Nested Quantifiers Goals: Explain how to work with nested quantiﬁers

Slides:

Advertisements

Similar presentations

Nested Quantifiers Section 1.4.

Advertisements

1.3 Predicates and Quantifiers

Tutorial 2: First Order Logic and Methods of Proofs

Nested Quantifiers Needed to express statements with multiple variables Example 1: “x+y = y+x for all real numbers”  x  y(x+y = y+x) where the domains.

The Logic of Quantified Statements

22C:19 Discrete Structures Logic and Proof Spring 2014 Sukumar Ghosh.

A Quick Look at Quantified Statements. Why are Quantified Statements Important? The logical structure of quantified statements provides a basis for the.

CSE 311 Foundations of Computing I Lecture 6 Predicate Logic Autumn 2011 CSE 3111.

CSE115/ENGR160 Discrete Mathematics 01/25/11 Ming-Hsuan Yang UC Merced 1.

Sets. Copyright © Peter Cappello Definition Visualize a dictionary as a directed graph. Nodes represent words If word w is defined in terms of word.

Nested Quantifiers Goals: Explain how to work with nested quantiﬁers

Discrete Structures Chapter 3: The Logic of Quantified Statements

Discrete Structures Chapter 3: The Logic of Quantified Statements

Propositional Equivalence Goal: Show how propositional equivalences are established & introduce the most important such equivalences.

Predicates and Quantifiers

Predicates & Quantifiers Goal: Introduce predicate logic, including existential & universal quantiﬁcation Introduce translation between English sentences.

Chapter 1: The Foundations: Logic and Proofs

CSci 2011 Discrete Mathematics Lecture 3 CSci 2011.

The Foundations: Logic and Proofs

CSCI 3328 Object Oriented Programming in C# Chapter 5: C# Control Statement – Part II UTPA – Fall

Computer Science 210 Computer Organization Introduction to Boolean Algebra.

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.

Chapter 1, Part II: Predicate Logic With Question/Answer Animations.

ㅎㅎ logical operator if if else switch while do while for Third step for Learning C++ Programming Repetition Control Structures.

Chapter 1, Part II: Predicate Logic With Question/Answer Animations.

Chapter 1: The Foundations: Logic and Proofs 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers.

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.

Math 51/COEN 19 Day 3, 1.4 Quantifiers 1. 3 Predicates A lot like functions that return booleans Let P(x) denote x

Chapter 2 The Logic of Quantified Statements. Section 2.1 Intro to Predicates & Quantified Statements.

Nested Quantifiers Section 1.5.

Sets Goal: Introduce the basic terminology of set theory.

1 Sections 1.3 and 1.4 Predicates & Quantifiers. 2 Propositional Functions In a mathematical assertion, such as x < 3, there are two parts: –the subject,

Lecture Predicates and Quantifiers 1.4 Nested Quantifiers.

Lecture 7 – Jan 28, Chapter 2 The Logic of Quantified Statements.

Discrete Structures – CNS 2300

Discrete Structures Predicate Logic 1 Dr. Muhammad Humayoun Assistant Professor COMSATS Institute of Computer Science, Lahore.

Predicates and Quantifiers Dr. Yasir Ali. 1.Predicates 2.Quantifiers a.Universal Quantifiers b.Existential Quantifiers 3.Negation of Quantifiers 4.Universal.

22C:19 Discrete Structures Logic and Proof Fall 2014 Sukumar Ghosh.

Copyright © Peter Cappello 2011 Predicates & Quantifiers.

PREDICATES AND QUANTIFIERS COSC-1321 Discrete Structures 1.

Section 1.4. Propositional Functions Propositional functions become propositions (and have truth values) when their variables are each replaced by a value.

Section 1.5. Section Summary Nested Quantifiers Order of Quantifiers Translating from Nested Quantifiers into English Translating Mathematical Statements.

Conditional Statements A conditional statement lets us choose which statement will be executed next Conditional statements give us the power to make basic.

رياضيات متقطعة لعلوم الحاسب MATH 226. Chapter 1 Predicates and Quantifiers 1.4.

