# Section 2.4 Sequences and Summations

Size: px
Start display at page:

## Transcription

1 Section 2.4 Sequences and Summations Definition: A sequence is a function from a subset of the natural numbers (usually of the form {0, 1, 2,... } to a set S. Note: the sets and {0, 1, 2, 3,..., k} {1, 2, 3, 4,..., k} are called initial segments of N. Notation: if f is a function from {0, 1, 2,...} to S we usually denote f(i) by a i and we write k {a 0, a 1, a 2, a 3,... } = {a i } i=0 = {a i } 0 k where k is the upper limit (usually ). Examples: Using zero-origin indexing, if f(i) = 1/(i + 1). then the sequence f = {1, 1/'2,1/3,1/4,... } = {a 0, a 1, a 2, a 3,.. } Prepared by: David F. McAllister TP , 2007 McGraw-Hill

2 Using one-origin indexing the sequence f becomes {1/2, 1/3,...} = {a 1, a 2, a 3,...} Summation Notation Given a sequence {a i } 0 k we can add together a subset of the sequence by using the summation and function notation or more generally Examples: a g(m) + a g(m+1) a g(n) = a g( j) a j j S n j=m r 0 + r 1 + r 2 + r r n = n 0 r j = 1 1 i a 2m + a 2(m+1) a 2(n) = n a 2 j j=m If S = {2, 5, 7, 10} then a j = a 2 + a 5 + a 7 + a 10 j S Similarly for the product notation: Prepared by: David F. McAllister TP , 2007 McGraw-Hill

3 n a j = a m a m+1...a n j=m Definition: A geometric progression is a sequence of the form Your book has a proof that a, ar, ar2, ar3, ar4,.... n r i i=0 = rn+1 1 r 1 if r 1 (you can figure out what it is if r = 1). You should also be able to determine the sum if the index starts at k vs. 0 if the index ends at something other than n (e.g., n- 1, n+1, etc.). Cardinality Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if there exists a bijection from A to B. Prepared by: David F. McAllister TP , 2007 McGraw-Hill

4 Definition: If a set has the same cardinality as a subset of the natural numbers N, then the set is called countable. If A = N, the set A is countably infinite. The (transfinite) cardinal number of the set N is aleph null = ℵ 0. If a set is not countable we say it is uncountable. Examples: The following sets are uncountable (we show later) The real numbers in [0, 1] P(N), the power set of N Note: With infinite sets proper subsets can have the same cardinality. This cannot happen with finite sets. Countability carries with it the implication that there is a listing of the elements of the set. Definition: A B if there is an injection from A to B. Note: as you would hope, Prepared by: David F. McAllister TP , 2007 McGraw-Hill

5 Theorem: If A B and B A then A = B. This implies then if there is an injection from A to B if there is an injection from B to A there must be a bijection from A to B This is difficult to prove but is an example of demonstrating existence without construction. It is often easier to build the injections and then conclude the bijection exists. Example: Theorem: If A is a subset of B then A B. Proof: the function f(x) = x is an injection from A to B. Prepared by: David F. McAllister TP , 2007 McGraw-Hill

6 Example: {0, 2, 5} ℵ 0 The injection f: {0, 2, 4} N defined by f(x) = x is shown below: Some Countably Infinite Sets The set of even integers E ( 0 is considered even) is countably infinite. Note that E is a proper subset of N! Proof: Let f(x) = 2x. Then f is a bijection from N to E Z+, the set of positive integers is countably infinite. Prepared by: David F. McAllister TP , 2007 McGraw-Hill

7 The set of positive rational numbers Q+ is countably infinite. Proof: Z+ is a subset of Q+ so Z+ = ℵ 0 Q+. Now we have to show that Q+ ℵ 0. To do this we show that the positive rational numbers with repetitions, Q R, is countably infinite. Then, since Q+ is a subset of Q R, it follows that Q+ ℵ 0 and hence Q+ = ℵ 0. x y /1 2/1 3/1 4/1 5/1 6/1 7/1 1/2 2/2 3/2 4/2 5/2 6/2 7/2 1/3 2/3 3/3 4/3 5/3 6/3 7/3 1/4 2/4 3/4 4/4 5/4 6/4 7/4 5 The position on the path (listing) indicates the image of the bijective function f from N to Q R : f(0) = 1/1, f(1) = 1/2, f(2) = 2/1, f(3) = 3/1, and so forth. Every rational number appears on the list at least once, some many times (repetitions). Prepared by: David F. McAllister TP , 2007 McGraw-Hill

