Statement A proposition is either true or false but not both The sky is blue Lisa is a Math major x == y Not propositions: Are you Bob? x := 7 2 Boolean variables
Chapter 4: Classical Propositional Semantics
Chapter 4: Classical Propositional Semantics Language : L {,,, }. Classical Semantics assumptions: TWO VALUES: there are only two logical values: truth (T) and false (F), and EXTENSIONALITY: the 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
Why Learning Logic? Logic. Propositional Logic. Compound Propositions
Logic Objectives Propositions and compound propositions Negation, conjunction, disjunction, and exclusive or Implication and biconditional Logic equivalence and satisfiability Application of propositional
Propositional Logic Logical Implication (4A) Young W. Lim 4/11/17
Propositional Logic Logical Implication (4A) Young W. Lim Copyright (c) 2016-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
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
Discrete Mathematics and Applications COT3100
Discrete Mathematics and Applications CO3100 Dr. Ungor Sources: Slides are based on Dr. G. Bebis material. uesday, January 7, 2014 oundations of Logic: Overview Propositional logic: (Sections 1.1-1.3)
The Foundations: Logic and Proofs. Part I
The Foundations: Logic and Proofs Part I Chapter Summary Propositional Logic n The Language of Propositions n Applications n Logical Equivalences Predicate Logic n The Language of Quantifiers n Logical
CSE 311: Foundations of Computing I. Lecture 1: Propositional Logic
CSE 311: Foundations of Computing I Lecture 1: Propositional Logic Some Perspective Computer Science and Engineering Programming CSE 14x Theory Hardware CSE 311 About the Course We will study the theory
About the course. CSE 321 Discrete Structures. Why this material is important. Topic List. Propositional Logic. Administration. From the CSE catalog:
About the course CSE 321 Discrete Structures Winter 2008 Lecture 1 Propositional Logic From the CSE catalog: CSE 321 Discrete Structures (4) Fundamentals of set theory, graph theory, enumeration, and algebraic
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
Logic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies
Logic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies Valentin Stockholm University September 2016 Propositions Proposition:
Examples. Example (1) Example (2) Let x, y be two variables, and denote statements p : x = 0 and q : y = 1. Solve. x 2 + (y 1) 2 = 0.
Examples Let x, y be two variables, and denote statements p : x = 0 and q : y = 1. Example (1) Solve x 2 + (y 1) 2 = 0. The solution is x = 0 AND y = 1. [p q.] Example (2) Solve x(y 1) = 0. Examples Let
2/13/2012. Logic: Truth Tables. CS160 Rosen Chapter 1. Logic?
Logic: Truth Tables CS160 Rosen Chapter 1 Logic? 1 What is logic? Logic is a truth-preserving system of inference Truth-preserving: If the initial statements are true, the inferred statements will be true
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,
Truth-Functional Logic
Truth-Functional Logic Syntax Every atomic sentence (A, B, C, ) is a sentence and are sentences With ϕ a sentence, the negation ϕ is a sentence With ϕ and ψ sentences, the conjunction ϕ ψ is a sentence
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
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 :
1 Propositional Logic
1 Propositional Logic Required reading: Foundations of Computation. Sections 1.1 and 1.2. 1. Introduction to Logic a. Logical consequences. If you know all humans are mortal, and you know that you are
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
Today s Topic: Propositional Logic
Today s Topic: Propositional Logic What is a proposition? Logical connectives and truth tables Translating between English and propositional logic Logic is the basis of all mathematical and analytical
Propositional Logic and Semantics
Propositional Logic and Semantics English is naturally ambiguous. For example, consider the following employee (non)recommendations and their ambiguity in the English language: I can assure you that no
Note: The area of logic that deals with propositions is called the propositional calculus or propositional logic.
