Foundations of Computing Discrete Mathematics Solutions to exercises for week 2

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Foundations of Computing Discrete Mathematics Solutions to exercises for week 2"

Transcription

1 Foundations of Computing Discrete Mathematics Solutions to exercises for week 2 Agata Murawska September 16, 2013 Note. The solutions presented here are usually one of many possiblities. The fact that your solution is different does not make it bad. Note. The level of detail in the solutions presented here is similar to the level we expect you to present for the exam. Exercise (2.1.2). Use set builder notation to give a description of each of these sets. a) {0, 3, 6, 9, 12} = {3n n N n 4} b) { 3, 2, 1, 0, 1, 2, 3} = {x Z x 3} c) {m, n, o, p} = {c c is a letter m c p} Exercise (2.1.12). Suppose that A, B and C are sets such that A B, B C. Show that A C. Let us first expand the definitions of subsets. We know: A 1 : A B = x, x A x B We want to show: A 2 : B C = x, x B x C S : A C = x, x A x C Let us fix some x. We assume that this x is an element of A and we want to show that it is also an element of C. This changes S to S : x C and introduces another assumption, A 3 : x A. To show that x 1

2 is in C we will want to use the assumption A 2 instantiated with x. This creates another assumption, A 2 : x B x C. Now to conclude x C we need to know that x B. In order to show this, we can again instantiate one of our assumptions, this time A 1. Assumption created this way is A 1 : x A x B. These are the assumptions we have at this point of the proof: A 1 : x, x A x B A 2 : x, x B x C A 3 : x A A 1 : x A x B A 2 : x B x C The goal we want to achieve is showing S : x C Now what we can do is use rule of inference: modus ponens 1 two times. First we use it on A 1 and A 3. This produces another assumption: A 4 : x B Finally, to finish the proof we use modus ponens again, this time on A 2 and A 4. The conclusion is x C and that is exactly what we were supposed to show. Exercise (2.1.16). Can you conclude that A = B if A and B are two sets with the same power set? Yes, we can. First note the following: P(A) = P(B) means that X, X P(A) X P(B), so (by the definition of powerset) also X, X A X B. We know that A A, therefore also A B, similarly we know B B and so B A. From the two: A B and B A we can conclude A = B. Exercise (2.2.2). Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g, h}. Find a) A B = {a, b, c, d, e, f, g, h} b) A B = {a, b, c, d, e} c) A B = 1 From A B and A we can conclude B 2

3 d) B A = {f, g, h} Exercise (2.2.8). Let A and B be sets. Show that 2 : a) A B A A B = {x x A x B}, A B = x, x A x B So A B A means x, x {x x A x B} x A; we can simplify this definition into: x, x A x B x A. To prove this, we take an arbitrary x. We want to show that x A x B x A. Let us assume that x A x B. From that we want to conclude that x A. But if x A x B then surely x A. d) A (B A) = This translates to x, x A (B A) x. Equivalently, we can write it as x, x A x (B A) x x, x A (x B x / A) x. Let us prove this by taking an arbitrary x and showing that: x A (x B x / A) x. Now, by the definition of empty set ( ), we know that no element is in it: x, x /, so in particular x /. This means that x is always false. By the definition of biconditional, this means that x A (x B x / A) must also be false. This is in fact the case, as by regrouping we get x A (x B x / A) x A x B x / A x A x / A x B (x A x / A) x B false x B false Exercise (2.3.4). Find the domain and range of these functions. Note that in each case (...) a) the function that assigns to each nonnegative integer its last digit domain: (nonnegative integers) Z (or: Z + {0}, or: N) range: (the set of all digits) {n Z 0 n 9} b) the function that assigns the next largest integer to a positive integer domain: Z + range: {n N 2 n} 2 We only show two examples here, the rest can be solved using similar methods 3

4 c) the function that assigns to a bit string the number of one bits in the string domain: (the set of all bit strings) {0, 1} range: N d) the function that asigns to a bit string the number of bits in the string domain: (the set of all bit strings) {0, 1} range: N Exercise (2.3.8). Determine whether each of these functions from Z to Z is one-to-one. a) f(n) = n 1 it is one-to-one: there are no two different elements x 1, x 2 such that x 1 1 = x 2 1, as this would imply x 1 = x 2 b) f(n) = n is not one-to-one: for example take 1 and 1, for both of them the image is 2 c) f(n) = n 3 is one-to-one; to see this informally notice that sign of f(n) is the same as the sign of n and the values are increasing for n 0 and decreasing for n < 0. d) f(n) = n/2 is not one-to-one; for example take 1 and 2, then f(1) = f(2) = 1. Exercise (2.4.2). What are the terms a 0, a 1, a 2 and a 3 of the sequence {a n }, where a n equals a) 2 n + 1 : 1, 3, 5, 9 b) (n + 1) n+1 : 1, 4, 81, 256 c) n/2 : 0, 0, 1, 1 d) n/2 + n/2 : 0, 1, 2, 3 Exercise (2.4.20). Compute each of these double sums. a) (i + j) = j=1 ((i + 1) + (i + 2) + (i + 3)) = (3i + 6) = (3 + 6) + (6 + 6) = = 21 4

5 b) (2i + 3j) = ((2i + 0) + (2i + 3) + (2i + 6) + (2i + 9)) = j=0 (8i + 18) = (0 + 18) + (8 + 18) + ( ) = 78 c) i = j=0 (i + i + i) = (3i) = = 18 d) (ij) = j=1 (i + i 2 + i 3) = (6i) = = 18 5

Climbing an Infinite Ladder

Climbing an Infinite Ladder Section 5.1 Climbing an Infinite Ladder Suppose we have an infinite ladder and the following capabilities: 1. We can reach the first rung of the ladder. 2. If we can reach a particular rung of the ladder,

More information

Chapter I Logic and Proofs

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

More information

Mathematical induction. Niloufar Shafiei

Mathematical induction. Niloufar Shafiei Mathematical induction Niloufar Shafiei Mathematical induction Mathematical induction is an extremely important proof technique. Mathematical induction can be used to prove results about complexity of

More information

Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set.

Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set. Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square

More information

Sequences and Summations. Niloufar Shafiei

Sequences and Summations. Niloufar Shafiei Sequences and Summations Niloufar Shafiei Sequences A sequence is a discrete structure used to represent an ordered list. 1 Sequences A sequence is a function from a subset of the set integers (usually

More information

Mathematical induction & Recursion

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,

More information

Chapter Prove or disprove: A (B C) = (A B) (A C). Ans: True, since

Chapter Prove or disprove: A (B C) = (A B) (A C). Ans: True, since Chapter 2 1. Prove or disprove: A (B C) = (A B) (A C)., since A ( B C) = A B C = A ( B C) = ( A B) ( A C) = ( A B) ( A C). 2. Prove that A B= A B by giving a containment proof (that is, prove that the

More information

3. Recurrence Recursive Definitions. To construct a recursively defined function:

3. Recurrence Recursive Definitions. To construct a recursively defined function: 3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a

More information

Mathematical Induction

Mathematical Induction Mathematical Induction MAT30 Discrete Mathematics Fall 016 MAT30 (Discrete Math) Mathematical Induction Fall 016 1 / 19 Outline 1 Mathematical Induction Strong Mathematical Induction MAT30 (Discrete Math)

More information

3. Mathematical Induction

3. Mathematical Induction 3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

More information

DISCRETE MATHEMATICS W W L CHEN

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

More information

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

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

More information

NOTES: Justify your answers to all of the counting problems (give explanation or show work).

NOTES: Justify your answers to all of the counting problems (give explanation or show work). Name: Solutions Student Number: CISC 203 Discrete Mathematics for Computing Science Test 3, Fall 2010 Professor Mary McCollam This test is 50 minutes long and there are 40 marks. Please write in pen and

More information

MATH / Assignment 2. November 6, 2002 Late penalty: 5% for each school day.

MATH / Assignment 2. November 6, 2002 Late penalty: 5% for each school day. MATH 260 2002/20013 Assignment 2 November 6, 2002 Late penalty: 5% for each school day. 1. 1.6 #. (2 points each part) Find the domain and range of the following functions. (a) the function that assigns

More information

1.3 Induction and Other Proof Techniques

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.

