Sets, Relations and Functions


 Silvia Hensley
 2 years ago
 Views:
Transcription
1 Sets, Relations and Functions Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org ugust 26, 2014 These notes provide a very brief background in discrete mathematics. 1 asic Set Theory We often groups things together. Everyone in this class, your group of friends, your family. These are all collections of people. Set theory is a mathematical language to talk about collections. The easiest way to visualize a set is to use a Venn diagram. Venn Diagram is a geometrical visualization of a set, or collection of sets. bstractly, a set is just a collection of objects that have something in common. We will denote sets with uppercase letters, and elements of sets will be denoted with lowercase letters. There are two ways to write down the contents of a set: 1. List all the elements of the set. Each element should be separated by a comma and there entire list of elements is written between curly brackets: { and }. For example suppose is the set of the first 5 letters of the alphabet. Then = {a, b, c, d, e}. 2. Write down a property that all elements of the set have in common. For example, if is the set of all positive integers, then we can describe as follows = {x x 0 and x is an integer}. This is read is the set of all x such that x is an integer that is greater than or equal to zero. 1
2 Definition 1.1 (Set) ny collection of objects (formally, a set is a collection of elements of a universal set 1 ). Definition 1.2 (Element) member of a set. We write x to mean x is an element of the set. Definition 1.3 (Subset) set is a subset of a set, denoted, provided every element of is also an element of. Notice that every set is a subset of the universal set. The notion of subset can be pictured as follows: Definition 1.4 (Union) The union of two (or more) sets is a set that contains all the elements of each set. For two sets and, the union of and is the set = {x x or x }. The union of two sets can be pictured as follows (the gray shaded region is the union of and ): Definition 1.5 (Intersection) The intersection of two (or more) sets is the set of all items in common to each set. If and are two sets, then the intersection of and is the set = {x x and x }. 1 Typically, we start by fixing a universal set that defines the domain of discourse. 2
3 The intersection can be pictured as follows (the gray shaded region is the intersection of and ): Definition 1.6 (Set Difference) The difference between two sets and ( minus ) is all elements in but not in. The difference between and is the set = {x x and x }. The differences between and can be pictured as follows: Definition 1.7 (Complement) The Complement of a set is the set of all elements not contained in that set. Formally, the complement of the set is C = {x x is in the universal set and x } Why is the notion of a universal set necessary for this definition? Exercise 1.8 Using Venn diagrams, convince yourself that for any sets and, = C Definition 1.9 (Symmetric Difference) The symmetric difference of two sets is all the elements in either set but not in both. The symmetric difference is the set ( ) ( ). The symmetric difference can be pictured as follows: 3
4 ( ) ( ) Definition 1.10 (Product) The Cartesian product of two or more sets is all possible pairings of each elements of the sets. If and are two sets, then the cartesian product of and is the set = {(x, y) x and y } Definition 1.11 (Null Set) The null or empty set is a set that contains no elements. We write to denote the set containing no elements. Definition 1.12 (Power Set) The power set is the set of all subsets. If is a set, the power set of is the set () = { }. Notice that the empty set is a subset of every set. Definition 1.13 (Cardinality of a Set) The cardinality of a finite 2 set is the total number of elements in, and is denoted. Definition 1.14 (Partition) partition of a set S is a collection of sets S = {S 1, S 2,...} (possibly infinite) such that the sets are pairwise disjoint: if S i, S j S with i j, then S i S j = their union is S, that is, S = Si SS i. 2 Relations Definition 2.1 (inary Relation) binary relation R on two sets and is a subset of the cross product of and, i.e., R 2 The notion of cardinality can be applied to infinite sets as well. However, a discussion of this is beyond the scope of these introductory notes. 4
5 You have already studied binary relations during your mathematical eduction: =,,, <, and > are all relations on numbers (eg., the natural numbers N, real numbers R, rational numbers Q, etc.) and is a relation on the power set of a set S. For example, the binary relation N N is the set {(a, b) a, b N and a is less than or equal to b} Suppose that R is a relation. We often write a R b to mean (a, b) R. The following are properties of relations that may be of interest: Reflexivity for all a, a R a Transitivity for all a, b, c, if a R b and b R c then a R c Completeness for all a, b, either a R b or b R a (or a = b) Symmetry for all a, b, if a R b then b R a ntisymmetric for all a, b, if a R b and b R a then a = b Definition 2.2 (Equivalence Relation) relation R that is reflexive, symmetric and transitive is said to be an equivalence relation Definition 2.3 (Equivalence Class) If R is an equivalence relation on, then for each a, the equivalence class of a, denoted by [a], is the following set [a] = {b arb}. Definition 2.4 (Partial Order) relation that is reflexive, antisymmetric and transitive is said to be a partial order. The standard example of a partial order is the relation. The following is our first theorem. It is somewhat technical, but illustrates a fundamental idea about equivalence classes and partitions. Namely, that every partition has an equivalence relation associated with it, and every equivalence class has a partition associated with it. Theorem 2.5 The equivalence classes of any equivalence relation R on a set forms a partition of, and any partition of determines an equivalence relation on for which the sets in the partition are the equivalence classes. Proof. Suppose R is an equivalence relation on. We must show that the equivalence classes of R forms a partition of. 5
6 1. Each equivalence class is nonempty, since a R a for all a. 2. Clearly is the union of all the equivalence classes (since each element of belongs to at least one equivalence class) 3. We must show any two equivalence classes are disjoint. