Mathematics for Computer Science


 Malcolm Cross
 1 years ago
 Views:
Transcription
1 Mathematics for Computer Science Lecture 2: Functions and equinumerous sets Areces, Blackburn and Figueira TALARIS team INRIA Nancy Grand Est Contact: Course website:
2 Functions: some intuitions A function (or mapping or transformation) is one of the most basic of all mathematical concepts. Intuitively, a function is something that takes input and turns it into some kind of output. There is a heavy computational flavour to the intuition. Indeed, when we introduce Turing Machines next week, we will think of Turing Machines as simple devices for computing functions. In this lecture we are going to see how this concept is made made mathematical precise, and will then use certain kinds of functions (namely bijective functions) to compare the sizes of sets. Next week we will look at functions from a computational perspective.
3 Where we are going today We begin by introducing the notion of an ordered pair and of a relations. We then define functions to be certain kinds of relations, and define three special types of functions: injective functions, surjective functions, and bijective functions. We then define a fundamental concept: when two sets have the same number of elements or (to use the official terminology) when they are equinumerous. We then compare the sizes of a number of infinite sets, with some surprising results.
4 Ordered pairs In set theory, the notation x, y is used to indicate that x and y are are a pair of items with x coming first and y second. That is, this notation specifies both the items involved (x and y) and the order in which they occur. That is x, y is not the same as {x, y}. The order is not specified in {x, y}; after all {x, y} = {y, x}. For ordered pairs we must have that: x, y = u, v iff x = u and y = v. This is the characteristic property of ordered pairs.
5 Aside on ordered pairs As I remarked last week, in set theory everything is defined purely in terms of sets and indeed, the notation x, y is usually defined as follows: x, y = def {{x}, {x, y}} This is not an obvious definition; it was first discovered by Kazimierz Kuratowski in 1921, replacing a more complex definition given by Norbert Wiener in The crucial thing to notice about this definition is that it works as we would expect ordered pairs to work. That is, it is not difficult to show that under this definition we have that: x, y = u, v iff x = u and y = v, which is what we require of any successful definition.
6 Binary relations A binary relation is defined to be a set of ordered pairs. More precisely, given a set X and a set Y, we define X Y to be the set of all ordered pairs x, y such that x X and y Y. The set X Y is called the Cartesian product of X and Y. A binary relation is simply a subset of X Y. X is called the domain of the relation, and Y is called the range.
7 Binary Relation Examples I Let X = {1, 2, 3} and Y = {a, b}. Then R 1 = { 1, a, 2, a } is a binary relation on these sets. Another (bigger) binary relation on these sets is R 2 = { 1, b, 2, b, 2, a }. Another bigger (and fact the biggest possible) binary relation on these sets is simply X Y itself. When dealing with small relations like these, it is common (and useful) to draw them using dots (to represent the entities) and arrows (to indicate which entities are related).
8 Binary Relation Examples II When defining binary relations it is very common for the domain and range of the relation to be the same set. That is, it is common to consider binary relations on X X (often written as X 2 ) for some set X. For example, let P be the set {Mia, Vincent, Marcellus}. Then L = { Vincent, Mia, Marcellus, Mia } is a binary relation (perhaps the loves relation) on X X. To give a more mathematical example, { x, y R R x y} is a binary relation on R R. It consists of all pairs of real numbers such the the first is less than or equal to the second. It can be represented on the Cartesian plane in the familiar way.
9 But what about functions? OK, but what about functions? At some stage in your career you have probably seem functions like f(x) = x 2, and are used to notation like f(2) = 4 and f(3) = 9. That is, you are probably used to being given definitions of functions f, g, h and so on, and used to the idea that such functions can be given arguments to produce results (or output). But what does this have to do with our discussion of relations? The answer is very simple...
10 Function A function with domain X and range Y is simply a special kind of binary relation. What makes it special? This: each x X is related to exactly one element y of Y. Suppose f is a function with domain X and range Y. Then we write f(x) to denote the unique element of Y that x is related to (that is, we simply sue the standard notation that you are used to from school maths). The image of a function f is {y Y : y = f(x) for some x X}.
