# CS 2336 Discrete Mathematics

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CS 2336 Discrete Mathematics Lecture 2 Logic: Predicate Calculus 1

2 Outline Predicates Quantifiers Binding Applications Logical Equivalences 2

3 Predicates In mathematics arguments, we will often see sentences containing variables, such as: x > 0 x = y + 3 Computer x is functioning properly For each sentence, we call the variables as the subject of the sentence the other part, which describes the property of the variables, as the predicate of the sentence 3

4 Predicates Take x > 0 as an example x : the subject > 0: the predicate Once the value of x is assigned, the above sentence becomes a proposition and has a truth value We can denote it as some function P(x ) of x P is called a propositional function 4

5 Predicates Example 1: Let P(x ) denote the sentence x > 0. What are the truth values of P(0) and P(1)? Example 2: Let Q(x, y ) denote the sentence x = y + 3. What are the truth values of Q(1,2) and Q(3,0)? 5

6 Quantifiers In English, the words all, some, many, none, few are used to express some property (predicate) is true over a range of subjects These words are called quantifiers In mathematics, two important quantifiers are commonly used to create a proposition from a propositional function: universal quantifier and existential quantifier 6

7 Universal Quantifier Many mathematical statements say that a property is true for all values of a variable, when values are chosen from some domain Examples: z(z + 1)(z + 2) is divisible by 6 for all integer z q 2 is rational for all rational number q r 3 > 0 for all positive real number r Important Note: Domain needs to be specified! 7

8 Universal Quantifier Universal Quantifier The universal quantification of P(x ) is the proposition P(x ) for all values of x in the domain. The notation x P(x ) represents the above proposition. A value of x making the proposition false is called a counter-example. 8

9 Universal Quantifier If all values in the domain can be listed, say x 1, x 2,, x k, then x P(x ) is the same as P(x 1 ) P(x 2 ) P(x k ) Example: What is the truth value of x (x 10 ) when the domain consists of all positive integers not exceeding 3? What is the truth value of P(1) P(2) P(3)? 9

10 Test Your Understanding What is the truth value of x (x 2 x ) if the domain consists of all real numbers? if the domain consists of all integers? 10

11 Test Your Understanding (Solution) False, if the domain consists of all real numbers. In particular, the case x = 0.5 is a counter-example. True, if the domain consists of all integers. To see this, we notice the following equivalences: x 2 x x (x 1) 0 x 0 or x 1 Thus, x 2 x cannot be false, since there are no integers x with 0 x 1. 11

12 Existential Quantifier Many mathematical statements say that a property is true for some value of a variable, when values are chosen from some domain Examples: z is a prime for some non-negative integer z r s is rational for some irrational numbers r and s Important Note: Domain needs to be specified! 12

13 Existential Quantifier Existential Quantifier The existential quantification of P(x ) is the proposition P(x ) for some value of x in the domain. The notation x P(x ) represents the above proposition. The proposition is false if and only if P(x ) is false for all values of x. 13

14 Existential Quantifier If all values in the domain can be listed, say x 1, x 2,, x k, then x P(x ) is the same as P(x 1 ) P(x 2 ) P(x k ) Example: What is the truth value of x (x 0 ) when the domain consists of all positive integers not exceeding 3? What is the truth value of P(1) P(2) P(3)? 14

15 Test Your Understanding What is the truth value of z (z 2 10 ) if the domain consists of all positive integers not exceeding 3? if the domain consists of all integers not exceeding 3? 15

16 Quantifiers with Restricted Domain Sometimes, we want to simplify the writing by using short-hand notation Assuming the domain consists of all integers, guess what does each of the following mean? x 0 (x 2 0 ) y 0 (y 3 0 ) z 0 (z 2 = 10 ) 16

17 Quantifiers with Restricted Domain x 0 (x 2 0 ) means For every x in the domain with x 0, x 2 0. The proposition is the same as: x (x 0 x 2 0 ) z 0 (z 2 = 10 ) means There is some z in the domain with z 0, z 2 = 10. The proposition is the same as: z (z 0 z 2 = 10 ) 17

18 Binding Variables If there is a quantifier used on a variable x, we say the variable is bound. Else it is free. Ex: In x (x + y = 1 ), x is bound and y is free If all variables in a propositional function are bound, the function becomes a proposition Ex: y x (x + y = 1 ) is a proposition 18

19 Multiple Quantifiers In the last example, we have a proposition y x (x + y = 1 ) with two quantifiers, where y is applied to x (x + y = 1 ), and x is applied to x + y = 1 The part of the logical expression where a quantifier is applied is called the scope of that quantifier 19

20 Multiple Quantifiers How about this? y 0 (y 3 0 ) x (x = 1 ) What is the scope of y? Quantifier is assumed to have a higher precedence than logical operators, so the above is the same as: x 0 (x 3 0 ) x (x = 1 ) 20

21 The Order of Quantifiers Order in which quantifiers appear is important Example: Suppose that the domain for both x and y are integers. What are the truth values of the following? 1. y x (x + y = 1 ) 2. x y (x + y = 1 ) 21

22 The Order of Quantifiers Two special cases where the order of quantifiers is not important are: 1. All quantifiers are universal quantifiers 2. All quantifiers are existential quantifiers Example: x y (x + y = 1 ) means the same as y x (x + y = 1 ) 22

23 Applications: English Translation How to translate the following sentence Every student in this class has studied Calculus. into a logical expression, if Q(x ) denotes x has studied Calculus, and the domain of x is all students in this class? What if the domain of x consists of all students in NTHU? 23

24 Applications: English Translation How to translate the following sentences 1. All lions are fierce. 2. Some lion does not drink coffee. 3. Some fierce creatures do not drink coffee. into logical expressions, if P(x ) := x is a lion, Q(x ) := x is fierce, R(x ) := x drinks coffee, and the domain of x consists of all creatures? 24

25 Applications: English Translation How to translate the following sentence If a person is a female and is a parent, then this person is someone s mother into a logical expression, if F(x ) := x is a female, P(x ) := x is a parent, M(x, y ) := x is a mother of y, and the domain consists of all people? 25

26 Applications: English Translation How to translate the following sentence Every person has exactly one best friend into a logical expression, if B(x, y ) := y is a best friend of x, and the domain consists of all people? 26

27 Applications: English Translation How to translate the following sentence There is a woman who has taken a flight on every airline in the world into a logical expression, if P(w, f ) := w has taken a flight f, Q( f, a ) := f is a flight of airline a, and the domains of w, f, a consist of all women in the world, all airplane flights, and all airlines, respectively? 27

28 Applications: Math Translation How to translate the statements 1. The sum of two positive integers is always positive 2. Every real number, except 0, can find some real number such that their product is 1 into logical expressions? 28

29 Applications: Translating Expression How to translate the following expression x y z (( F(x, y ) F(x, z ) (y z )) F(y, z ) ) into English, if F(x, y ) := x is a friend of y, and the domain consists of all people? 29

30 Logical Equivalences As in the case of propositions, some common logical equivalences have been derived for predicates and quantifiers Examples: 1. x y P(x, y ) y x P(x, y ) 2. x y P(x, y ) y x P(x, y ) hold for any predicate P, and any domain 30

31 Logical Equivalences De Morgan s Laws 1. x P(x ) x P(x ) 2. x P(x ) x P(x ) Can you show that x (P(x ) Q(x ) ) is equivalent to x (P(x ) Q(x ) )? 31

### Predicate Logic. Lucia Moura. Winter Predicates and Quantifiers Nested Quantifiers Using Predicate Calculus

Predicate Logic Winter 2010 Predicates A Predicate is a declarative sentence whose true/false value depends on one or more variables. The statement x is greater than 3 has two parts: the subject: x is

### Discrete Mathematics, Chapter : Predicate Logic

Discrete Mathematics, Chapter 1.4-1.5: Predicate Logic Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 1 / 23 Outline 1 Predicates

### Predicates and Quantifiers. Niloufar Shafiei

Predicates and Quantifiers Niloufar Shafiei Review Proposition: 1. It is a sentence that declares a fact. 2. It is either true or false, but not both. Examples: 2 + 1 = 3. True Proposition Toronto is the

### 1.4 Predicates and Quantifiers

Can we express the following statement in propositional logic? Every computer has a CPU No 1 Can we express the following statement in propositional logic? No Every computer has a CPU Let's see the new

### DISCRETE MATH: LECTURE 4

DISCRETE MATH: LECTURE 4 DR. DANIEL FREEMAN 1. Chapter 3.1 Predicates and Quantified Statements I A predicate is a sentence that contains a finite number of variables and becomes a statement when specific

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

Chapter 1, Part II: Predicate Logic With Question/Answer Animations Summary Predicate Logic (First-Order Logic (FOL) The Language of Quantifiers Logical Equivalences Nested Quantifiers Translation from

### 1.4/1.5 Predicates and Quantifiers

1.4/1.5 Predicates and Quantifiers Can we express the following statement in propositional logic? Every computer has a CPU No Outline: Predicates Quantifiers Domain of Predicate Translation: Natural Languauge

### DISCRETE MATHEMATICS W W L CHEN

DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free

### Propositional Logic. A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both.

irst Order Logic Propositional Logic A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both. Are the following sentences propositions? oronto

### 3. Predicates and Quantifiers

3. PREDICATES AND QUANTIFIERS 45 3. Predicates and Quantifiers 3.1. Predicates and Quantifiers. Definition 3.1.1. A predicate or propositional function is a description of the property (or properties)

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

Chapter 1, Part II: Predicate Logic With Question/Answer Animations Summary Predicate Logic (First Order Logic (FOL), Predicate Calculus (not covered)) The Language of Quantifiers Logical Equivalences

### Predicate Logic. Example: All men are mortal. Socrates is a man. Socrates is mortal.

Predicate Logic Example: All men are mortal. Socrates is a man. Socrates is mortal. Note: We need logic laws that work for statements involving quantities like some and all. In English, the predicate is

### Chapter I Logic and Proofs

MATH 1130 1 Discrete Structures Chapter I Logic and Proofs Propositions A proposition is a statement that is either true (T) or false (F), but or both. s Propositions: 1. I am a man.. I am taller than

### 1 Proposition, Logical connectives and compound statements

Discrete Mathematics: Lecture 4 Introduction to Logic Instructor: Arijit Bishnu Date: July 27, 2009 1 Proposition, Logical connectives and compound statements Logic is the discipline that deals with the

### The Foundations: Logic and Proofs. Chapter 1, Part III: Proofs

The Foundations: Logic and Proofs Chapter 1, Part III: Proofs Rules of Inference Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments

### Rules of Inference Friday, January 18, 2013 Chittu Tripathy Lecture 05

Rules of Inference Today s Menu Rules of Inference Quantifiers: Universal and Existential Nesting of Quantifiers Applications Old Example Re-Revisited Our Old Example: Suppose we have: All human beings

### Section 7.1 First-Order Predicate Calculus predicate Existential Quantifier Universal Quantifier.

Section 7.1 First-Order Predicate Calculus Predicate calculus studies the internal structure of sentences where subjects are applied to predicates existentially or universally. A predicate describes a

### Logic will get you from A to B. Imagination will take you everywhere.

Chapter 3 Predicate Logic Logic will get you from A to B. Imagination will take you everywhere. A. Einstein In the previous chapter, we studied propositional logic. This chapter is dedicated to another

### Predicate in English. Predicates and Quantifiers. Propositional Functions: Prelude. Predicate in Logic. Propositional Function

Predicates and Quantifiers By Chuck Cusack Predicate in English In English, a sentence has 2 parts: the subject and the predicate. The predicate is the part of the sentence that states something about

### Even Number: An integer n is said to be even if it has the form n = 2k for some integer k. That is, n is even if and only if n divisible by 2.

MATH 337 Proofs Dr. Neal, WKU This entire course requires you to write proper mathematical proofs. All proofs should be written elegantly in a formal mathematical style. Complete sentences of explanation

### MATH 55: HOMEWORK #2 SOLUTIONS

MATH 55: HOMEWORK # SOLUTIONS ERIC PETERSON * 1. SECTION 1.5: NESTED QUANTIFIERS 1.1. Problem 1.5.8. Determine the truth value of each of these statements if the domain of each variable consists of all

### Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.

M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider

### Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.

Math 232 - Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the

### What is logic? Propositional Logic. Negation. Propositions. This is a contentious question! We will play it safe, and stick to:

Propositional Logic This lecture marks the start of a new section of the course. In the last few lectures, we have had to reason formally about concepts. This lecture introduces the mathematical language

### All programs that passed test 1 were written in Java.

. Consider the following statements: (S) Programs that passed test also passed test. (S) Programs passed test unless they failed test. (S) Programs passed test only if they passed test. (a) Rewrite statements

### Midterm Examination 1 with Solutions - Math 574, Frank Thorne Thursday, February 9, 2012

Midterm Examination 1 with Solutions - Math 574, Frank Thorne (thorne@math.sc.edu) Thursday, February 9, 2012 1. (3 points each) For each sentence below, say whether it is logically equivalent to the sentence

### 1.5 Arguments & Rules of Inference

1.5 Arguments & Rules of Inference Tools for establishing the truth of statements Argument involving a seuence of propositions (premises followed by a conclusion) Premises 1. If you have a current password,

### CS 441 Discrete Mathematics for CS Lecture 4. Predicate logic. CS 441 Discrete mathematics for CS. Announcements

CS 441 Discrete Mathematics for CS Lecture 4 Predicate logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Announcements Homework assignment 1 due today Homework assignment 2: posted on the course

### Predicate Calculus. There are certain arguments that seem to be perfectly logical, yet they cannot be expressed by using propositional calculus.

Predicate Calculus (Alternative names: predicate logic, first order logic, elementary logic, restricted predicate calculus, restricted functional calculus, relational calculus, theory of quantification,

### Problems on Discrete Mathematics 1

Problems on Discrete Mathematics 1 Chung-Chih Li 2 Kishan Mehrotra 3 L A TEX at July 18, 2007 1 No part of this book can be reproduced without permission from the authors 2 Illinois State University, Normal,

### Quantifiers are used to describe variables in statements. - The universal quantifier means for all. - The existential quantifier means there exists.

11 Quantifiers are used to describe variables in statements. - The universal quantifier means for all. - The existential quantifier means there exists. The phrases, for all x in R if x is an arbitrary

MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

### To formulate more complex mathematical statements, we use the quantifiers there exists, written, and for all, written. If P(x) is a predicate, then

Discrete Math in CS CS 280 Fall 2005 (Kleinberg) Quantifiers 1 Quantifiers To formulate more complex mathematical statements, we use the quantifiers there exists, written, and for all, written. If P(x)

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### First Order Predicate Logic. Lecture 4

First Order Predicate Logic Lecture 4 First Order Predicate Logic Limitation of Propositional Logic The facts: peter is a man, paul is a man, john is a man can be symbolized by P, Q and R respectively

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### 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

### 31 is a prime number is a mathematical statement (which happens to be true).

Chapter 1 Mathematical Logic In its most basic form, Mathematics is the practice of assigning truth to welldefined statements. In this course, we will develop the skills to use known true statements to

### Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 1, due Wedneday, January 25 1.1.10 Let p and q be the propositions The election is decided and The votes have been counted, respectively.

