Statements Containing Multiple Quantifiers

Slides:



Advertisements
Similar presentations
Nested Quantifiers Section 1.4.
Advertisements

1.3 Predicates and Quantifiers
The "if structure" is used to execute statement(s) only if the given condition is satisfied.
RMIT University; Taylor's College This is a story about four people named Everybody, Somebody, Anybody and Nobody. There was an important job to be done.
22C:19 Discrete Structures Logic and Proof Spring 2014 Sukumar Ghosh.
Discrete Mathematics Lecture 2 Alexander Bukharovich New York University.
Nested Quantifiers Goals: Explain how to work with nested quantifiers
Discrete Structures Chapter 3: The Logic of Quantified Statements
Quantifiers and Negation Predicates. Predicate Will define function. The domain of a function is the set from which possible values may be chosen for.
Section 1.3: Predicates and Quantifiers
Lecture 8 Introduction to Logic CSCI – 1900 Mathematics for Computer Science Fall 2014 Bill Pine.
CSCI 115 Chapter 2 Logic. CSCI 115 §2.1 Propositions and Logical Operations.
Mathematical Structures A collection of objects with operations defined on them and the accompanying properties form a mathematical structure or system.
A Brief Summary for Exam 1 Subject Topics Propositional Logic (sections 1.1, 1.2) –Propositions Statement, Truth value, Proposition, Propositional symbol,
Nested Quantifiers. 2 Nested Iteration Let the domain be {1, 2, …, 10}. Let P(x, y) denote x > y.  x,  y, P(x, y) means  x, (  y, P(x, y) ) Is the.
Predicates and Quantified Statements M , 3.2.
Multiplying and Dividing Integers When you MULTIPLY: Two positives equal a positive Two negatives equal a positive One positive & one negative equal.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.1, Slide Logic The Study of What’s True or False or Somewhere in Between.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.1, Slide Logic The Study of What’s True or False or Somewhere in Between.
Nesting Quantifiers And Their Manipulation Copyright © Curt Hill.
Statements with Multiple Quantifiers. When a statement contains more than one quantifier, we imagine the actions suggested by the quantifiers as being.
Chapter 2 The Logic of Quantified Statements. Section 2.1 Intro to Predicates & Quantified Statements.
Multiplication of Real Numbers Section 2.5. Multiplying Rules 1) If the numbers have the same signs then the answer is positive. (-7) (-4) = 28 2) If.
4.1 Proofs and Counterexamples. Even Odd Numbers Find a property that describes each of the following sets E={…, -4, -2, 0, 2, 4, 6, …} O={…, -3, -1,
Divide Rational Numbers. Objective The student will be able to:
CSS342: Quantifiers1 Professor: Munehiro Fukuda. CSS342: Quantifiers2 Review of Propositions Proposition: a statement that is either true or false, but.
Lecture Predicates and Quantifiers 1.4 Nested Quantifiers.
Lecture 7 – Jan 28, Chapter 2 The Logic of Quantified Statements.
COMP 170 L2 L08: Quantifiers. COMP 170 L2 Outline l Quantifiers: Motivation and Concepts l Quantifiers: Notations and Meaning l Saying things with Quantified.
Logical Operations – Page 1CSCI 1900 – Discrete Structures CSCI 1900 Discrete Structures Logical Operations Reading: Kolman, Section 2.1.
Discrete Structures – CNS 2300
Basic Definitions of Set Theory Lecture 24 Section 5.1 Fri, Mar 2, 2007.
Statements Containing Multiple Quantifiers Lecture 11 Section 2.3 Mon, Feb 5, 2007.
321 Section, Week 2 Natalie Linnell. Extra Credit problem.
Section 1.5 and 1.6 Predicates and Quantifiers. Vocabulary Predicate Domain Universal Quantifier Existential Quantifier Counterexample Free variable Bound.
Direct Proof and Counterexample I Lecture 11 Section 3.1 Fri, Jan 28, 2005.
Section 1.4. Propositional Functions Propositional functions become propositions (and have truth values) when their variables are each replaced by a value.
Introduction to Predicates and Quantified Statements I Lecture 9 Section 2.1 Wed, Jan 31, 2007.
Logic Hubert Chan [O1 Abstract Concepts] [O2 Proof Techniques]
Conditional Statements
3. The Logic of Quantified Statements Summary
CSNB 143 Discrete Mathematical Structures
COT 3100, Spring 2001 Applications of Discrete Structures
Theory of Computation Lecture 12: A Universal Program III
Nested Quantifiers Goals: Explain how to work with nested quantifiers
Chapter 3 Introduction to Logic 2012 Pearson Education, Inc.
Logic Hubert Chan [O1 Abstract Concepts] [O2 Proof Techniques]
Negations of Quantified Statements
Chapter 3 Introduction to Logic 2012 Pearson Education, Inc.
Copyright © Cengage Learning. All rights reserved.
1.4 Predicates and Quantifiers
Introduction to Predicates and Quantified Statements II
Chapter 3 Introduction to Logic © 2008 Pearson Addison-Wesley.
Mathematics for Computer Science MIT 6.042J/18.062J
Introduction to Predicates and Quantified Statements II
Statements Containing Multiple Quantifiers
A Brief Summary for Exam 1
Chapter 1 Logic and Proof.
(1.4) An Introduction to Logic
Introduction to Predicates and Quantified Statements I
Direct Proof and Counterexample I
Discrete Mathematics Lecture 4 Logic of Quantified Statements
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Chapter 3: Selection Structures: Making Decisions
Chapter 2 Sets Active Learning Lecture Slides
Properties of Operations
Theory of Computation Lecture 11: A Universal Program III
Lecture 1.3: Predicate Logic
Presentation transcript:

Statements Containing Multiple Quantifiers Lecture 9 Section 2.3 Fri, Feb 3, 2006

Multiply Quantified Statements Multiple universal statements x  S, y  T, P(x, y) y  T, x  S, P(x, y) The order does not matter. Multiple existential statements x  S, y  T, P(x, y) y  T, x  S, P(x, y)

Mixed Quantifiers Mixed universal and existential statements Compare x  S, y  T, P(x, y) y  T, x  S, P(x, y) The order does matter. What is the difference? Compare x  R, y  R, x + y = 0. y  R, x  R, x + y = 0.

Mixed Quantifiers “There is a woman who is right for every man.” “For every man, there is a woman who is right for him.”

Examples Which of the following statements are true? x  N, y  N, y < x. x  Q, y  Q, y < x. x  R, y  R, y < x. x  Q, y  Q, z  Q, x < z < y. For those that are false, what is their negation?

Negation of Multiply Quantified Statements Negate the statement x  R, y  R, z  R, x + y + z = 0. (x  R, y  R, z  R, x + y + z = 0)

Negation of Multiply Quantified Statements Negate the statement x  R, y  R, z  R, x + y + z = 0. (x  R, y  R, z  R, x + y + z = 0)  x  R, (y  R, z  R, x + y + z = 0)

Negation of Multiply Quantified Statements Negate the statement x  R, y  R, z  R, x + y + z = 0. (x  R, y  R, z  R, x + y + z = 0)  x  R, (y  R, z  R, x + y + z = 0)  x  R, y  R, (z  R, x + y + z = 0)

Negation of Multiply Quantified Statements Negate the statement x  R, y  R, z  R, x + y + z = 0. (x  R, y  R, z  R, x + y + z = 0)  x  R, (y  R, z  R, x + y + z = 0)  x  R, y  R, (z  R, x + y + z = 0)  x  R, y  R, z  R, (x + y + z = 0)

Negation of Multiply Quantified Statements Negate the statement x  R, y  R, z  R, x + y + z = 0. (x  R, y  R, z  R, x + y + z = 0)  x  R, (y  R, z  R, x + y + z = 0)  x  R, y  R, (z  R, x + y + z = 0)  x  R, y  R, z  R, (x + y + z = 0)  x  R, y  R, z  R, x + y + z  0

Multiply Quantified Statements In the statement x  R, y  R, z  R, x + y + z  0 the predicate x + y + z  0 must be true for every y and for some x and for some z. However, the choice of x must not depend on y, while the choice of z may depend on y.

Negation of Multiply Quantified Statements Consider the statement n  N, r, s, t  N, n = r2 + s2 + t2. Its negation is n  N, r, s, t  N, n  r2 + s2 + t2. Which statement is true? How would you prove it?

Example Write the following statement using quantifiers. “There exists a computer program that can read the code of any computer program and determine whether that program will eventually halt when it is executed.”

Example Write the following statement using quantifiers. “For every computer program, there exists a computer program that can read its code and determine whether it will eventually halt when it is executed.”

Superbowl Predictor I have a computer program that will correctly predict whether the Seattle Seahawks will win the Superbowl on Sunday.

Superbowl Predictor Example1.cpp Example2.cpp