More information

COMP232 - Mathematics for Computer Science

COMP232 - Mathematics for Computer Science COMP3 - Mathematics for Computer Science Tutorial 10 Ali Moallemi moa ali@encs.concordia.ca Iraj Hedayati h iraj@encs.concordia.ca Concordia University, Winter 016 Ali Moallemi, Iraj Hedayati COMP3 - Mathematics

More information

ICS141: Discrete Mathematics for Computer Science I

ICS141: Discrete Mathematics for Computer Science I ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Jan Stelovsky based on slides by Dr. Baek and Dr. Still Originals by Dr. M. P. Frank and Dr. J.L. Gross Provided by

More information

1.5 Rules of Inference

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

More information

Finish Set Theory Nested Quantifiers

Finish Set Theory Nested Quantifiers Finish Set Theory Nested Quantifiers Margaret M. Fleck 18 February 2009 This lecture does a final couple examples of set theory proofs. It then fills in material on quantifiers, especially nested ones,

More information

(Refer Slide Time: 1:41)

(Refer Slide Time: 1:41) Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You must

More information

Math/CSE 1019: Discrete Mathematics for Computer Science Fall Suprakash Datta

Math/CSE 1019: Discrete Mathematics for Computer Science Fall Suprakash Datta Math/CSE 1019: Discrete Mathematics for Computer Science Fall 2011 Suprakash Datta datta@cse.yorku.ca Office: CSEB 3043 Phone: 416-736-2100 ext 77875 Course page: http://www.cse.yorku.ca/course/1019 1

More information

Problems on Discrete Mathematics 1

Problems on Discrete Mathematics 1 Problems on Discrete Mathematics 1 Chung-Chih Li Kishan Mehrotra 3 L A TEX at July 18, 007 1 No part of this book can be reproduced without permission from the authors Illinois State University, Normal,

More information

(Refer Slide Time 1.50)

(Refer Slide Time 1.50) Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Module -2 Lecture #11 Induction Today we shall consider proof

More information

Discrete Mathematics. Some related courses. Assessed work. Motivation: functions. Motivation: sets. Exercise. Motivation: relations

Discrete Mathematics. Some related courses. Assessed work. Motivation: functions. Motivation: sets. Exercise. Motivation: relations Discrete Mathematics Philippa Gardner This course is based on previous lecture notes by Iain Phillips. K.H. Rosen. Discrete Mathematics and its Applications, McGraw Hill 1995. J.L. Gersting. Mathematical

More information

n logical not (negation) n logical or (disjunction) n logical and (conjunction) n logical exclusive or n logical implication (conditional)

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

More information

Theorem 2. If x Q and y R \ Q, then. (a) x + y R \ Q, and. (b) xy Q.

Theorem 2. If x Q and y R \ Q, then. (a) x + y R \ Q, and. (b) xy Q. Math 305 Fall 011 The Density of Q in R The following two theorems tell us what happens when we add and multiply by rational numbers. For the first one, we see that if we add or multiply two rational numbers

More information

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

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

More information

Mathematical Induction

Mathematical Induction Mathematical Induction Victor Adamchik Fall of 2005 Lecture 1 (out of three) Plan 1. The Principle of Mathematical Induction 2. Induction Examples The Principle of Mathematical Induction Suppose we have

More information

Worksheet on induction Calculus I Fall 2006 First, let us explain the use of for summation. The notation

Worksheet on induction Calculus I Fall 2006 First, let us explain the use of for summation. The notation Worksheet on induction MA113 Calculus I Fall 2006 First, let us explain the use of for summation. The notation f(k) means to evaluate the function f(k) at k = 1, 2,..., n and add up the results. In other

More information

Logic, Sets, and Proofs

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

More information

3.3 Proofs Involving Quantifiers

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

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 1 9/3/2008 PROBABILISTIC MODELS AND PROBABILITY MEASURES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 1 9/3/2008 PROBABILISTIC MODELS AND PROBABILITY MEASURES MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 1 9/3/2008 PROBABILISTIC MODELS AND PROBABILITY MEASURES Contents 1. Probabilistic experiments 2. Sample space 3. Discrete probability

More information

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

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

More information

This section demonstrates some different techniques of proving some general statements.

