Discrete Mathematics and its Applications
|
|
|
- Madlyn Randall
- 5 years ago
- Views:
Transcription
1 Discrete Mathematics and its Applications Ngày 14 tháng 9 năm 2011 () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
2 Logic-2 (Applications) Applications of logic are abundent. Systems specifications where complex systems are designed use logic. Chips, web searches, programming languages, puzzle solving all use logic. For instance, Sudoku puzzles can be solved using logic. Just introduce 729 propositions p(m, n, k) which will mean the number k goes in row m column n then construct compound propositions that state that in every row, in every column and in the nine 3 3 blocks every integer from 1 to 9 must appear. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
3 Logic-2 (Applications) Applications of logic are abundent. Systems specifications where complex systems are designed use logic. Chips, web searches, programming languages, puzzle solving all use logic. For instance, Sudoku puzzles can be solved using logic. Just introduce 729 propositions p(m, n, k) which will mean the number k goes in row m column n then construct compound propositions that state that in every row, in every column and in the nine 3 3 blocks every integer from 1 to 9 must appear. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
4 Logic-2 (Applications) Applications of logic are abundent. Systems specifications where complex systems are designed use logic. Chips, web searches, programming languages, puzzle solving all use logic. For instance, Sudoku puzzles can be solved using logic. Just introduce 729 propositions p(m, n, k) which will mean the number k goes in row m column n then construct compound propositions that state that in every row, in every column and in the nine 3 3 blocks every integer from 1 to 9 must appear. One of the most important development that had a profound impact on our life, is our ability to implement logical operations by electronic devices. The generic name of electronic devices that implement logic operations is logic gates. Imagine, withoout logic gates there would be no Facebook. Can you imagine living in a world without facebook? () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
5 How important are logic gates? Walther Bothe received the Nobel Price in physics in 1954 for his development of the first electronic device that implemented the gate. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
6 How important are logic gates? Walther Bothe received the Nobel Price in physics in 1954 for his development of the first electronic device that implemented the gate. Today, we know how to implement logic gates at the molecular level. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
7 How important are logic gates? Walther Bothe received the Nobel Price in physics in 1954 for his development of the first electronic device that implemented the gate. Today, we know how to implement logic gates at the molecular level. Most programming languages implement four logical operators: 1 not () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
8 How important are logic gates? Walther Bothe received the Nobel Price in physics in 1954 for his development of the first electronic device that implemented the gate. Today, we know how to implement logic gates at the molecular level. Most programming languages implement four logical operators: 1 not 2 and () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
9 How important are logic gates? Walther Bothe received the Nobel Price in physics in 1954 for his development of the first electronic device that implemented the gate. Today, we know how to implement logic gates at the molecular level. Most programming languages implement four logical operators: 1 not 2 and 3 or () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
10 How important are logic gates? Walther Bothe received the Nobel Price in physics in 1954 for his development of the first electronic device that implemented the gate. Today, we know how to implement logic gates at the molecular level. Most programming languages implement four logical operators: 1 not 2 and 3 or 4 implies (usually through the reserved word if) () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
11 How important are logic gates? Walther Bothe received the Nobel Price in physics in 1954 for his development of the first electronic device that implemented the gate. Today, we know how to implement logic gates at the molecular level. Most programming languages implement four logical operators: 1 not 2 and 3 or 4 implies (usually through the reserved word if) Definition Two propositions are logically equivalent if they have the same truth table. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
12 How important are logic gates? Walther Bothe received the Nobel Price in physics in 1954 for his development of the first electronic device that implemented the gate. Today, we know how to implement logic gates at the molecular level. Most programming languages implement four logical operators: 1 not 2 and 3 or 4 implies (usually through the reserved word if) Definition Two propositions are logically equivalent if they have the same truth table. Example p (q r) and (p q) (p r) are logically equivalent. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
13 There are a few surprises when considering equivalence of propositions. Here is a list of some of them: 1 Are (p q) s and p (q s) equivalent? () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
14 There are a few surprises when considering equivalence of propositions. Here is a list of some of them: 1 Are (p q) s and p (q s) equivalent? 2 Are p q and q p equivalent? () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
15 There are a few surprises when considering equivalence of propositions. Here is a list of some of them: 1 Are (p q) s and p (q s) equivalent? 2 Are p q and q p equivalent? 3 What about p (q s) and (p q) s? () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
16 There are a few surprises when considering equivalence of propositions. Here is a list of some of them: 1 Are (p q) s and p (q s) equivalent? 2 Are p q and q p equivalent? 3 What about p (q s) and (p q) s? Definition A set of logical operators is a functionally complete collection if every truth table has an equivalent proposition using only operators from this set. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
17 There are a few surprises when considering equivalence of propositions. Here is a list of some of them: 1 Are (p q) s and p (q s) equivalent? 2 Are p q and q p equivalent? 3 What about p (q s) and (p q) s? Definition A set of logical operators is a functionally complete collection if every truth table has an equivalent proposition using only operators from this set. Comment In the next slide we shall see some examples of functionally complete collections. A bit surprising is that the and each by itself is a functionally complete collection. It is also noteworthy that the nand gate is the cheapest to build. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
18 The proposition (p q r s t) is false only if the five propositions inside it are all false. In all other cases it will be true. Alternatively, its truth table has 32 entries, 31 of them are true and only one is false. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
19 The proposition (p q r s t) is false only if the five propositions inside it are all false. In all other cases it will be true. Alternatively, its truth table has 32 entries, 31 of them are true and only one is false. Similary, the proposition (p q r s t) is true if and only if the five propositions inside are all true. In all other cases it will be false. Alternatively, its truth table has 32 entries, 31 of them are false and onlyone is true. Definition () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
20 The proposition (p q r s t) is false only if the five propositions inside it are all false. In all other cases it will be true. Alternatively, its truth table has 32 entries, 31 of them are true and only one is false. Similary, the proposition (p q r s t) is true if and only if the five propositions inside are all true. In all other cases it will be false. Alternatively, its truth table has 32 entries, 31 of them are false and onlyone is true. Definition 1 A proposition of the form (p 1 p 2... p k ) is called a disjunction. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
21 The proposition (p q r s t) is false only if the five propositions inside it are all false. In all other cases it will be true. Alternatively, its truth table has 32 entries, 31 of them are true and only one is false. Similary, the proposition (p q r s t) is true if and only if the five propositions inside are all true. In all other cases it will be false. Alternatively, its truth table has 32 entries, 31 of them are false and onlyone is true. Definition 1 A proposition of the form (p 1 p 2... p k ) is called a disjunction. 2 A proposition of the form (p 1 p 2... p k ) is called a conjunction. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
22 The proposition (p q r s t) is false only if the five propositions inside it are all false. In all other cases it will be true. Alternatively, its truth table has 32 entries, 31 of them are true and only one is false. Similary, the proposition (p q r s t) is true if and only if the five propositions inside are all true. In all other cases it will be false. Alternatively, its truth table has 32 entries, 31 of them are false and onlyone is true. Definition 1 A proposition of the form (p 1 p 2... p k ) is called a disjunction. 2 A proposition of the form (p 1 p 2... p k ) is called a conjunction. 3 A proposition of the form (D 1 D 2... D m ) where each D i is a disjunction is called a conjunction of disjunctions. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
23 The proposition (p q r s t) is false only if the five propositions inside it are all false. In all other cases it will be true. Alternatively, its truth table has 32 entries, 31 of them are true and only one is false. Similary, the proposition (p q r s t) is true if and only if the five propositions inside are all true. In all other cases it will be false. Alternatively, its truth table has 32 entries, 31 of them are false and onlyone is true. Definition 1 A proposition of the form (p 1 p 2... p k ) is called a disjunction. 2 A proposition of the form (p 1 p 2... p k ) is called a conjunction. 3 A proposition of the form (D 1 D 2... D m ) where each D i is a disjunction is called a conjunction of disjunctions. 4 Similarly, a proposition of the form (C 1 C 2... C k ) where each C i is a conjunction is called a disjunction of conjunctions. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