8 Hence, N = Q R = ℵ 0. Q. E. D. The set of all rational numbers Q, positive and negative, is countably infinite. The set of (finite length) strings S over a finite alphabet A is countably infinite. To show this we assume that - A is nonvoid - There is an alphabetical ordering of the symbols in A Proof: List the strings in lexicographic order: order, order, - all the strings of zero length, - then all the strings of length 1 in alphabetical - then all the strings of length 2 in alphabetical etc. This implies a bijection from N to the list of strings and hence it is a countably infinite set. Prepared by: David F. McAllister TP , 2007 McGraw-Hill

9 For example: Let A = {a, b, c}. Then the lexicographic ordering of A is {λ, a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc, aaa, aab, aac, aba,...} = {f(0), f(1), f(2), f(3), f(4),....} The set of all C programs is countable. Proof: Let S be the set of legitimate characters which can appear in a C program. - A C compiler will determine if an input program is a syntactically correct C program (the program doesn't have to do anything useful). - Use the lexicographic ordering of S and feed the strings into the compiler. If the compiler says YES, this is a syntactically correct C program, we add the program to the list. Else we move on to the next string. In this way we construct a list or an implied bijection from N to the set of C programs. Hence, the set of C programs is countable. Q. E. D. Prepared by: David F. McAllister TP , 2007 McGraw-Hill

10 Cantor Diagonalization - An important technique used to construct an object which is not a member of a countable set of objects with (possibly) infinite descriptions Theorem: The set of real numbers between 0 and 1 is uncountable. Proof: We assume that it is countable and derive a contradiction. If it is countable we can list them (i.e., there is a bijection from a subset of N to the set). We show that no matter what list you produce we can construct a real number between 0 and 1 which is not in the list. Hence, there cannot exist a list and therefore the set is not countable It's actually much bigger than countable. It is said to have the cardinality of the continuum, c. Represent each real number in the list using its decimal expansion. e.g., 1/3 = /2 = = If there is more than one expansion for a number, it doesn't matter as long as our construction takes this into account. Prepared by: David F. McAllister TP , 2007 McGraw-Hill

11 THE LIST... r 1 =.d 11 d 12 d 13 d 14 d 15 d r 2 =.d 21 d 22 d 23 d 24 d 25 d r 3 =.d 31 d 32 d 33 d 34 d 35 d Now construct the number x =.x 1 x 2 x 3 x 4 x 5 x 6 x x i = 3 if d ii 3 x i = 4 if d ii = 3 (Note: choosing 0 and 9 is not a good idea because of the non uniqueness of decimal expansions.) Then x is not equal to any number in the list. Hence, no such list can exist and hence the interval (0,1) is uncountable. Q. E. D. An extra goody: Definition: a number x between 0 and 1 is computable if there is a C program which when given the input i, will produce the ith digit in the decimal expansion of x. Prepared by: David F. McAllister TP , 2007 McGraw-Hill

12 Example: The number 1/3 is computable. The C program which always outputs the digit 3, regardless if the input, computes the number. Theorem: There is exists a number x between 0 and 1 which is not computable. There does not exist a C program (or a program in any other language) which will compute it! Why? Because there are more numbers between 0 and 1 than there are C programs to compute them. (in fact there are c such numbers!) Our second example of the nonexistence of programs to compute things! Prepared by: David F. McAllister TP , 2007 McGraw-Hill

### Section 1.7 Sequences, Summations Cardinality of Infinite Sets

Section 1.7 Sequences, Summations Cardinality of Infinite Sets Definition: A sequence is a function from a subset of the natural numbers (usually of the form {0, 1, 2,... } to a set S. Note: the sets and

### A B. bijection. injection. Section 2.4: Countability. a b c d e g

Section 2.4: Countability We can compare the cardinality of two sets. A = B means there is a bijection between A and B. A B means there is an injection from A to B. A < B means A B and A B Example: Let