This section demonstrates some different techniques of proving some general statements. Section 4. Number Theory 4.. Introduction This section demonstrates some different techniques of proving some general statements. Examples: Prove that the sum of any two odd numbers is even. Firstly you

More information

61. REARRANGEMENTS 119

61. REARRANGEMENTS 119 61. REARRANGEMENTS 119 61. Rearrangements Here the difference between conditionally and absolutely convergent series is further refined through the concept of rearrangement. Definition 15. (Rearrangement)

More information

CS 2336 Discrete Mathematics

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

More information

Sample Induction Proofs

Sample Induction Proofs Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given

More information

CS 2336 Discrete Mathematics

CS 2336 Discrete Mathematics CS 2336 Discrete Mathematics Lecture 2 Logic: Predicate Calculus 1 Outline Predicates Quantifiers Binding Applications Logical Equivalences 2 Predicates In mathematics arguments, we will often see sentences

More information

Chapter Three. Functions. In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics.

Chapter Three. Functions. In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics. Chapter Three Functions 3.1 INTRODUCTION In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics. Definition 3.1: Given sets X and Y, a function from X to

More information

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 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

More information

1. R In this and the next section we are going to study the properties of sequences of real numbers.

1. R In this and the next section we are going to study the properties of sequences of real numbers. +a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

More information

Applications of Methods of Proof

Applications of Methods of Proof CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

More information

MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I. A brief introduction to proofs, sets, and functions MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

More information

2.1.1 Examples of Sets and their Elements

2.1.1 Examples of Sets and their Elements Chapter 2 Set Theory 2.1 Sets The most basic object in Mathematics is called a set. As rudimentary as it is, the exact, formal definition of a set is highly complex. For our purposes, we will simply define

More information

Notes: Chapter 2 Section 2.2: Proof by Induction

Notes: Chapter 2 Section 2.2: Proof by Induction Notes: Chapter 2 Section 2.2: Proof by Induction Basic Induction. To prove: n, a W, n a, S n. (1) Prove the base case - S a. (2) Let k a and prove that S k S k+1 Example 1. n N, n i = n(n+1) 2. Example

More information

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

More information

WUCT121. Discrete Mathematics. Logic

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,

More information

Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014

Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the

More information

Discrete Mathematics (2009 Spring) Induction and Recursion (Chapter 4, 3 hours)

Discrete Mathematics (2009 Spring) Induction and Recursion (Chapter 4, 3 hours) Discrete Mathematics (2009 Spring) Induction and Recursion (Chapter 4, 3 hours) Chih-Wei Yi Dept. of Computer Science National Chiao Tung University April 17, 2009 4.1 Mathematical Induction 4.1 Mathematical

More information

Finite and Infinite Sets

Finite and Infinite Sets Chapter 9 Finite and Infinite Sets 9. Finite Sets Preview Activity (Equivalent Sets, Part ). Let A and B be sets and let f be a function from A to B..f W A! B/. Carefully complete each of the following

More information

Discrete Mathematics: Homework 6 Due:

Discrete Mathematics: Homework 6 Due: Discrete Mathematics: Homework 6 Due: 2011.05.20 1. (3%) How many bit strings are there of length six or less? We use the sum rule, adding the number of bit strings of each length to 6. If we include the

More information

Solutions for Practice problems on proofs

Solutions for Practice problems on proofs Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some

More information

Reading 7 : Program Correctness

Reading 7 : Program Correctness CS/Math 240: Introduction to Discrete Mathematics Fall 2015 Instructors: Beck Hasti, Gautam Prakriya Reading 7 : Program Correctness 7.1 Program Correctness Showing that a program is correct means that

More information

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. 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)

More information

LOGICAL INFERENCE & PROOFs. Debdeep Mukhopadhyay Dept of CSE, IIT Madras

LOGICAL INFERENCE & PROOFs. Debdeep Mukhopadhyay Dept of CSE, IIT Madras LOGICAL INFERENCE & PROOFs Debdeep Mukhopadhyay Dept of CSE, IIT Madras Defn A theorem is a mathematical assertion which can be shown to be true. A proof is an argument which establishes the truth of a

More information

We give a basic overview of the mathematical background required for this course.

We give a basic overview of the mathematical background required for this course. 1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The

More information

If f is a 1-1 correspondence between A and B then it has an inverse, and f 1 isa 1-1 correspondence between B and A.

