# Sets and set operations: cont. Functions.

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CS 441 Discrete Mathematics for CS Lecture 8 Sets and set operations: cont. Functions. Milos Hauskrecht 5329 Sennott Square Set Definition: set is a (unordered) collection of objects. These objects are sometimes called elements or members of the set. (Cantor's naive definition) Examples: Vowels in the English alphabet V = { a, e, i, o, u } First seven prime numbers. X = { 2, 3, 5, 7, 11, 13, 17 } 1

2 Sets - review subset of B: is a subset of B if all elements in are also in B. Proper subset: is a proper subset of B, if is a subset of B and B power set: The power set of is a set of all subsets of Sets - review Cardinality of a set : The number of elements of in the set n n-tuple n ordered collection of n elements Cartesian product of and B set of all pairs such that the first element is in and the second in B 2

3 Set operations Set union: = {1,2,3,6} B = { 2,4,6,9} B = { 1,2,3,4,6,9 } Set intersection: = {1,2,3,6} B = { 2, 4, 6, 9} B = { 2, 6 } Set difference: = {1,2,3,6} B = { 2, 4, 6, 9} - B = { 1, 3} B = {4, 9} Complement of a set Definition: Let U be the universal set: the set of all objects under the consideration. Definition: The complement of the set, denoted by, is the complement of with respect to U. lternate: = { x x } U Example: U={1,2,3,4,5,6,7,8} ={1,3,5,7} = {2,4,6,8} 3

4 Set identities Set Identities (analogous to logical equivalences) Identity = U = Domination U = U = Idempotent = = Set identities Double complement = Commutative B = B B = B ssociative (B C) = ( B) C (B C) = ( B) C Distributive (B C) = ( B) ( C) (B C) = ( B) ( C) 4

5 DeMorgan ( B) = B ( B) = B bsorbtion Laws ( B) = ( B) = Complement Laws = U = Set identities Set identities Set identities can be proved using membership tables. List each combination of sets that an element can belong to. Then show that for each such a combination the element either belongs or does not belong to both sets in the identity. Prove: ( B) = B _ B B B B

6 Generalized unions and itersections Definition: The union of a collection of sets is the set that contains those elements that are members of at least one set in the collection. n i1 i { } n Example: Let i = {1,2,...,i} i =1,2,...,n n i 1 i {1, 2,..., n } Generalized unions and intersections Definition: The intersection of a collection of sets is the set that contains those elements that are members of all sets in the collection. n i 1 i { n } Example: Let i = {1,2,...,i} i =1,2,...,n n i 1 i { 1 } 6

7 Computer representation of sets How to represent sets in the computer? One solution: Data structures like a list better solution: ssign a bit in a bit string to each element in the universal set and set the bit to 1 if the element is present otherwise use 0 Example: ll possible elements: U={ } ssume ={2,5} Computer representation: = ssume B={1,5} Computer representation: B = Example: = B = Computer representation of sets The union is modeled with a bitwise or B = The intersection is modeled with a bitwise and B = The complement is modeled with a bitwise negation =

8 Functions Functions Definition: Let and B be two sets. function from to B, denoted f : B, is an assignment of exactly one element of B to each element of. We write f(a) = b to denote the assignment of b to an element a of by the function f. f: B B 8

9 Functions Definition: Let and B be two sets. function from to B, denoted f : B, is an assignment of exactly one element of B to each element of. We write f(a) = b to denote the assignment of b to an element a of by the function f. f: B B Not allowed!!! Representing functions Representations of functions: 1. Explicitly state the assignments in between elements of the two sets 2. Compactly by a formula. (using standard functions) Example1: Let = {1,2,3} and B = {a,b,c} ssume f is defined as: 1 c 2 a 3 c Is f a function? Yes. since f(1)=c, f(2)=a, f(3)=c. each element of is assigned an element from B 9

10 Representing functions Representations of functions: 1. Explicitly state the assignments in between elements of the two sets 2. Compactly by a formula. (using standard functions) Example 2: Let = {1,2,3} and B = {a,b,c} ssume g is defined as 1 c 1 b 2 a 3 c Is g a function? No. g(1) = is assigned both c and b. Representing functions Representations of functions: 1. Explicitly state the assignments in between elements of the two sets 2. Compactly by a formula. (using standard functions) Example 3: = {0,1,2,3,4,5,6,7,8,9}, B = {0,1,2} Define h: B as: h(x) = x mod 3. (the result is the remainder after the division by 3) ssignments:

11 Important sets Definitions: Let f be a function from to B. We say that is the domain of f and B is the codomain of f. If f(a) = b, b is the image of a and a is a pre-image of b. The range of f is the set of all images of elements of. lso, if f is a function from to B, we say f maps to B. Example: Let = {1,2,3} and B = {a,b,c} ssume f is defined as: 1 c, 2 a, 3 c What is the image of 1? 1 c c is the image of 1 What is the pre-image of a? 2 a 2 is a pre-image of a. Domain of f? {1,2,3} Codomain of f? {a,b,c} Range of f? {a,c} Image of a subset Definition: Let f be a function from set to set B and let S be a subset of. The image of S is a subset of B that consists of the images of the elements of S. We denote the image of S by f(s), so that f(s) = { f(s) s S }. S f: B B f(s) Example: Let = {1,2,3} and B = {a,b,c} and f: 1 c, 2 a, 3 c Let S = {1,3} then image f(s) =? 11

12 Image of a subset Definition: Let f be a function from set to set B and let S be a subset of. The image of S is a subset of B that consists of the images of the elements of S. We denote the image of S by f(s), so that f(s) = { f(s) s S }. S f: B B f(s) Example: Let = {1,2,3} and B = {a,b,c} and f: 1 c, 2 a, 3 c Let S = {1,3} then image f(s) = {c}. Injective function Definition: function f is said to be one-to-one, or injective, if and only if f(x) = f(y) implies x = y for all x, y in the domain of f. function is said to be an injection if it is one-to-one. lternate: function is one-to-one if and only if f(x) f(y), whenever x y. This is the contrapositive of the definition. f: B B f: B B Not injective Injective function 12

13 Injective functions Example 1: Let = {1,2,3} and B = {a,b,c} Define f as 1 c 2 a 3 c Is f one to one? No, it is not one-to-one since f(1) = f(3) = c, and 1 3. Example 2: Let g : Z Z, where g(x) = 2x - 1. Is g is one-to-one (why?) Yes. Suppose g(a) = g(b), i.e., 2a - 1 = 2b - 1 => 2a = 2b `` => a = b. Surjective function Definition: function f from to B is called onto, or surjective, if and only if for every b B there is an element a such that f(a) = b. lternative: all co-domain elements are covered f: B B 13

14 Surjective functions Example 1: Let = {1,2,3} and B = {a,b,c} Define f as 1 c 2 a 3 c Is f an onto? No. f is not onto, since b B has no pre-image. Example 2: = {0,1,2,3,4,5,6,7,8,9}, B = {0,1,2} Define h: B as h(x) = x mod 3. Is h an onto function? Yes. h is onto since a pre-image of 0 is 6, a pre-image of 1 is 4, a pre-image of 2 is 8. Bijective functions Definition: function f is called a bijection if it is both one-toone and onto. f: B B 14

15 Bijective functions Example 1: Let = {1,2,3} and B = {a,b,c} Define f as 1 c 2 a 3 b Is f is a bijection? Yes. It is both one-to-one and onto. Note: Let f be a function from a set to itself, where is finite. f is one-to-one if and only if f is onto. This is not true for an infinite set. Define f : Z Z, where f(z) = 2 * z. f is one-to-one but not onto (3 has no pre-image). Bijective functions Example 2: Define g : W W (whole numbers), where g(n) = [n/2] (floor function). 0 [0/2] = [0] = 0 1 [1/2] = [1/2] = 0 2 [2/2] = [1] = 1 3 [3/2] = [3/2] = 1... Is g a bijection? No. g is onto but not 1-1 (g(0) = g(1) = 0 however

### Sets and set operations

CS 441 Discrete Mathematics for CS Lecture 7 Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square asic discrete structures Discrete math = study of the discrete structures used

### Definition 14 A set is an unordered collection of elements or objects.

Chapter 4 Set Theory Definition 14 A set is an unordered collection of elements or objects. Primitive Notation EXAMPLE {1, 2, 3} is a set containing 3 elements: 1, 2, and 3. EXAMPLE {1, 2, 3} = {3, 2,

### 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

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### The Language of Mathematics

CHPTER 2 The Language of Mathematics 2.1. Set Theory 2.1.1. Sets. set is a collection of objects, called elements of the set. set can be represented by listing its elements between braces: = {1, 2, 3,

### 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

### Discrete Mathematics

Discrete Mathematics Chih-Wei Yi Dept. of Computer Science National Chiao Tung University March 16, 2009 2.1 Sets 2.1 Sets 2.1 Sets Basic Notations for Sets For sets, we ll use variables S, T, U,. We can

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

### 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

### Integers and division

CS 441 Discrete Mathematics for CS Lecture 12 Integers and division Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Symmetric matrix Definition: A square matrix A is called symmetric if A = A T.