### Infinity and Uncountability. Countable Countably infinite. Enumeration

Infinity and Uncountability. Countable Countably infinite. Enumeration How big is the set of reals or the set of integers? Infinite! Is one bigger or smaller? Same size? Same number? Make a function f

### Cardinality of Sets. Washington University Math Circle 10/30/2016

Cardinality of Sets Washington University Math Circle 0/0/06 The cardinality of a finite set A is just the number of elements of A, denoted by A. For example, A = {a, b, c, d}, B = {n Z : n } = {,,, 0,,,

### Material from Recitation 1

Material from Recitation 1 Darcey Riley Frank Ferraro January 18, 2011 1 Introduction In CSC 280 we will be formalizing computation, i.e. we will be creating precise mathematical models for describing

### Cardinality of Sets MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Cardinality of Sets Fall / 15

Cardinality of Sets MAT Transition to Higher Mathematics Fall 0 MAT (Transition to Higher Math) Cardinality of Sets Fall 0 / Outline Sets with Equal Cardinality Countable and Uncountable Sets MAT (Transition

### Comparing sizes of sets

Comparing sizes of sets Sets A and B are the same size if there is a bijection from A to B. (That was a definition!) For finite sets A, B, it is not difficult to verify that there is a bijection from A

Oxford University Press 2013 Slides for Faculty Assistance Preliminaries Author: Vivek Kulkarni vivek_kulkarni@yahoo.com Outline Following topics are covered in the slides: Basic concepts, namely, symbols,

### The Size of the Cantor Set

The Size of the Cantor Set Washington University Math Circle November 6, 2016 In mathematics, a set is a collection of things called elements. For example, {1, 2, 3, 4}, {a,b,c,...,z}, and {cat, dog, chicken}

### Cantor s Diagonal Argument for Different Levels of Infinity

JANUARY 2015 1 Cantor s Diagonal Argument for Different Levels of Infinity Michael J. Neely University of Southern California http://www-bcf.usc.edu/ mjneely Abstract These notes develop the classic Cantor

### SET DEFINITION 1 elements members

SETS SET DEFINITION 1 Unordered collection of objects, called elements or members of the set. Said to contain its elements. We write a A to denote that a is an element of the set A. The notation a A denotes

### 2. Functions, sets, countability and uncountability. Let A, B be sets (often, in this module, subsets of R).

2. Functions, sets, countability and uncountability I. Functions Let A, B be sets (often, in this module, subsets of R). A function f : A B is some rule that assigns to each element of A a unique element

### MITOCW watch?v=4dj1oguwtem

MITOCW watch?v=4dj1oguwtem PROFESSOR: So it's time to examine uncountable sets. And that's what we're going to do in this segment. So Cantor's question was, are all sets the same size? And he gives a definitive

### Phil 320 Chapter 1: Sets, Functions and Enumerability I. Sets Informally: a set is a collection of objects. The objects are called members or

Phil 320 Chapter 1: Sets, Functions and Enumerability I. Sets Informally: a set is a collection of objects. The objects are called members or elements of the set. a) Use capital letters to stand for sets

### Source of Slides: Introduction to Automata Theory, Languages, and Computation By John E. Hopcroft, Rajeev Motwani and Jeffrey D.

Source of Slides: Introduction to Automata Theory, Languages, and Computation By John E. Hopcroft, Rajeev Motwani and Jeffrey D. Ullman And Introduction to Languages and The by J. C. Martin Basic Mathematical

### Proof Techniques Alphabets, Strings, and Languages. Foundations of Computer Science Theory

Proof Techniques Alphabets, Strings, and Languages Foundations of Computer Science Theory Proof By Case Enumeration Sometimes the most straightforward way to prove that a property holds for all elements

### EDAA40 At home exercises 1

EDAA40 At home exercises 1 1. Given, with as always the natural numbers starting at 1, let us define the following sets (with iff ): Give the number of elements in these sets as follows: 1. 23 2. 6 3.

### .Math 0450 Honors intro to analysis Spring, 2009 Notes #4 corrected (as of Monday evening, 1/12) some changes on page 6, as in .