11 Examples of functions (and nonfunctions) I Let X be {1, 2, 3, 4}, and let f be the following binary relation on X X: { 1, 2, 2, 3, 3, 2, 4, 3 }. It is easy to check that this binary relation is indeed a function. On the other hand suppose we let g be the following binary relation on X X: { 1, 2, 2, 3, 1, 3, 4, 3 }. This is clearly not a function. The binary relation { x, y R R y = x 2 } is a function. As you probably all saw at school, it looks like a parabola when drawn in the Cartesian plane. On the other hand, the binary relation { x, y R R y = x} is not a function (as positive real numbers have two square roots, one positive, the other negative).
12 Identity functions 1. A special (and very simple function) called the identity function can be defined on any set X. 2. The identity function id X, with domain X and range X is simply that function that maps each element of X to itself. 3. That is, for all x X, id X (x) = x.
13 Function composition The fundamental operation of functions is called function composition. In essence, this is the operation of plugging together two functions with compatible domains and ranges. More precisely, let A, B and C be sets, let g be a function with domain A and range B, and let f be a function with domain B and range C. Then fog, the composition of f with g is the function with domain A and range C defined by fog(a) = f(g(a)). That is, we first apply g to an elementa of A, and then apply f to the result, yielding an element of C.
14 Injective functions A function f is injective if and only if distinct elements of the domain map to distinct elements of the range. That is, a function is injective if and only if for all x, y in its domain such that x y, we have that f(x) f(y). Injective functions are often called onetoone functions.
15 Examples Let A be {a, b, c}, let f be { a, b, b, c, c, a }, and let g be { a, b, b, c, c, b }. Then both f and g are functions, but f is injective and g is not. The function with domain and range R defined by y = x + 1 is injective. The function with domain and range R defined by y = x 2 is not injective.
16 Surjective functions A function f is said to be surjective if its image is its range. To put it another way, a surjective function is one where every element of the range has some element of the domain that is mapped to it. Surjective functions are often called onto functions.
17 Examples Let A be {a, b, c}, let f be { a, b, b, c, c, a }, and let g be { a, b, b, c, c, b }. Then both f and g are functions, but f is surjective and g is not. The function with domain and range R defined by y = x 3 is surjective. The function with domain and range R defined by y = x 2 is not surjective.
18 Bijective functions A function is bijective if and only if it is both injective and surjective. That is, a function is a bijection if and only if it is both one to one and onto. Bijections have the following property: they can be turned around. More precisely, if f is a bijection with domain A and range B, then there is a bijection (called f 1 ) with domain B and range A such that f 1 of = id A. What does this mean and why is it it true?
19 Examples Let A be {a, b, c}, let f be { a, b, b, c, c, a }. As we have already seen, f is both injective and surjective. That is, f is bijective. What is f 1? The function with domain and range R defined by g(x) = x + 1 is bijective. What is g 1? For any set X, the identity function id X is a bijection from X to X.
20 Bijections and having the same size We are now ready to turn to the link between bijections and counting, and more generally, what it means for two sets to have the same size. Intuitively, when we count objects, what we are doing is establishing a bijection between a set of numbers and a set of objects. For example, if there are three doughnuts on the table, and I count them, what I have done is establish a bijection between the three doughnuts and the set {1, 2, 3}. More generally, if I show that there is a bijection between two sets, I have (even without counting) shown that they have the same size. This prompts the following definition...
21 Equinumerous sets Two sets are said to be equinumerous (that is, to have the same number of elements) if and only if there is a bijection between them. To put it another way, two sets are said to have the same size (the same cardinality) if and only if it is possible to map one to the other via a bijection. Sometimes we say that two sets can be put into one to one correspondence instead of saying that they are equinumerous. That is, this expression is just another common way of saying that there is a bijection between the two sets.