### CS250: Discrete Math for Computer Science. L1: Course Overview & Intro to Language of Math

CS250: Discrete Math for Computer Science L1: Course Overview & Intro to Language of Math What is this course about? Rigorous introduction to discrete mathematics and language of math. Structures and concepts

### CmSc 175 Discrete Mathematics Lesson 10: SETS A B, A B

CmSc 175 Discrete Mathematics Lesson 10: SETS Sets: finite, infinite, : empty set, U : universal set Describing a set: Enumeration = {a, b, c} Predicates = {x P(x)} Recursive definition, e.g. sequences

### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

### 3.3. INFERENCE 105. Table 3.5: Another look at implication. p q p q T T T T F F F T T F F T

3.3. INFERENCE 105 3.3 Inference Direct Inference (Modus Ponens) and Proofs We concluded our last section with a proof that the sum of two even numbers is even. That proof contained several crucial ingredients.

### Predicate Logic. For example, consider the following argument:

Predicate Logic The analysis of compound statements covers key aspects of human reasoning but does not capture many important, and common, instances of reasoning that are also logically valid. For example,

### Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called

### LECTURE 10: FIRST-ORDER LOGIC. Software Engineering Mike Wooldridge

LECTURE 10: FIRST-ORDER LOGIC Mike Wooldridge 1 Why not Propositional Logic? Consider the following statements: all monitors are ready; X12 is a monitor. We saw in an earlier lecture that these statements

### A set is an unordered collection of objects.

Section 2.1 Sets A set is an unordered collection of objects. the students in this class the chairs in this room The objects in a set are called the elements, or members of the set. A set is said to contain

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### Example. Statements in Predicate Logic. Predicate Logic. Predicate Logic. (Rosen, Chapter ) P(x,y) TOPICS. Predicate Logic

Predicate Logic Predicate Logic (Rosen, Chapter 1.4-1.6) TOPICS Predicate Logic Quantifiers Logical Equivalence Predicate Proofs n Some statements cannot be expressed in propositional logic, such as: n

### Homework Set 2 (Due in class on Thursday, Sept. 25) (late papers accepted until 1:00 Friday)

Math 202 Homework Set 2 (Due in class on Thursday, Sept. 25) (late papers accepted until 1:00 Friday) Jerry L. Kazdan The problem numbers refer to the D Angelo - West text. 1. [# 1.8] In the morning section

### Announcements. CompSci 230 Discrete Math for Computer Science Sets. Introduction to Sets. Sets

CompSci 230 Discrete Math for Computer Science Sets September 12, 2013 Prof. Rodger Slides modified from Rosen 1 nnouncements Read for next time Chap. 2.3-2.6 Homework 2 due Tuesday Recitation 3 on Friday

### The set consisting of all natural numbers that are in A and are in B is the set f1; 3; 5g;

Chapter 5 Set Theory 5.1 Sets and Operations on Sets Preview Activity 1 (Set Operations) Before beginning this section, it would be a good idea to review sets and set notation, including the roster method

### MTH 06 LECTURE NOTES (Ojakian) Topic 2: Functions

MTH 06 LECTURE NOTES (Ojakian) Topic 2: Functions OUTLINE (References: Iyer Textbook - pages 4,6,17,18,40,79,80,81,82,103,104,109,110) 1. Definition of Function and Function Notation 2. Domain and Range

### Week 5: Quantifiers. Number Sets. Quantifier Symbols. Quantity has a quality all its own. attributed to Carl von Clausewitz

Week 5: Quantifiers Quantity has a quality all its own. attributed to Carl von Clausewitz Number Sets Many people would say that mathematics is the science of numbers. This is a common misconception among

### Review for Final Exam

Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics. Introductory Notes in Discrete Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics Introductory Notes in Discrete Mathematics Marcel B. Finan c All Rights Reserved Last Updated April 6, 2016 Preface

### Inference Rules and Proof Methods

Inference Rules and Proof Methods Winter 2010 Introduction Rules of Inference and Formal Proofs Proofs in mathematics are valid arguments that establish the truth of mathematical statements. An argument