24 Theorem Every truth table represents a proposition which has an equivalent conjunction of disjunctions. This implies that,, is a functionally complete collection of logic operators. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
25 Theorem Every truth table represents a proposition which has an equivalent conjunction of disjunctions. This implies that,, is a functionally complete collection of logic operators. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
26 Theorem Every truth table represents a proposition which has an equivalent conjunction of disjunctions. This implies that,, is a functionally complete collection of logic operators. Every truth table represents a proposition which has an equivalent disjunction of conjunctions. Chứng minh. Start writing... () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
27 Theorem Every truth table represents a proposition which has an equivalent conjunction of disjunctions. This implies that,, is a functionally complete collection of logic operators. Every truth table represents a proposition which has an equivalent disjunction of conjunctions. Chứng minh. Start writing... Comment This explains why programming languages implement only four logical operators. All others can be easily implemented using them. See for example the Sage example. Even though as we shall soon see, we can use less operators, the choice to implement those is to make it more convenient for the programmer. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
28 Definition A proposition which is true for every assertion (assigning truth values to the variables) is called a tautology. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
29 Definition A proposition which is true for every assertion (assigning truth values to the variables) is called a tautology. Example The simplest example is the proposition p p. We will see other, more complicated examples in the exercises and drills. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
30 Definition A proposition which is true for every assertion (assigning truth values to the variables) is called a tautology. Example The simplest example is the proposition p p. We will see other, more complicated examples in the exercises and drills. Definition A proposition which is false for all possible assertions is called a contradiction. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
31 Definition A proposition which is true for every assertion (assigning truth values to the variables) is called a tautology. Example The simplest example is the proposition p p. We will see other, more complicated examples in the exercises and drills. Definition A proposition which is false for all possible assertions is called a contradiction. Example Again, the simplest example is the proposition p p. () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm / 6
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
Sec$on Summary. Tautologies, Contradictions, and Contingencies. Logical Equivalence. Normal Forms (optional, covered in exercises in text)
Section 1.3 1 Sec$on Summary Tautologies, Contradictions, and Contingencies. Logical Equivalence Important Logical Equivalences Showing Logical Equivalence Normal Forms (optional, covered in exercises
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,...)
Syntax of propositional logic. Syntax tree of a formula. Semantics of propositional logic (I) Subformulas
Syntax of propositional logic Syntax tree of a formula An atomic formula has the form A i where i =, 2, 3,.... Formulas are defined by the following inductive process: Every formula can be represented
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
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
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
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:
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,
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)
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
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
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
Lecture 02: Propositional Logic
Lecture 02: Propositional Logic CSCI 358 Discrete Mathematics, Spring 2016 Hua Wang, Ph.D. Department of Electrical Engineering and Computer Science January 19, 2015 Propositional logic Propositional logic
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
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
Announcements CompSci 102 Discrete Math for Computer Science
Announcements CompSci 102 Discrete Math for Computer Science Read for next time Chap. 1.4-1.6 Recitation 1 is tomorrow Homework will be posted by Friday January 19, 2012 Today more logic Prof. Rodger Most
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
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
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
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
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
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 Truth Tables, Equivalences and the Contrapositive
2 Truth Tables, Equivalences and the Contrapositive 12 2 Truth Tables, Equivalences and the Contrapositive 2.1 Truth Tables In a mathematical system, true and false statements are the statements of the
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?