### Automata Theory. Şubat 2006 Tuğrul Yılmaz Ankara Üniversitesi

Automata Theory Automata theory is the study of abstract computing devices. A. M. Turing studied an abstract machine that had all the capabilities of today s computers. Turing s goal was to describe the

### 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

### You know from calculus that functions play a fundamental role in mathematics.

CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.

### Basic Concepts of Set Theory, Functions and Relations

March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory...1 1.1. Sets and elements...1 1.2. Specification of sets...2 1.3. Identity and cardinality...3

### Mathematics for Computer Science

Mathematics for Computer Science Lecture 2: Functions and equinumerous sets Areces, Blackburn and Figueira TALARIS team INRIA Nancy Grand Est Contact: patrick.blackburn@loria.fr Course website: http://www.loria.fr/~blackbur/courses/math

### Practice with Proofs

Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

### What is a set? Sets. Specifying a Set. Notes. The Universal Set. Specifying a Set 10/29/13

What is a set? Sets CS 231 Dianna Xu set is a group of objects People: {lice, ob, Clara} Colors of a rainbow: {red, orange, yellow, green, blue, purple} States in the S: {labama, laska, Virginia, } ll

### Notes on counting finite sets

Notes on counting finite sets Murray Eisenberg February 26, 2009 Contents 0 Introduction 2 1 What is a finite set? 2 2 Counting unions and cartesian products 4 2.1 Sum rules......................................

### Cardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.

Section 2.5 Cardinality (another) Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a bijection from A to B. If there is an injection

### Florida State University Course Notes MAD 2104 Discrete Mathematics I

Florida State University Course Notes MAD 2104 Discrete Mathematics I Florida State University Tallahassee, Florida 32306-4510 Copyright c 2011 Florida State University Written by Dr. John Bryant and Dr.

### 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)

### Basic Set Theory. Chapter Set Theory. can be written: A set is a Many that allows itself to be thought of as a One.

Chapter Basic Set Theory A set is a Many that allows itself to be thought of as a One. - Georg Cantor This chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical

### 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

### 1 The Concept of a Mapping

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 1 The Concept of a Mapping The concept of a mapping (aka function) is important throughout mathematics. We have been dealing

### Discrete Mathematics Set Operations

Discrete Mathematics 1-3. Set Operations Introduction to Set Theory A setis a new type of structure, representing an unordered collection (group, plurality) of zero or more distinct (different) objects.

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

### SETS AND FUNCTIONS, MATH 215 FALL 2015 (WHYTE)

SETS AND FUNCTIONS, MATH 215 FALL 2015 (WHYTE) 1. Intro to Sets After some work with numbers, we want to talk about sets. For our purposes, sets are just collections of objects. These objects can be anything

### 7 Relations and Functions

7 Relations and Functions In this section, we introduce the concept of relations and functions. Relations A relation R from a set A to a set B is a set of ordered pairs (a, b), where a is a member of A,

### CPSC 121: Models of Computation Assignment #4, due Wednesday, July 22nd, 2009 at 14:00

CPSC 2: Models of Computation ssignment #4, due Wednesday, July 22nd, 29 at 4: Submission Instructions Type or write your assignment on clean sheets of paper with question numbers prominently labeled.

### Recursion Theory in Set Theory

Contemporary Mathematics Recursion Theory in Set Theory Theodore A. Slaman 1. Introduction Our goal is to convince the reader that recursion theoretic knowledge and experience can be successfully applied

### 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

### This chapter is all about cardinality of sets. At first this looks like a

CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

### Logic in Computer Science: Logic Gates

Logic in Computer Science: Logic Gates Lila Kari The University of Western Ontario Logic in Computer Science: Logic Gates CS2209, Applied Logic for Computer Science 1 / 49 Logic and bit operations Computers

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### 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

### Clicker Question. Theorems/Proofs and Computational Problems/Algorithms MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES

MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES Tuesday, 1/21/14 General course Information Sets Reading: [J] 1.1 Optional: [H] 1.1-1.7 Exercises: Do before next class; not to hand in [J] pp. 12-14:

### Lecture Notes on Discrete Mathematics

Lecture Notes on Discrete Mathematics A. K. Lal September 26, 2012 2 Contents 1 Preliminaries 5 1.1 Basic Set Theory.................................... 5 1.2 Properties of Integers.................................