### CSE 191, Class Note 01 Propositional Logic Computer Sci & Eng Dept SUNY Buffalo

Propositional Logic CSE 191, Class Note 01 Propositional Logic Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 37 Discrete Mathematics What is Discrete

### MAT Discrete Mathematics

RHODES UNIVERSITY Grahamstown 6140, South Africa Lecture Notes CCR MAT 102 - Discrete Mathematics Claudiu C. Remsing DEPT. of MATHEMATICS (Pure and Applied) 2005 Mathematics is not about calculations but

### CS 2336 Discrete Mathematics

CS 2336 Discrete Mathematics Lecture 5 Proofs: Mathematical Induction 1 Outline What is a Mathematical Induction? Strong Induction Common Mistakes 2 Introduction What is the formula of the sum of the first

### Lecture 16 : Relations and Functions DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

### Predicate Logic. M.A.Galán, TDBA64, VT-03

Predicate Logic 1 Introduction There are certain arguments that seem to be perfectly logical, yet they cannot be specified by using propositional logic. All cats have tails. Tom is a cat. From these two

### Math 3000 Running Glossary

Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (

### Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.

Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole

### Logic, Sets, and Proofs

Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical

### Review Name Rule of Inference

CS311H: Discrete Mathematics Review Name Rule of Inference Modus ponens φ 2 φ 2 Modus tollens φ 2 φ 2 Inference Rules for Quantifiers Işıl Dillig Hypothetical syllogism Or introduction Or elimination And

### Sections 2.1, 2.2 and 2.4

SETS Sections 2.1, 2.2 and 2.4 Chapter Summary Sets The Language of Sets Set Operations Set Identities Introduction Sets are one of the basic building blocks for the types of objects considered in discrete

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 1

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 1 Course Outline CS70 is a course on Discrete Mathematics and Probability for EECS Students. The purpose of the course

### Semantics for the Predicate Calculus: Part I

Semantics for the Predicate Calculus: Part I (Version 0.3, revised 6:15pm, April 14, 2005. Please report typos to hhalvors@princeton.edu.) The study of formal logic is based on the fact that the validity

### 4. FIRST-ORDER LOGIC

4. FIRST-ORDER LOGIC Contents 4.1: Splitting the atom: names, predicates and relations 4.2: Structures 4.3: Quantification:, 4.4: Syntax 4.5: Semantics 4.6: De Morgan s laws for and 4.7: Truth trees for

### Section 1.3: Predicate Logic

1 Section 1.3: Purpose of Section: To introduce predicate logic (or first-order logic) which the language of mathematics. We see how predicate logic extends the language of sentential calculus studied

### LESSON 1 PRIME NUMBERS AND FACTORISATION

LESSON 1 PRIME NUMBERS AND FACTORISATION 1.1 FACTORS: The natural numbers are the numbers 1,, 3, 4,. The integers are the naturals numbers together with 0 and the negative integers. That is the integers

### CS 441 Discrete Mathematics for CS Lecture 5. Predicate logic. CS 441 Discrete mathematics for CS. Negation of quantifiers

CS 441 Discrete Mathematics for CS Lecture 5 Predicate logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Negation of quantifiers English statement: Nothing is perfect. Translation: x Perfect(x)

### CS 441 Discrete Mathematics for CS Lecture 2. Propositional logic. CS 441 Discrete mathematics for CS. Course administration

CS 441 Discrete Mathematics for CS Lecture 2 Propositional logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Course administration Homework 1 First homework assignment is out today will be posted

### Logical Inference and Mathematical Proof

Logical Inference and Mathematical Proof CSE 191, Class Note 03: Logical Inference and Mathematical Proof Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures

### A set is a Many that allows itself to be thought of as a One. (Georg Cantor)

Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains

### 3.3 Proofs Involving Quantifiers

3.3 Proofs Involving Quantifiers 1. In exercise 6 of Section 2.2 you use logical equivalences to show that x(p (x) Q(x)) is equivalent to xp (x) xq(x). Now use the methods of this section to prove that