THE LOGIC OF COMPOUND STATEMENTS
CHAPTER 2 THE LOGIC OF COMPOUND STATEMENTS Copyright Cengage Learning. All rights reserved. SECTION 2.1 Logical Form and Logical Equivalence Copyright Cengage Learning. All rights reserved. Logical Form
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
CSE 20 DISCRETE MATH WINTER
CSE 20 DISCRETE MATH WINTER 2016 http://cseweb.ucsd.edu/classes/wi16/cse20-ab/ Reminders Exam 1 in one week One note sheet ok Review sessions Saturday / Sunday Assigned seats: seat map on Piazza shortly
Part Two: The Basic Components of the SOFL Specification Language
Part Two: The Basic Components of the SOFL Specification Language SOFL logic Module Condition Data Flow Diagrams Process specification Function definition and specification Process decomposition Other
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
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
We last time we began introducing equivalency laws.
Monday, January 14 MAD2104 Discrete Math 1 Course website: www/mathfsuedu/~wooland/mad2104 Today we will continue in Course Notes Chapter 22 We last time we began introducing equivalency laws Today we
Discrete Structures of Computer Science Propositional Logic I
Discrete Structures of Computer Science Propositional Logic I Gazihan Alankuş (Based on original slides by Brahim Hnich) July 26, 2012 1 Use of Logic 2 Statements 3 Logic Connectives 4 Truth Tables Use
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
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
Discrete Mathematics & Mathematical Reasoning Predicates, Quantifiers and Proof Techniques
Discrete Mathematics & Mathematical Reasoning Predicates, Quantifiers and Proof Techniques Colin Stirling Informatics Some slides based on ones by Myrto Arapinis Colin Stirling (Informatics) Discrete Mathematics
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
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
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
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
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,
Boolean Algebra. Philipp Koehn. 9 September 2016
Boolean Algebra Philipp Koehn 9 September 2016 Core Boolean Operators 1 AND OR NOT A B A and B 0 0 0 0 1 0 1 0 0 1 1 1 A B A or B 0 0 0 0 1 1 1 0 1 1 1 1 A not A 0 1 1 0 AND OR NOT 2 Boolean algebra Boolean
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
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
PHIL12A Section answers, 16 February 2011
PHIL12A Section answers, 16 February 2011 Julian Jonker 1 How much do you know? 1. Show that the following sentences are equivalent. (a) (Ex 4.16) A B A and A B A B (A B) A A B T T T T T T T T T T T F
Logic and Proofs. Jan COT3100: Applications of Discrete Structures Jan 2007
COT3100: Propositional Equivalences 1 Logic and Proofs Jan 2007 COT3100: Propositional Equivalences 2 1 Translating from Natural Languages EXAMPLE. Translate the following sentence into a logical expression:
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
Introduction Propositional Logic
Discrete Mathematics for CSE of KU Introduction Propositional Logic Instructor: Kangil Kim (CSE) E-mail: kikim01@konkuk.ac.kr Tel. : 02-450-3493 Room : New Milenium Bldg. 1103 Lab : New Engineering Bldg.
Discrete Mathematics
Discrete Mathematics Discrete mathematics is devoted to the study of discrete or distinct unconnected objects. Classical mathematics deals with functions on real numbers. Real numbers form a continuous
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
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.