0.1 More on innity.math 0450 Honors intro to analysis Spring, 2009 Notes #4 corrected (as of Monday evening, 1/12) some changes on page 6, as in email. 0.1.1 If you haven't read 1.3, do so now! In notes#1

### Cardinality Lectures

Cardinality Lectures Enrique Treviño March 8, 014 1 Definition of cardinality The cardinality of a set is a measure of the size of a set. When a set A is finite, its cardinality is the number of elements

### CSE 120. Computer Science Principles

Adam Blank Lecture 17 Winter 2017 CSE 120 Computer Science Principles CSE 120: Computer Science Principles Proofs & Computation e w h e q 0 q 1 q 2 q 3 h,e w,e w,h w,h q garbage w,h,e CSE = Abstraction

### 1.3 Functions and Equivalence Relations 1.4 Languages

CSC4510 AUTOMATA 1.3 Functions and Equivalence Relations 1.4 Languages Functions and Equivalence Relations f : A B means that f is a function from A to B To each element of A, one element of B is assigned

### Let A(x) be x is an element of A, and B(x) be x is an element of B.

Homework 6. CSE 240, Fall, 2014 Due, Tuesday October 28. Can turn in at the beginning of class, or earlier in the mailbox labelled Pless in Bryan Hall, room 509c. Practice Problems: 1. Given two arbitrary

### Calculating Cardinalities

Math Circle Monday March 20, 2017 Calculating Cardinalities Martin Zeman To say that a set A has 5 elements means that we can write the elements of A as a list a 1, a 2, a 3, a 4, a 5 in a way that (a)

### Point-Set Topology 1. TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS

Point-Set Topology 1. TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS Definition 1.1. Let X be a set and T a subset of the power set P(X) of X. Then T is a topology on X if and only if all of the following

### MATHEMATICS 191, FALL 2004 MATHEMATICAL PROBABILITY Outline #1 (Countability and Uncountability)

MATHEMATICS 191, FALL 2004 MATHEMATICAL PROBABILITY Outline #1 (Countability and Uncountability) Last modified: September 16, 2004 Reference: Apostol, Calculus, Vol. 2, section 13.19 (attached). The aim

### MATH 139 W12 Review 1 Checklist 1. Exam Checklist. 1. Introduction to Predicates and Quantified Statements (chapters ).

MATH 139 W12 Review 1 Checklist 1 Exam Checklist 1. Introduction to Predicates and Quantified Statements (chapters 3.1-3.4). universal and existential statements truth set negations of universal and existential

### Chapter 18: Decidability

Chapter 18: Decidability Peter Cappello Department of Computer Science University of California, Santa Barbara Santa Barbara, CA 93106 cappello@cs.ucsb.edu Please read the corresponding chapter before

### Taibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103. Chapter 2. Sets

Taibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103 Chapter 2 Sets Slides are adopted from Discrete Mathematics and It's Applications Kenneth H.

### Functions. How is this definition written in symbolic logic notation?

functions 1 Functions Def. Let A and B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by

### Review of Sets. Review. Philippe B. Laval. Current Semester. Kennesaw State University. Philippe B. Laval (KSU) Sets Current Semester 1 / 16

Review of Sets Review Philippe B. Laval Kennesaw State University Current Semester Philippe B. Laval (KSU) Sets Current Semester 1 / 16 Outline 1 Introduction 2 Definitions, Notations and Examples 3 Special

### To illustrate what is intended the following are three write ups by students. Diagonalization

General guidelines: You may work with other people, as long as you write up your solution in your own words and understand everything you turn in. Make sure to justify your answers they should be clear

### Assignment No.4 solution. Pumping Lemma Version I and II. Where m = n! (n-factorial) and n = 1, 2, 3

Assignment No.4 solution Question No.1 a. Suppose we have a language defined below, Pumping Lemma Version I and II a n b m Where m = n! (n-factorial) and n = 1, 2, 3 Some strings belonging to this language

### Limitations of Algorithmic Solvability In this Chapter we investigate the power of algorithms to solve problems Some can be solved algorithmically and

Computer Language Theory Chapter 4: Decidability 1 Limitations of Algorithmic Solvability In this Chapter we investigate the power of algorithms to solve problems Some can be solved algorithmically and

### 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

### 9.5 Equivalence Relations

9.5 Equivalence Relations You know from your early study of fractions that each fraction has many equivalent forms. For example, 2, 2 4, 3 6, 2, 3 6, 5 30,... are all different ways to represent the same