Ch. 1.1 Logic Logic 1 Def. A Proposition is a statement that is either true or false. Example 1: Which of the following are propositions? Statement Proposition (yes or no) UHD is a University 1 + 3 = 0
~ p is always false. Based on the basic truth table for disjunction, if q is true then p ~
MAT 101 Solutions Exam 2 (Logic, Part I) Multiple-Choice Questions 1. D Because this sentence contains exactly ten words, it is stating that it is false. But if it is taken to be false, then it has to
Section 1.1 Propositional Logic. proposition : true = T (or 1) or false = F (or 0) (binary logic) the moon is made of green cheese
Section 1.1 Propositional Logic proposition : true = T (or 1) or false = F (or 0) (binary logic) the moon is made of green cheese go to town! X - imperative What time is it? X - interrogative propositional
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. Def. A Proposition is a statement that is either true or false.
Logic Logic 1 Def. A Proposition is a statement that is either true or false. Examples: Which of the following are propositions? Statement Proposition (yes or no) If yes, then determine if it is true or
Part I: Propositional Calculus
Logic Part I: Propositional Calculus Statements Undefined Terms True, T, #t, 1 False, F, #f, 0 Statement, Proposition Statement/Proposition -- Informal Definition Statement = anything that can meaningfully
Propositional Logic Logical Implication (4A) Young W. Lim 4/21/17
Propositional Logic Logical Implication (4A) Young W. Lim Copyright (c) 2016-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
2/2/2018. CS 103 Discrete Structures. Chapter 1. Propositional Logic. Chapter 1.1. Propositional Logic
CS 103 Discrete Structures Chapter 1 Propositional Logic Chapter 1.1 Propositional Logic 1 1.1 Propositional Logic Definition: A proposition :is a declarative sentence (that is, a sentence that declares
Solutions to Homework I (1.1)
Solutions to Homework I (1.1) Problem 1 Determine whether each of these compound propositions is satisable. a) (p q) ( p q) ( p q) b) (p q) (p q) ( p q) ( p q) c) (p q) ( p q) (a) p q p q p q p q p q (p
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
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.
CSE20: Discrete Mathematics for Computer Science. Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication
CSE20: Discrete Mathematics for Computer Science Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication Disjunctive normal form Example: Let f (x, y, z) =xy z. Write this function in DNF. Minterm
Propositional natural deduction
Propositional natural deduction COMP2600 / COMP6260 Dirk Pattinson Australian National University Semester 2, 2016 Major proof techniques 1 / 25 Three major styles of proof in logic and mathematics Model
Propositional Logic Resolution (6A) Young W. Lim 12/12/16
Propositional Logic Resolution (6A) 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,
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
Propositional Logic. Yimei Xiang 11 February format strictly follow the laws and never skip any step.
Propositional Logic Yimei Xiang yxiang@fas.harvard.edu 11 February 2014 1 Review Recursive definition Set up the basis Generate new members with rules Exclude the rest Subsets vs. proper subsets Sets of
Logic: Propositional Logic (Part I)
Logic: Propositional Logic (Part I) Alessandro Artale Free University of Bozen-Bolzano Faculty of Computer Science http://www.inf.unibz.it/ artale Descrete Mathematics and Logic BSc course Thanks to Prof.
Discrete Mathematics and Its Applications
Discrete Mathematics and Its Applications Lecture 1: Proposition logic MING GAO DASE @ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Sep. 12, 2017 Outline 1 Propositions 2 Connectives
Propositional Logic: Semantics
Propositional Logic: Semantics Alice Gao Lecture 4, September 19, 2017 Semantics 1/56 Announcements Semantics 2/56 The roadmap of propositional logic Semantics 3/56 FCC spectrum auction an application
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
Announcements. CS243: Discrete Structures. Propositional Logic II. Review. Operator Precedence. Operator Precedence, cont. Operator Precedence Example
Announcements CS243: Discrete Structures Propositional Logic II Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Tuesday 09/11 Weilin s Tuesday office hours are
Unit 1. Propositional Logic Reading do all quick-checks Propositional Logic: Ch. 2.intro, 2.2, 2.3, 2.4. Review 2.9
Unit 1. Propositional Logic Reading do all quick-checks Propositional Logic: Ch. 2.intro, 2.2, 2.3, 2.4. Review 2.9 Typeset September 23, 2005 1 Statements or propositions Defn: A statement is an assertion