22 Equinumerous sets For example, {a, b, c} and {x, y, z} have the same number of elements. Why? Because it is easy to show that there is a bijection between these two sets. The elements can be paired off. Of course, we could say that these two sets have the same size by counting them, and saying that they both have three elements. But we don t need to count. The idea of having the same number of elements does not depend on the notion of number.
23 Comparing infinite sets Now that we know what having the same size means, we can start comparing sets. Now, this is a little boring for finite sets (we all learned to count finite sets when we were young). But matters are more interesting when we compare infinite sets. The most fundamental infinite set in set theory is N, the set of natural numbers. This is the set that contains all our usual counting numbers (0, 1, 2, 3, 4,... ) and nothing else. To get the ball rolling, let us call a set X countably infinite if it can be put into one to one correspondence with N (that is, if it is equinumerous with N).
24 The set of even numbers is countably infinite The set E of even numbers, that is, {0, 2, 4, 6, 8,...} is countably infinite. That is, there is a bijection with domain E and range N. The function is the obvious one: simply divide each even number by two. That is: we use the function f such that f(e) = e/2. It is easy to check that f is both injective and surjective. Thus while it is true that E N, it is also true that E and N have the same number of elements. E is countably infinite.
25 The set of integers is countably infinite The set of integers Z (that is, {... 3, 2, 1, 0, 1, 2, 3...} is also countably infinite. That is, there is a bijection with domain Z and range N. Here are the first few values of a suitable bijection f: f(0) = 0, f(1) = 1, f( 1) = 2, f(2) = 3, f( 2) = 4, f(3) = 5, f( 3) = 6, f(4) = 7, f( 4) = 8, Draw a diagram to see the pattern. That is, we have the funtion defined by f(z) = 2z, for all z 0, and f(z) = 2z 1 for z > 0. Thus while it is true that N Z, it is also true that Z and N have the same number of elements. Z is countably infinite.
26 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 0. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
27 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 0. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
28 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 1. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
29 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 2. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
30 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 3. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
31 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 4. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
32 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 5. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
33 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 6. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
34 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 6. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
35 Lots of interesting sets are countably infinite In particular, the set of rational numbers Q is countably infinite. This can be seen by an argument rather like the one we just saw for N N. In fact this might lead us to wonder are all infinite sets countably infinite? The answer to this is no. There are many bigger infinite sets. In fact, hre infinitely many sizes of ever everbigger infinities! Let s take a closer look...
36 Bigger infinities We are going to prove two main results The set of real numbers R (an infinite set( is not countably infinite it is bigger. For any set X, the set P(X) is bigger. We saw last week that this result holds for finite sets. This week we will see that it holds for arbitrary sets. A consequence of this result is if we start with an infinite set (say N) and then form its power set, we get a bigger infinite set. If we take the power set of that, we get a still bigger infinite set. And so am and so on and so on... the infinities keep getting bigger. The countably infinite is merely the bottom rung of the ladder of infinity.
37 The Diagonal argument Both arguments are proved using what have become known as diagonal arguments. This celebrated technique, was originally due to Georg Cantor, the father of set theory, who used it to prove the two results we shall now prove. As you will see, the two arguments have a slightly different form. However the underlying idea is the same. Nowadays the diagonal argument is used in many different forms in many different parts fo mathematics and computer science.
38 The real numbers R Recall from that every real number r can be represented by an infinite decimal number. For example: Can all these numbers be counted? That is, can we define a bijection form N to these numbers. The diagonal argument gives a systematic way of showing that no such function exists. If someone proposes such a function, we have a systematic method for showing it does not work.
39 The heart of the argument Suppose we have a function f with domain N that does count the real numbers. For example, suppose we have that: f(0) = f(1) = f(2) = f(3) = Define the following number. Before the decimal point it is 0. After the decimal point, its n + 1th decimal place is always 3, unless the n+1th place of f(n) is 3, in which case the is is 4. In this case, the number we build is: This number is not in the image of f, so f cannot be surjective, so it not a bijection.