CSE15 Discrete Mathematics 01/25/17

3. The Logic of Quantified Statements Summary

Computer Science 210 Computer Organization

Propositional Equivalence

Chapter 3 The Logic of Quantified Statements

Predicates & Quantifiers

Rules of inference for quantifiers

Negations of Quantified Statements

Chapter 1 The Foundations: Logic and Proofs

1.4 Predicates and Quantifiers

Computer Science 210 Computer Organization

Mathematics for Computer Science MIT 6.042J/18.062J

Nested Quantifiers Nested quantifiers are often necessary to express the meaning of sentences in English as well as important concepts in computer science.

Introduction to Predicates and Quantified Statements II

Statements Containing Multiple Quantifiers

Introduction to Predicates and Quantified Statements I

Discrete Mathematics Lecture 4 & 5: Predicate and Quantifier

Discrete Mathematics Lecture 4 & 5: Predicate and Quantifier

ICS 253: Discrete Structures I

Thinking Mathematically

Predicates and Quantifiers

Discrete Mathematics Lecture 4 Logic of Quantified Statements

Presentation transcript:

Nested Quantifiers Goals: Explain how to work with nested quantifiers Show that the order of quantification matters. Work with logical expressions involving multiple quantifiers.

Copyright © Peter Cappello 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 above statement true? Copyright © Peter Cappello

Copyright © Peter Cappello boolean axEyP() // x y P( x, y ) { for ( int x = 1; x <= 10; x++ ) boolean b = false; for ( int y = 1; y <= 10; y++ ) // y P( x, y ) if ( x > y ) b = true; break; // finding 1 y value is enough } if ( ! b ) return false; return true; Computational Interpretation Copyright © Peter Cappello

Copyright © Peter Cappello Multiple Quantifiers Legend: A B is valid x  y P(x, y) y  x P(x, y) y x P(x, y) x y P(x, y) x y P(x, y) y x P(x, y) y x P(x, y)  x y P(x, y) Copyright © Peter Cappello

Copyright © Peter Cappello Translate to English Let the domain be the real numbers. x y ( ( x ≥ 0  y < 0 )  x – y > 0 ) Is there something wrong with x ( ( x ≥ 0  y ( y < 0 ) )  x – y > 0 ) Copyright © Peter Cappello

Translate to a Logical Expression Let Q( s, q ) denote “s has been a contestant on quiz show q” I( s1, s2 ) denote “student s1 is student s2” The domain for s, s1, s2 is students at UCSB. The domain for q is quiz shows on TV. Give a logical expression for: Every TV quiz show has had a student from UCSB as a contestant. At least 2 students from UCSB have been contestants on Jeopardy. Copyright © Peter Cappello

Copyright © Peter Cappello Translations 1. q s Q( s, q ) Copyright © Peter Cappello

Copyright © Peter Cappello 2. s1 s2 ( I( s1, s2 )  Q( s1, Jeopardy )  Q( s2 , Jeopardy ) ) Copyright © Peter Cappello

Negating Nested Quantifiers Negate x y ( P( x, y )  Q( x, y ) ) so that no quantifiers are negated. x y ( P( x, y )  Q( x, y ) ). Copyright © Peter Cappello 9

Negating Nested Quantifiers Negate x y ( P( x, y )  Q( x, y ) ) so that no quantifiers are negated. x y ( P( x, y )  Q( x, y ) ). x y ( P( x, y )  Q( x, y ) ). Copyright © Peter Cappello

Negating Nested Quantifiers Negate x y ( P( x, y )  Q( x, y ) ) so that no quantifiers are negated. x y ( P( x, y )  Q( x, y ) ). x y ( P( x, y )  Q( x, y ) ). x y  ( P( x, y )  Q( x, y ) ). Copyright © Peter Cappello 11

Negating Nested Quantifiers Negate x y ( P( x, y )  Q( x, y ) ) so that no quantifiers are negated. x y ( P( x, y )  Q( x, y ) ). x y ( P( x, y )  Q( x, y ) ). x y  ( P( x, y )  Q( x, y ) ). x y (  P( x, y )   Q( x, y ) ). Copyright © Peter Cappello 12 12