Automata Theory TEST 1 Answers Max points: 156 Grade basis: 150 Median grade: 81% 1. (2 pts) See text. You can t be sloppy defining terms like this. You must show a bijection between the natural numbers

### Lec-5-HW-1, TM basics

Lec-5-HW-1, TM basics (Problem 0)-------------------- Design a Turing Machine (TM), T_sub, that does unary decrement by one. Assume a legal, initial tape consists of a contiguous set of cells, each containing

### Chapter 18 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal.

Chapter 8 out of 7 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal 8 Matrices Definitions and Basic Operations Matrix algebra is also known

### TOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3.

TOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3. 301. Definition. Let m be a positive integer, and let X be a set. An m-tuple of elements of X is a function x : {1,..., m} X. We sometimes use x i instead

### Hinged Sets And the Answer to the Continuum Hypothesis

Hinged Sets And the Answer to the Continuum Hypothesis by R. Webster Kehr 1st Version: January 20, 1999 Dedicated to: Marit Olaug Liset Kehr, my wife of 27 years and mother of our 7 children and grandmother

### Binary Relations McGraw-Hill Education

Binary Relations A binary relation R from a set A to a set B is a subset of A X B Example: Let A = {0,1,2} and B = {a,b} {(0, a), (0, b), (1,a), (2, b)} is a relation from A to B. We can also represent

### Functions 2/1/2017. Exercises. Exercises. Exercises. and the following mathematical appetizer is about. Functions. Functions

Exercises Question 1: Given a set A = {x, y, z} and a set B = {1, 2, 3, 4}, what is the value of 2 A 2 B? Answer: 2 A 2 B = 2 A 2 B = 2 A 2 B = 8 16 = 128 Exercises Question 2: Is it true for all sets

### 9/19/12. Why Study Discrete Math? What is discrete? Sets (Rosen, Chapter 2) can be described by discrete math TOPICS

What is discrete? Sets (Rosen, Chapter 2) TOPICS Discrete math Set Definition Set Operations Tuples Consisting of distinct or unconnected elements, not continuous (calculus) Helps us in Computer Science

### Functions. Def. Let A and B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A.

Functions functions 1 Def. Let A and B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A. a A! b B b is assigned to a a A! b B f ( a) = b Notation: If

### CHAPTER 4. COMPUTABILITY AND DECIDABILITY

CHAPTER 4. COMPUTABILITY AND DECIDABILITY 1. Introduction By definition, an n-ary function F on N assigns to every n-tuple k 1,...,k n of elements of N a unique l N which is the value of F at k 1,...,k

### CS3102 Theory of Computation Solutions to Problem Set 1, Spring 2012 Department of Computer Science, University of Virginia

CS3102 Theory of Computation Solutions to Problem Set 1, Spring 2012 Department of Computer Science, University of Virginia Gabriel Robins Please start solving these problems immediately, and work in study

### CS402 - Theory of Automata Glossary By

CS402 - Theory of Automata Glossary By Acyclic Graph : A directed graph is said to be acyclic if it contains no cycles. Algorithm : A detailed and unambiguous sequence of instructions that describes how

About the Tutorial Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics

### Lecture 17: Continuous Functions

Lecture 17: Continuous Functions 1 Continuous Functions Let (X, T X ) and (Y, T Y ) be topological spaces. Definition 1.1 (Continuous Function). A function f : X Y is said to be continuous if the inverse

### Introduction to Sets and Logic (MATH 1190)

Introduction to Sets and Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics and Statistics York University Dec 4, 2014 Outline 1 2 3 4 Definition A relation R from a set A to a set

### THREE LECTURES ON BASIC TOPOLOGY. 1. Basic notions.

THREE LECTURES ON BASIC TOPOLOGY PHILIP FOTH 1. Basic notions. Let X be a set. To make a topological space out of X, one must specify a collection T of subsets of X, which are said to be open subsets of

### CMPSCI 250: Introduction to Computation. Lecture #28: Regular Expressions and Languages David Mix Barrington 2 April 2014

CMPSCI 250: Introduction to Computation Lecture #28: Regular Expressions and Languages David Mix Barrington 2 April 2014 Regular Expressions and Languages Regular Expressions The Formal Inductive Definition