40 Remark And that s the heart of diagonalisation. That is, someone proposes a suitable function that allegedly counts the set we are interested. We construct something (here a real number) that is not in the image of the function, thereby showing that the function cannot be surjective, and hence not bijective. We do so by working down the diagonal, systematically changing things. The technique takes it names from this idea.
41 Power sets are big! We are now going to show that any set X is always smaller than it power set P(X). The key word to observe here is the any ; we make no assumption that X is finite. We again show this by means of a (slightly different form) of the diagonal argument.
42 Some preliminary observations... First, recall that the result is true for finite sets. We proved last week that if a finite set X has n elements, then P(X) has 2 n elements. Second, it straightforward to see that for any X at all, whether X is finite or infinite, there is a bijection from X to a subset P(X). We simply use the function f that maps every y X to {y}. This means that P(X) can t possibly be smaller than X there is a clear sense in which X is embedded inside it. Still, none of this shows that there it is impossible (at least for infinite sets) to find a cunning way of bijectively mapping X to P(X). To prove this is impossible we need to be a bit smarter.
43 What we will do Let X be any set, and let g be a function from X to P(X). Again we will use diagonalisation to define a subset D g of X, called the diagonal set of g, that is is not in the image of g. That is we will show that g is not surjective, It s not obvious that we can do this after all, we know very little about X and g. But we can, as follows...
44 Diagonalisation Given any such function we define D g, the diagonal of g as follows: D g = {z X : z g(z)} Clearly D g X. That is, D g belongs to P(X). But for each z X: z D g iff z g(z). Hence D g g(z) for any z. Think about this carefully! And look at today s worksheet!
45 Consequences This argument shows that no function g from X to P(X) can be surjective, and hence no such function g can be a bijection. That is, for any set X we have that P(X) is bigger than X. In particular, P(N) is bigger than N. P(N) is certainly not countably infinite it is bigger. And P(P(N)) is bigger than P(N). And P(P(P(N))) is bigger than P(P(N)). And so on and so on: the infinities just keep on getting bigger and bigger and the process never stops...
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 informationCARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify
More informationMAT2400 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 informationSETS, RELATIONS, AND FUNCTIONS
September 27, 2009 and notations Common Universal Subset and Power Set Cardinality Operations A set is a collection or group of objects or elements or members (Cantor 1895). the collection of the four
More information13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcsftl 2010/9/8 0:40 page 379 #385
mcsftl 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
More informationSets and functions. {x R : x > 0}.
Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.
More informationIn mathematics you don t understand things. You just get used to them. (Attributed to John von Neumann)
Chapter 1 Sets and Functions We understand a set to be any collection M of certain distinct objects of our thought or intuition (called the elements of M) into a whole. (Georg Cantor, 1895) In mathematics
More informationBasic 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
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationDiscrete 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 informationLecture 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
More informationWe 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 informationSets, Relations and Functions
Sets, Relations and Functions Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org epacuit@umd.edu ugust 26, 2014 These notes provide a very brief background in discrete
More informationSets and Cardinality Notes for C. F. Miller
Sets and Cardinality Notes for 620111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use
More informationSets and Subsets. Countable and Uncountable
Sets and Subsets Countable and Uncountable Reading Appendix A Section A.6.8 Pages 788792 BIG IDEAS Themes 1. There exist functions that cannot be computed in Java or any other computer language. 2. There
More informationGeorg Cantor and Set Theory
Georg Cantor and Set Theory. Life Father, Georg Waldemar Cantor, born in Denmark, successful merchant, and stock broker in St Petersburg. Mother, Maria Anna Böhm, was Russian. In 856, because of father
More informationFinite Sets. Theorem 5.1. Two nonempty 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 informationGeorg Cantor (18451918):
Georg Cantor (84598): 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,
More informationThis 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 },
More informationSets and set operations: cont. Functions.