### Fixed Point Theorems in Topology and Geometry

Fixed Point Theorems in Topology and Geometry A Senior Thesis Submitted to the Department of Mathematics In Partial Fulfillment of the Requirements for the Departmental Honors Baccalaureate By Morgan Schreffler

### 1 / Basic Structures: Sets, Functions, Sequences, and Sums - definition of a set, and the use of the intuitive notion that any property whatever there

C H A P T E R Basic Structures: Sets, Functions, Sequences, and Sums.1 Sets. Set Operations.3 Functions.4 Sequences and Summations Much of discrete mathematics is devoted to the study of discrete structures,

### 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

### Chapter 10. Abstract algebra

Chapter 10. Abstract algebra C.O.S. Sorzano Biomedical Engineering December 17, 2013 10. Abstract algebra December 17, 2013 1 / 62 Outline 10 Abstract algebra Sets Relations and functions Partitions and

### 2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair

2.1 J.A.Beachy 1 2.1 Functions from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 21. The Vertical Line Test from calculus says that a curve in the xy-plane

### 1. Determine all real numbers a, b, c, d that satisfy the following system of equations.

altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### Lecture Notes on Topology for MAT3500/4500 following J. R. Munkres textbook. John Rognes

Lecture Notes on Topology for MAT3500/4500 following J. R. Munkres textbook John Rognes November 29th 2010 Contents Introduction v 1 Set Theory and Logic 1 1.1 ( 1) Fundamental Concepts..............................

### Automata and Formal Languages

Automata and Formal Languages Winter 2009-2010 Yacov Hel-Or 1 What this course is all about This course is about mathematical models of computation We ll study different machine models (finite automata,

### H/wk 13, Solutions to selected problems

H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.

### Math 223 Abstract Algebra Lecture Notes

Math 223 Abstract Algebra Lecture Notes Steven Tschantz Spring 2001 (Apr. 23 version) Preamble These notes are intended to supplement the lectures and make up for the lack of a textbook for the course

### CHAPTER 2. Set, Whole Numbers, and Numeration

CHAPTER 2 Set, Whole Numbers, and Numeration 2.1. Sets as a Basis for Whole Numbers A set is a collection of objects, called the elements or members of the set. Three common ways to define sets: (1) A

### Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett

Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.

### A Little Set Theory (Never Hurt Anybody)

A Little Set Theory (Never Hurt Anybody) Matthew Saltzman Department of Mathematical Sciences Clemson University Draft: August 21, 2013 1 Introduction The fundamental ideas of set theory and the algebra

### CS 3719 (Theory of Computation and Algorithms) Lecture 4

CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a

### 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

### Set theory as a foundation for mathematics

Set theory as a foundation for mathematics Waffle Mathcamp 2011 In school we are taught about numbers, but we never learn what numbers really are. We learn rules of arithmetic, but we never learn why these

### Introduction to Topology

Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

### ( ) which must be a vector

MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

### Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

### Discrete Maths. Philippa Gardner. These lecture notes are based on previous notes by Iain Phillips.

Discrete Maths Philippa Gardner These lecture notes are based on previous notes by Iain Phillips. This short course introduces some basic concepts in discrete mathematics: sets, relations, and functions.

### Lecture 17 : Equivalence and Order Relations DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

### Chapter 7. Homotopy. 7.1 Basic concepts of homotopy. Example: z dz. z dz = but

Chapter 7 Homotopy 7. Basic concepts of homotopy Example: but γ z dz = γ z dz γ 2 z dz γ 3 z dz. Why? The domain of /z is C 0}. We can deform γ continuously into γ 2 without leaving C 0}. Intuitively,

### 2.1 Symbols and Terminology

2.1 Symbols and Terminology Definitions: set is a collection of objects. The objects belonging to the set are called elements, ormembers, oftheset. Sets can be designated in one of three different ways:

### 13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385

mcs-ftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite

### Geometric Transformations

Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### 1 Semantic Competence

24.903 Language & Structure II: Semantics and Pragmatics Spring 2003, 2-151, MW 1-2.30 1 Semantic Competence The goal of semantics is to properly characterize semantic competence. What is semantic competence?