### Recursive Definitions Structural Induction Recursive Algorithms

Chapter 4 1 4.3-4.4 Recursive Definitions Structural Induction Recursive Algorithms 2 Section 4.1 3 Principle of Mathematical Induction Principle of Mathematical Induction: To prove that P(n) is true for

### Point-Set Topology II

Point-Set Topology II Charles Staats September 14, 2010 1 More on Quotients Universal Property of Quotients. Let X be a topological space with equivalence relation. Suppose that f : X Y is continuous and

### Computation, Computers, and Programs. Administrivia. Resources. Course texts: Kozen: Introduction to Computability Hickey: Introduction to OCaml

CS20a: Computation, Computers, Programs Instructor: Jason Hickey Email: jyh@cs.caltech.edu Office hours: TR 10-11am TAs: Nathan Gray (n8gray@cs.caltech.edu) Brian Aydemir (emre@cs.caltech.edu) Jason Frantz

### Quadratic Equations over Matrices over the Quaternions. By Diana Oliff Mentor: Professor Robert Wilson

Quadratic Equations over Matrices over the Quaternions By Diana Oliff Mentor: Professor Robert Wilson Fields A field consists of a set of objects S and two operations on this set. We will call these operations

### Lamé s Theorem. Strings. Recursively Defined Sets and Structures. Recursively Defined Sets and Structures

Lamé s Theorem Gabriel Lamé (1795-1870) Recursively Defined Sets and Structures Lamé s Theorem: Let a and b be positive integers with a b Then the number of divisions used by the Euclidian algorithm to

### CSE 20 DISCRETE MATH WINTER

CSE 20 DISCRETE MATH WINTER 2016 http://cseweb.ucsd.edu/classes/wi16/cse20-ab/ Today's learning goals Explain the steps in a proof by (strong) mathematical induction Use (strong) mathematical induction

### Formal Languages and Automata

Mobile Computing and Software Engineering p. 1/3 Formal Languages and Automata Chapter 3 Regular languages and Regular Grammars Chuan-Ming Liu cmliu@csie.ntut.edu.tw Department of Computer Science and

### Discrete Mathematics. Kruskal, order, sorting, induction

Discrete Mathematics wwwmifvult/~algis Kruskal, order, sorting, induction Kruskal algorithm Kruskal s Algorithm for Minimal Spanning Trees The algorithm constructs a minimal spanning tree as follows: Starting

### 2. Sets. 2.1&2.2: Sets and Subsets. Combining Sets. c Dr Oksana Shatalov, Fall

c Dr Oksana Shatalov, Fall 2014 1 2. Sets 2.1&2.2: Sets and Subsets. Combining Sets. Set Terminology and Notation DEFINITIONS: Set is well-defined collection of objects. Elements are objects or members

### SYNERGY INSTITUTE OF ENGINEERING & TECHNOLOGY,DHENKANAL LECTURE NOTES ON DIGITAL ELECTRONICS CIRCUIT(SUBJECT CODE:PCEC4202)

Lecture No:5 Boolean Expressions and Definitions Boolean Algebra Boolean Algebra is used to analyze and simplify the digital (logic) circuits. It uses only the binary numbers i.e. 0 and 1. It is also called

### 13 th Annual Johns Hopkins Math Tournament Saturday, February 19, 2011 Automata Theory EUR solutions

13 th Annual Johns Hopkins Math Tournament Saturday, February 19, 011 Automata Theory EUR solutions Problem 1 (5 points). Prove that any surjective map between finite sets of the same cardinality is a

### CHAPTER TWO LANGUAGES. Dr Zalmiyah Zakaria

CHAPTER TWO LANGUAGES By Dr Zalmiyah Zakaria Languages Contents: 1. Strings and Languages 2. Finite Specification of Languages 3. Regular Sets and Expressions Sept2011 Theory of Computer Science 2 Strings

### Using Cantor s Diagonal Method to Show ζ(2) is Irrational

Using Cantor s Diagonal Method to Show ζ(2) is Irrational Timothy W. Jones November 1, 2018 Abstract We look at some of the details of Cantor s Diagonal Method and argue that the swap function given does

### LECTURE NOTES ON SETS

LECTURE NOTES ON SETS PETE L. CLARK Contents 1. Introducing Sets 1 2. Subsets 5 3. Power Sets 5 4. Operations on Sets 6 5. Families of Sets 8 6. Partitions 10 7. Cartesian Products 11 1. Introducing Sets

Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124

### 2.8. Connectedness A topological space X is said to be disconnected if X is the disjoint union of two non-empty open subsets. The space X is said to

2.8. Connectedness A topological space X is said to be disconnected if X is the disjoint union of two non-empty open subsets. The space X is said to be connected if it is not disconnected. A subset of

### COMPUTABILITY THEORY AND RECURSIVELY ENUMERABLE SETS

COMPUTABILITY THEORY AND RECURSIVELY ENUMERABLE SETS JOSHUA LENERS Abstract. An algorithm is function from ω to ω defined by a finite set of instructions to transform a given input x to the desired output

### PACKING DIGRAPHS WITH DIRECTED CLOSED TRAILS

PACKING DIGRAPHS WITH DIRECTED CLOSED TRAILS PAUL BALISTER Abstract It has been shown [Balister, 2001] that if n is odd and m 1,, m t are integers with m i 3 and t i=1 m i = E(K n) then K n can be decomposed

### Computability, Cantor s diagonalization, Russell s Paradox, Gödel s s Incompleteness, Turing Halting Problem.

Computability, Cantor s diagonalization, Russell s Paradox, Gödel s s Incompleteness, Turing Halting Problem. Advanced Algorithms By Me Dr. Mustafa Sakalli March, 06, 2012. Incompleteness. Lecture notes

### Power Set of a set and Relations

Power Set of a set and Relations 1 Power Set (1) Definition: The power set of a set S, denoted P(S), is the set of all subsets of S. Examples Let A={a,b,c}, P(A)={,{a},{b},{c},{a,b},{b,c},{a,c},{a,b,c}}

### Final Exam 2, CS154. April 25, 2010

inal Exam 2, CS154 April 25, 2010 Exam rules. he exam is open book and open notes you can use any printed or handwritten material. However, no electronic devices are allowed. Anything with an on-off switch

### 2 Review of Set Theory

2 Review of Set Theory Example 2.1. Let Ω = {1, 2, 3, 4, 5, 6} 2.2. Venn diagram is very useful in set theory. It is often used to portray relationships between sets. Many identities can be read out simply

### DVA337 HT17 - LECTURE 4. Languages and regular expressions

DVA337 HT17 - LECTURE 4 Languages and regular expressions 1 SO FAR 2 TODAY Formal definition of languages in terms of strings Operations on strings and languages Definition of regular expressions Meaning

### M3P1/M4P1 (2005) Dr M Ruzhansky Metric and Topological Spaces Summary of the course: definitions, examples, statements.

M3P1/M4P1 (2005) Dr M Ruzhansky Metric and Topological Spaces Summary of the course: definitions, examples, statements. Chapter 1: Metric spaces and convergence. (1.1) Recall the standard distance function

### Chapter 3. Set Theory. 3.1 What is a Set?

Chapter 3 Set Theory 3.1 What is a Set? A set is a well-defined collection of objects called elements or members of the set. Here, well-defined means accurately and unambiguously stated or described. Any

### HW 1 CMSC 452. Morally DUE Feb 7 NOTE- THIS HW IS THREE PAGES LONG!!! SOLUTIONS THROUGOUT THIS HW YOU CAN ASSUME:

HW 1 CMSC 452. Morally DUE Feb 7 NOTE- THIS HW IS THREE PAGES LONG!!! SOLUTIONS THROUGOUT THIS HW YOU CAN ASSUME: The union of a finite number of countable sets is countable. The union of a countable number

### Math 302 Introduction to Proofs via Number Theory. Robert Jewett (with small modifications by B. Ćurgus)

Math 30 Introduction to Proofs via Number Theory Robert Jewett (with small modifications by B. Ćurgus) March 30, 009 Contents 1 The Integers 3 1.1 Axioms of Z...................................... 3 1.