If f is a 1-1 correspondence between A and B then it has an inverse, and f 1 isa 1-1 correspondence between B and A. Chapter 5 Cardinality of sets 51 1-1 Correspondences A 1-1 correspondence between sets A and B is another name for a function f : A B that is 1-1 and onto If f is a 1-1 correspondence between A and B,

More information

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

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

More information

Sections 2.1, 2.2 and 2.4

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

More information

Review Name Rule of Inference

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

More information

Mathematical Induction. Lecture 10-11

Mathematical Induction. Lecture 10-11 Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach

More information

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

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

More information

Announcements. CompSci 230 Discrete Math for Computer Science. Test 1

Announcements. CompSci 230 Discrete Math for Computer Science. Test 1 CompSci 230 Discrete Math for Computer Science Sep 26, 2013 Announcements Exam 1 is Tuesday, Oct. 1 No class, Oct 3, No recitation Oct 4-7 Prof. Rodger is out Sep 30-Oct 4 There is Recitation: Sept 27-30.

More information

Propositional Logic and Methods of Inference SEEM

Propositional Logic and Methods of Inference SEEM Propositional Logic and Methods of Inference SEEM 5750 1 Logic Knowledge can also be represented by the symbols of logic, which is the study of the rules of exact reasoning. Logic is also of primary importance

More information

Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting

Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Colin Stirling Informatics Slides originally by Kousha Etessami Colin Stirling (Informatics) Discrete Mathematics (Chapter 6) Today 1 /

More information

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.

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

More information

2.1 Sets, power sets. Cartesian Products.

2.1 Sets, power sets. Cartesian Products. Lecture 8 2.1 Sets, power sets. Cartesian Products. Set is an unordered collection of objects. - used to group objects together, - often the objects with similar properties This description of a set (without

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 20

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 20 CS 70 Discrete Mathematics and Probability Theory Fall 009 Satish Rao, David Tse Note 0 Infinity and Countability Consider a function (or mapping) f that maps elements of a set A (called the domain of

More information

arxiv:math/ v1 [math.nt] 31 Mar 2002

arxiv:math/ v1 [math.nt] 31 Mar 2002 arxiv:math/0204006v1 [math.nt] 31 Mar 2002 Additive number theory and the ring of quantum integers Melvyn B. Nathanson Department of Mathematics Lehman College (CUNY) Bronx, New York 10468 Email: nathansn@alpha.lehman.cuny.edu

More information

COMPUTER SCIENCE 123. Foundations of Computer Science. 20. Mathematical induction

COMPUTER SCIENCE 123. Foundations of Computer Science. 20. Mathematical induction COMPUTER SCIENCE 123 Foundations of Computer Science 20. Mathematical induction Summary: This lecture introduces mathematical induction as a technique for proving the equivalence of two functions, or for

More information

3.3 MATHEMATICAL INDUCTION

3.3 MATHEMATICAL INDUCTION Section 3.3 Mathematical Induction 3.3.1 3.3 MATHEMATICAL INDUCTION From modus ponens: p p q q basis assertion conditional assertion conclusion we can easily derive double modus ponens : p 0 p 0 p 1 p

More information

Students in their first advanced mathematics classes are often surprised

Students in their first advanced mathematics classes are often surprised CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are

More information

The Basics of Counting. Niloufar Shafiei

The Basics of Counting. Niloufar Shafiei The Basics of Counting Niloufar Shafiei Counting applications Counting has many applications in computer science and mathematics. For example, Counting the number of operations used by an algorithm to

More information

Fundamentals of Mathematics Lecture 6: Propositional Logic

Fundamentals of Mathematics Lecture 6: Propositional Logic Fundamentals of Mathematics Lecture 6: Propositional Logic Guan-Shieng Huang National Chi Nan University, Taiwan Spring, 2008 1 / 39 Connectives Propositional Connectives I 1 Negation: (not A) A A T F

More information

Problems on Discrete Mathematics 1

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

More information

Deductive Systems. Marco Piastra. Artificial Intelligence. Artificial Intelligence - A.A Deductive Systems [1]

Deductive Systems. Marco Piastra. Artificial Intelligence. Artificial Intelligence - A.A Deductive Systems [1] Artificial Intelligence Deductive Systems Marco Piastra Artificial Intelligence - A.A. 2012- Deductive Systems 1] Symbolic calculus? A wff is entailed by a set of wff iff every model of is also model of

More information

Introducing Functions

Introducing Functions Functions 1 Introducing Functions A function f from a set A to a set B, written f : A B, is a relation f A B such that every element of A is related to one element of B; in logical notation 1. (a, b 1

More information

Chapter 1 Basic Number Concepts

Chapter 1 Basic Number Concepts Draft of September 2014 Chapter 1 Basic Number Concepts 1.1. Introduction No problems in this section. 1.2. Factors and Multiples 1. Determine whether the following numbers are divisible by 3, 9, and 11:

More information

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. 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.