### Foundations of Mathematics I Set Theory (only a draft)

Foundations of Mathematics I Set Theory (only a draft) Ali Nesin Mathematics Department Istanbul Bilgi University Kuştepe Şişli Istanbul Turkey anesin@bilgi.edu.tr February 12, 2004 2 Contents I Naive

### Chapter 7. Functions and onto. 7.1 Functions

Chapter 7 Functions and onto This chapter covers functions, including function composition and what it means for a function to be onto. In the process, we ll see what happens when two dissimilar quantifiers

### CS 580: Software Specifications

CS 580: Software Specifications Lecture 2: Sets and Relations Slides originally Copyright 2001, Matt Dwyer, John Hatcliff, and Rod Howell. The syllabus and all lectures for this course are copyrighted

### Multiplicity. Chapter 6

Chapter 6 Multiplicity The fundamental theorem of algebra says that any polynomial of degree n 0 has exactly n roots in the complex numbers if we count with multiplicity. The zeros of a polynomial are

### GRE Prep: Precalculus

GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach

### Relations and Functions

Section 5. Relations and Functions 5.1. Cartesian Product 5.1.1. Definition: Ordered Pair Let A and B be sets and let a A and b B. An ordered pair ( a, b) is a pair of elements with the property that:

### Algebra I Notes Relations and Functions Unit 03a

OBJECTIVES: F.IF.A.1 Understand the concept of a function and use function notation. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element

### Collinear Points in Permutations

Collinear Points in Permutations Joshua N. Cooper Courant Institute of Mathematics New York University, New York, NY József Solymosi Department of Mathematics University of British Columbia, Vancouver,

### An Introduction to Elementary Set Theory

An Introduction to Elementary Set Theory Guram Bezhanishvili and Eachan Landreth 1 Introduction In this project we will learn elementary set theory from the original historical sources by two key figures

### INTRODUCTION TO TOPOLOGY

INTRODUCTION TO TOPOLOGY ALEX KÜRONYA In preparation January 24, 2010 Contents 1. Basic concepts 1 2. Constructing topologies 13 2.1. Subspace topology 13 2.2. Local properties 18 2.3. Product topology

### LESSON SUMMARY. Set Operations and Venn Diagrams

LESSON SUMMARY CXC CSEC MATHEMATICS UNIT Three: Set Theory Lesson 4 Set Operations and Venn Diagrams Textbook: Mathematics, A Complete Course by Raymond Toolsie, Volumes 1 and 2. (Some helpful exercises

### We now explore a third method of proof: proof by contradiction.

CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### Introduction to Theory of Computation

Introduction to Theory of Computation Prof. (Dr.) K.R. Chowdhary Email: kr.chowdhary@iitj.ac.in Formerly at department of Computer Science and Engineering MBM Engineering College, Jodhpur Tuesday 28 th

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

### Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations

Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate

### Georg Cantor (1845-1918):

Georg Cantor (845-98): The man who tamed infinity lecture by Eric Schechter Associate Professor of Mathematics Vanderbilt University http://www.math.vanderbilt.edu/ schectex/ In papers of 873 and 874,

### A Course in Discrete Structures. Rafael Pass Wei-Lung Dustin Tseng

A Course in Discrete Structures Rafael Pass Wei-Lung Dustin Tseng Preface Discrete mathematics deals with objects that come in discrete bundles, e.g., 1 or 2 babies. In contrast, continuous mathematics

### Sets (DSLC 1.2) Intuitively, a set is a collection of things, called its elements, ormembers. To say that x is an element of S, wewrite

Sets (DSLC 1.2) Intuitively, a set is a collection of things, called its elements, ormembers. To say that x is an element of S, wewrite x 2 S. Other ways of saying this: x belongs to S, S contains x, x

### x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

### Set operations and Venn Diagrams. COPYRIGHT 2006 by LAVON B. PAGE

Set operations and Venn Diagrams Set operations and Venn diagrams! = { x x " and x " } This is the intersection of and. # = { x x " or x " } This is the union of and. n element of! belongs to both and,

### Proseminar on Semantic Theory Fall 2013 Ling 720. Problem Set on the Formal Preliminaries : Answers and Notes

1. Notes on the Answers Problem Set on the Formal Preliminaries : Answers and Notes In the pages that follow, I have copied some illustrative answers from the problem sets submitted to me. In this section,

### Regular Languages and Finite State Machines

Regular Languages and Finite State Machines Plan for the Day: Mathematical preliminaries - some review One application formal definition of finite automata Examples 1 Sets A set is an unordered collection

### To define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions

Chapter 7 Functions This unit defines and investigates functions as algebraic objects. First, we define functions and discuss various means of representing them. Then we introduce operations on functions

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but