### 1.5 Rules of Inference

1.5 Rules of Inference (Inference: decision/conclusion by evidence/reasoning) Introduction Proofs are valid arguments that establish the truth of statements. An argument is a sequence of statements that

### 2. Propositional Equivalences

2. PROPOSITIONAL EQUIVALENCES 33 2. Propositional Equivalences 2.1. Tautology/Contradiction/Contingency. Definition 2.1.1. A tautology is a proposition that is always true. Example 2.1.1. p p Definition

### List the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated

MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible

### CHAPTER 1 The Foundations: Logic and Proof, Sets, and Functions

Section 1.5 Methods of Proof 1 CHAPTER 1 The Foundations: Logic and Proof, Sets, and Functions SECTION 1.5 Methods of Proof Learning to construct good mathematical proofs takes years. There is no algorithm

### Foundations of Computing Discrete Mathematics Solutions to exercises for week 2

Foundations of Computing Discrete Mathematics Solutions to exercises for week 2 Agata Murawska (agmu@itu.dk) September 16, 2013 Note. The solutions presented here are usually one of many possiblities.

### WUCT121. Discrete Mathematics. Logic

WUCT121 Discrete Mathematics Logic 1. Logic 2. Predicate Logic 3. Proofs 4. Set Theory 5. Relations and Functions WUCT121 Logic 1 Section 1. Logic 1.1. Introduction. In developing a mathematical theory,

### Math 3000 Section 003 Intro to Abstract Math Homework 2

Math 3000 Section 003 Intro to Abstract Math Homework 2 Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Solutions (February 13, 2012) Please note that these

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### Symbolic Logic. 1. Consider the following statements:

Symbolic Logic 1. Consider the following statements: (S1) Programs that passed test 1 also passed test 2. (S2) Programs passed test 2 unless they failed test 1. (S3) Programs passed test 2 only if they

### Mathematical induction & Recursion

CS 441 Discrete Mathematics for CS Lecture 15 Mathematical induction & Recursion Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Proofs Basic proof methods: Direct, Indirect, Contradiction, By Cases,

### Discrete Mathematics Lecture 1 Logic of Compound Statements. Harper Langston New York University

Discrete Mathematics Lecture 1 Logic of Compound Statements Harper Langston New York University Administration Class Web Site http://cs.nyu.edu/courses/summer05/g22.2340-001/ Mailing List Subscribe at

### DISCRETE MATHEMATICS W W L CHEN. c W W L Chen, 1982.

DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 982. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic

Mathematics 0N1 Solutions 1 1. Write the following sets in list form. 1(i) The set of letters in the word banana. {a, b, n}. 1(ii) {x : x 2 + 3x 10 = 0}. 3(iv) C A. True 3(v) B = {e, e, f, c}. True 3(vi)

### Sets. A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object.

Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: real numbers, complex numbers, C integers, All students in

### CSI 2101 / Rules of Inference ( 1.5)

CSI 2101 / Rules of Inference ( 1.5) Introduction what is a proof? Valid arguments in Propositional Logic equivalence of quantified expressions Rules of Inference in Propositional Logic the rules using

### Problems on Discrete Mathematics 1

Problems on Discrete Mathematics 1 Chung-Chih Li 2 Kishan Mehrotra 3 Syracuse University, New York L A TEX at January 11, 2007 (Part I) 1 No part of this book can be reproduced without permission from

### Introduction to Predicate Calculus

Introduction to Predicate Calculus 1 Motivation Last time, we discuss the need for a representation language that was declarative, expressively adequate, unambiguous, context-independent, and compositional.

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### How does x differ in what it represents in the following statements? x is real. x represents no values

Section. redicate Logic Discussion: In Maths we use variables usuall ranging over numbers in various was. How does differ in what it represents in the following statements? is real. = 0 represents one

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.