Propositional Logic. Logical Expressions. Logic Minimization. CNF and DNF. Algebraic Laws for Logical Expressions CSC 173
Propositional Logic CSC 17 Propositional logic mathematical model (or algebra) for reasoning about the truth of logical expressions (propositions) Logical expressions propositional variables or logical
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
1.3 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
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
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
UNIT-I: Propositional Logic
1. Introduction to Logic: UNIT-I: Propositional Logic Logic: logic comprises a (formal) language for making statements about objects and reasoning about properties of these objects. Statements in a logical
Section 1.1: Propositions and Connectives
Section 1.1: Propositions and Connectives Definition : Proposition: A statement that has exactly one truth value, i.e., either true (T) or false (F). Axiom (or Postulate): A statement about the primitive
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
Phil Introductory Formal Logic
Phil 134 - Introductory Formal Logic Lecture 4: Formal Semantics In this lecture we give precise meaning to formulae math stuff: sets, pairs, products, relations, functions PL connectives as truth-functions
THE LOGIC OF COMPOUND STATEMENTS
CHAPTER 2 THE LOGIC OF COMPOUND STATEMENTS Copyright Cengage Learning. All rights reserved. SECTION 2.4 Application: Digital Logic Circuits Copyright Cengage Learning. All rights reserved. Application:
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:
~ 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
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
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)
ON1 Big Mock Test 4 October 2013 Answers This test counts 4% of assessment for the course. Time allowed: 10 minutes
ON1 Big Mock Test 4 October 2013 Answers This test counts 4% of assessment for the course. Time allowed: 10 minutes The actual test will contain only 2 questions. Marking scheme: In each multiple choice
Propositional Logic Not Enough
Section 1.4 Propositional Logic Not Enough If we have: All men are mortal. Socrates is a man. Does it follow that Socrates is mortal? Can t be represented in propositional logic. Need a language that talks
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 20 DISCRETE MATH WINTER
CSE 20 DISCRETE MATH WINTER 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Reminders Homework 3 due Sunday at noon Exam 1 in one week One note card can be used. Bring photo ID. Review sessions Thursday
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:
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
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
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
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.
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
Natural Deduction for Propositional Logic
Natural Deduction for Propositional Logic Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan September 10, 2018 Bow-Yaw Wang (Academia Sinica) Natural Deduction for Propositional Logic
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
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
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:
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,
Overview. I Review of natural deduction. I Soundness and completeness. I Semantics of propositional formulas. I Soundness proof. I Completeness proof.
Overview I Review of natural deduction. I Soundness and completeness. I Semantics of propositional formulas. I Soundness proof. I Completeness proof. Propositional formulas Grammar: ::= p j (:) j ( ^ )
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
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
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,
Named numbres. Ngày 25 tháng 11 năm () Named numbres Ngày 25 tháng 11 năm / 7
Named numbres Ngày 25 tháng 11 năm 2011 () Named numbres Ngày 25 tháng 11 năm 2011 1 / 7 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. () Named
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
Philosophy 220. Truth-Functional Equivalence and Consistency
Philosophy 220 Truth-Functional Equivalence and Consistency Review Logical equivalency: The members of a pair of sentences [of natural language] are logically equivalent if and only if it is not [logically]
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.
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
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,
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
Propositional logic (revision) & semantic entailment. p. 1/34
Propositional logic (revision) & semantic entailment p. 1/34 Reading The background reading for propositional logic is Chapter 1 of Huth/Ryan. (This will cover approximately the first three lectures.)
1. Propositions: Contrapositives and Converses
Preliminaries 1 1. Propositions: Contrapositives and Converses Given two propositions P and Q, the statement If P, then Q is interpreted as the statement that if the proposition P is true, then the statement
Chapter 1: Formal Logic
Chapter 1: Formal Logic Dr. Fang (Daisy) Tang ftang@cpp.edu www.cpp.edu/~ftang/ CS 130 Discrete Structures Logic: The Foundation of Reasoning Definition: the foundation for the organized, careful method
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
https://vu5.sfc.keio.ac.jp/slide/
1 FUNDAMENTALS OF LOGIC NO.2 PROPOSITION AND TRUTH VALUE Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ Proposition A Proposition is a statement of which truth does
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?
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
Propositional Logic Review
Propositional Logic Review UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane The task of describing a logical system comes in three parts: Grammar Describing what counts as a formula Semantics Defining
Logic Gate Level. Part 2
Logic Gate Level Part 2 Constructing Boolean expression from First method: write nonparenthesized OR of ANDs Each AND is a 1 in the result column of the truth table Works best for table with relatively