### Nagual and Tonal Maths: Numbers, Sets, and Processes. Lukas A. Saul Oxford, England May, 2015

Nagual and Tonal Maths: Numbers, Sets, and Processes Lukas A. Saul Oxford, England May, 2015 Abstract: Some definitions and elementary theorems are given here describing tonal and nagual numbers, sets,

### 1.7 The Heine-Borel Covering Theorem; open sets, compact sets

1.7 The Heine-Borel Covering Theorem; open sets, compact sets This section gives another application of the interval halving method, this time to a particularly famous theorem of analysis, the Heine Borel

### Topology notes. Basic Definitions and Properties.

Topology notes. Basic Definitions and Properties. Intuitively, a topological space consists of a set of points and a collection of special sets called open sets that provide information on how these points

### Therefore, after becoming familiar with the Matrix Method, you will be able to solve a system of two linear equations in four different ways.

Grade 9 IGCSE A1: Chapter 9 Matrices and Transformations Materials Needed: Straightedge, Graph Paper Exercise 1: Matrix Operations Matrices are used in Linear Algebra to solve systems of linear equations.

### CGF Lecture 2 Numbers

CGF Lecture 2 Numbers Numbers A number is an abstract entity used originally to describe quantity. i.e. 80 Students etc The most familiar numbers are the natural numbers {0, 1, 2,...} or {1, 2, 3,...},

### MA651 Topology. Lecture 4. Topological spaces 2

MA651 Topology. Lecture 4. Topological spaces 2 This text is based on the following books: Linear Algebra and Analysis by Marc Zamansky Topology by James Dugundgji Fundamental concepts of topology by Peter

### CSE 20 DISCRETE MATH. Winter

CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Final exam The final exam is Saturday March 18 8am-11am. Lecture A will take the exam in GH 242 Lecture B will take the exam

### THE TOPOLOGICAL ENTROPY OF HOMEOMORPHISMS ON ONE-DIMENSIONAL CONTINUA

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 108, Number 4, April 1990 THE TOPOLOGICAL ENTROPY OF HOMEOMORPHISMS ON ONE-DIMENSIONAL CONTINUA GERALD T. SEIDLER (Communicated by James E. West)

### The Language of Sets and Functions

MAT067 University of California, Davis Winter 2007 The Language of Sets and Functions Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (January 7, 2007) 1 The Language of Sets 1.1 Definition and Notation

### CSE 20 DISCRETE MATH. Fall

CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Final exam The final exam is Saturday December 16 11:30am-2:30pm. Lecture A will take the exam in Lecture B will take the exam

### On the Parikh-de-Bruijn grid

On the Parikh-de-Bruijn grid Péter Burcsi Zsuzsanna Lipták W. F. Smyth ELTE Budapest (Hungary), U of Verona (Italy), McMaster U (Canada) & Murdoch U (Australia) Words & Complexity 2018 Lyon, 19-23 Feb.

### Chapter 3: Theory of Modular Arithmetic 1. Chapter 3: Theory of Modular Arithmetic

Chapter 3: Theory of Modular Arithmetic 1 Chapter 3: Theory of Modular Arithmetic SECTION A Introduction to Congruences By the end of this section you will be able to deduce properties of large positive

### Denotational semantics

1 Denotational semantics 2 What we're doing today We're looking at how to reason about the effect of a program by mapping it into mathematical objects Specifically, answering the question which function

### Alphabets, strings and formal. An introduction to information representation

Alphabets, strings and formal languages An introduction to information representation 1 Symbols Definition: A symbol is an object endowed with a denotation (i.e. literal meaning) Examples: Variables are

### CS Bootcamp Boolean Logic Autumn 2015 A B A B T T T T F F F T F F F F T T T T F T F T T F F F

1 Logical Operations 1.1 And The and operator is a binary operator, denoted as, &,, or sometimes by just concatenating symbols, is true only if both parameters are true. A B A B F T F F F F The expression

### Recursively Defined Functions

Section 5.3 Recursively Defined Functions Definition: A recursive or inductive definition of a function consists of two steps. BASIS STEP: Specify the value of the function at zero. RECURSIVE STEP: Give

### CS321 Introduction To Numerical Methods

CS3 Introduction To Numerical Methods Fuhua (Frank) Cheng Department of Computer Science University of Kentucky Lexington KY 456-46 - - Table of Contents Errors and Number Representations 3 Error Types

### Notes on Minimum Cuts and Modular Functions

Notes on Minimum Cuts and Modular Functions 1 Introduction The following are my notes on Cunningham s paper [1]. Given a submodular function f and a set S, submodular minimisation is the problem of finding