CS 441 Discrete Mathematics for CS Lecture 8 Sets and set operations: cont. Functions. Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Set Definition: set is a (unordered) collection of objects.
More informationCartesian 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
More informationThis chapter describes set theory, a mathematical theory that underlies all of modern mathematics.
Appendix A Set Theory This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. A.1 Basic Definitions Definition A.1.1. A set is an unordered collection of elements.
More information2.1 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 informationIt is not immediately obvious that this should even give an integer. Since 1 < 1 5
Math 163  Introductory Seminar Lehigh University Spring 8 Notes on Fibonacci numbers, binomial coefficients and mathematical induction These are mostly notes from a previous class and thus include some
More information5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1
MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing
More informationProblem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS
Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection
More information1 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
More informationStudents 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 information1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.
CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if
More informationLESSON 1 PRIME NUMBERS AND FACTORISATION
LESSON 1 PRIME NUMBERS AND FACTORISATION 1.1 FACTORS: The natural numbers are the numbers 1,, 3, 4,. The integers are the naturals numbers together with 0 and the negative integers. That is the integers
More informationCardinality. 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
More informationThis asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.
3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but
More informationCHAPTER 5: MODULAR ARITHMETIC
CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called
More informationPythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers
Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Amnon Yekutieli Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures
More informationSet 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
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More informationNotes 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......................................
More informationSets 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
More informationCompactness in metric spaces
MATHEMATICS 3103 (Functional Analysis) YEAR 2012 2013, TERM 2 HANDOUT #2: COMPACTNESS OF METRIC SPACES Compactness in metric spaces The closed intervals [a, b] of the real line, and more generally the
More informationThe 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,
More informationx 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
More informationChapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twentyfold way
Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or pingpong balls)
More information6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )
6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a nonempty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points
More information3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.
SOLUTIONS TO HOMEWORK 2  MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid
More informationMathematics 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
More informationModule MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions
Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective
More information3. 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,
More informationCS 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 ChurchTuring thesis Let s recap how it all started. In 1990, Hilbert stated a
More informationYou 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.
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More informationDefinition 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,
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a realvalued function defined on the sample space of some experiment. For instance,
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationLecture 4  Sets, Relations, Functions 1
Lecture 4 Sets, Relations, Functions Pat Place Carnegie Mellon University Models of Software Systems 17651 Fall 1999 Lecture 4  Sets, Relations, Functions 1 The Story So Far Formal Systems > Syntax»
More informationLecture 1. Basic Concepts of Set Theory, Functions and Relations
September 7, 2005 p. 1 Lecture 1. Basic Concepts of Set Theory, Functions and Relations 0. Preliminaries...1 1. Basic Concepts of Set Theory...1 1.1. Sets and elements...1 1.2. Specification of sets...2
More informationClimbing 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 informationSETS 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
More informationMath 4310 Handout  Quotient Vector Spaces
Math 4310 Handout  Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationMathematical Induction
Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural
More informationWe 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
More informationvertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws
Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,
More informationCalculus for Middle School Teachers. Problems and Notes for MTHT 466
Calculus for Middle School Teachers Problems and Notes for MTHT 466 Bonnie Saunders Fall 2010 1 I Infinity Week 1 How big is Infinity? Problem of the Week: The Chess Board Problem There once was a humble
More informationComputational Models Lecture 8, Spring 2009
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 1 Computational Models Lecture 8, Spring 2009 Encoding of TMs Universal Turing Machines The Halting/Acceptance
More information(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 informationDomain and Range. Many problems will ask you to find the domain of a function. What does this mean?
Domain and Range The domain of a function is the set of values that we are allowed to plug into our function. This set is the x values in a function such as f(x). The range of a function is the set of
More informationPrime Numbers. Chapter Primes and Composites
Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are
More informationMATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
More information3. 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 informationLogic & Discrete Math in Software Engineering (CAS 701) Dr. Borzoo Bonakdarpour
Logic & Discrete Math in Software Engineering (CAS 701) Background Dr. Borzoo Bonakdarpour Department of Computing and Software McMaster University Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS
More information7 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,
More informationDatabase Management System Dr. S. Srinath Department of Computer Science & Engineering Indian Institute of Technology, Madras Lecture No.