More information

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b. Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

More information

4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.

4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set. Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,

More information

Sequences and Mathematical Induction. CSE 215, Foundations of Computer Science Stony Brook University

Sequences and Mathematical Induction. CSE 215, Foundations of Computer Science Stony Brook University Sequences and Mathematical Induction CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 Sequences A sequence is a function whose domain is all the integers

More information

def: An axiom is a statement that is assumed to be true, or in the case of a mathematical system, is used to specify the system.

def: An axiom is a statement that is assumed to be true, or in the case of a mathematical system, is used to specify the system. Section 1.5 Methods of Proof 1.5.1 1.5 METHODS OF PROOF Some forms of argument ( valid ) never lead from correct statements to an incorrect. Some other forms of argument ( fallacies ) can lead from true

More information

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

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

More information

Inference Rules and Proof Methods

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

More information

Chapter 1. Logic and Proof

Chapter 1. Logic and Proof Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known

More information

Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent. MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

More information

Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

More information

Lecture 6: Approximation via LP Rounding

Lecture 6: Approximation via LP Rounding Lecture 6: Approximation via LP Rounding Let G = (V, E) be an (undirected) graph. A subset C V is called a vertex cover for G if for every edge (v i, v j ) E we have v i C or v j C (or both). In other

More information

Mathematical Foundations of Computer Science Lecture Outline

Mathematical Foundations of Computer Science Lecture Outline Mathematical Foundations of Computer Science Lecture Outline September 21, 2016 Example. How many 8-letter strings can be constructed by using the 26 letters of the alphabet if each string contains 3,4,

More information

Mathematical Induction. Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition)

Mathematical Induction. Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition) Mathematical Induction Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition) Mathmatical Induction Mathmatical induction can be used to prove statements that assert that P(n) is true for all positive

More information

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

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

More information

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear: MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

More information

Solutions to In-Class Problems Week 4, Mon.

Solutions to In-Class Problems Week 4, Mon. Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics for Computer Science September 26 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised September 26, 2005, 1050 minutes Solutions

More information

COMPOSITES THAT REMAIN COMPOSITE AFTER CHANGING A DIGIT

COMPOSITES THAT REMAIN COMPOSITE AFTER CHANGING A DIGIT COMPOSITES THAT REMAIN COMPOSITE AFTER CHANGING A DIGIT Michael Filaseta Mathematics Department University of South Carolina Columbia, SC 2208-0001 Mark Kozek Department of Mathematics Whittier College

More information

Examination paper for MA0301 Elementær diskret matematikk

Examination paper for MA0301 Elementær diskret matematikk Department of Mathematical Sciences Examination paper for MA0301 Elementær diskret matematikk Academic contact during examination: Iris Marjan Smit a, Sverre Olaf Smalø b Phone: a 9285 0781, b 7359 1750

More information

Discrete Mathematics and its Applications Counting (2) Xiaocong ZHOU

Discrete Mathematics and its Applications Counting (2) Xiaocong ZHOU Discrete Mathematics and its Applications Counting (2) Xiaocong ZHOU Department of Computer Science Sun Yat-sen University Feb. 2016 http://www.cs.sysu.edu.cn/ zxc isszxc@mail.sysu.edu.cn Xiaocong ZHOU

More information

Notes. Sets. Notes. Introduction II. Notes. Definition. Definition. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry.

Notes. Sets. Notes. Introduction II. Notes. Definition. Definition. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Sets Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 1.6 1.7 of Rosen cse235@cse.unl.edu Introduction

More information

Inference in propositional logic

Inference in propositional logic C 1571 Introduction to AI Lecture 13 Inference in propositional logic Milos Hauskrecht milos@cs.pitt.edu 5329 ennott quare Logical inference problem Logical inference problem: Given: a knowledge base KB

More information

Math 55: Discrete Mathematics

Math 55: Discrete Mathematics Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What

More information