Database Management System Dr. S. Srinath Department of Computer Science & Engineering Indian Institute of Technology, Madras Lecture No. # 7 ER Model to Relational Mapping Hello and welcome to the next
More informationIntroduction 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.......................................
More informationMathematics 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,
More informationTopology and Convergence by: Daniel Glasscock, May 2012
Topology and Convergence by: Daniel Glasscock, May 2012 These notes grew out of a talk I gave at The Ohio State University. The primary reference is [1]. A possible error in the proof of Theorem 1 in [1]
More informationS(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive.
MA 274: Exam 2 Study Guide (1) Know the precise definitions of the terms requested for your journal. (2) Review proofs by induction. (3) Be able to prove that something is or isn t an equivalence relation.
More informationCHAPTER 1. Basic Ideas
CHPTER 1 asic Ideas In the end, all mathematics can be boiled down to logic and set theory. ecause of this, any careful presentation of fundamental mathematical ideas is inevitably couched in the language
More informationTHE TURING DEGREES AND THEIR LACK OF LINEAR ORDER
THE TURING DEGREES AND THEIR LACK OF LINEAR ORDER JASPER DEANTONIO Abstract. This paper is a study of the Turing Degrees, which are levels of incomputability naturally arising from sets of natural numbers.
More information1. 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 informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationRelations 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:
More information3. Examples of discrete probability spaces.. Here α ( j)
3. EXAMPLES OF DISCRETE PROBABILITY SPACES 13 3. Examples of discrete probability spaces Example 17. Toss n coins. We saw this before, but assumed that the coins are fair. Now we do not. The sample space
More informationSTRAND B: Number Theory. UNIT B2 Number Classification and Bases: Text * * * * * Contents. Section. B2.1 Number Classification. B2.
STRAND B: Number Theory B2 Number Classification and Bases Text Contents * * * * * Section B2. Number Classification B2.2 Binary Numbers B2.3 Adding and Subtracting Binary Numbers B2.4 Multiplying Binary
More informationCOMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012
Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
More informationClassical Analysis I
Classical Analysis I 1 Sets, relations, functions A set is considered to be a collection of objects. The objects of a set A are called elements of A. If x is an element of a set A, we write x A, and if
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More information2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.
2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then
More informationPractice 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
More information3. 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 informationRecall. (IISER Pune, Fall 2014.) MTH101 1 / 22
Recall Recall from the previous lecture Rational numbers have recurring decimal expansion Real numbers are numbers which have infinite decimal expansion which may or may not be recurring. N Z Q R. An infinite
More informationA set is a Many that allows itself to be thought of as a One. (Georg Cantor)
Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains
More informationMATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
More informationMath 563 Measure Theory Project 1 (Funky Functions Group) Luis Zerón, Sergey Dyachenko
Math 563 Measure Theory Project (Funky Functions Group) Luis Zerón, Sergey Dyachenko 34 Let C and C be any two Cantor sets (constructed in Exercise 3) Show that there exists a function F: [,] [,] with
More informationGROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.
Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the
More informationRecursion 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
More informationChapter 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
More informationAndrew McLennan January 19, Winter Lecture 5. A. Two of the most fundamental notions of the dierential calculus (recall that
Andrew McLennan January 19, 1999 Economics 5113 Introduction to Mathematical Economics Winter 1999 Lecture 5 Convergence, Continuity, Compactness I. Introduction A. Two of the most fundamental notions
More informationPolish spaces and standard Borel spaces
APPENDIX A Polish spaces and standard Borel spaces We present here the basic theory of Polish spaces and standard Borel spaces. Standard references for this material are the books [143, 231]. A.1. Polish
More informationNotes on Complexity Theory Last updated: August, 2011. Lecture 1
Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look
More information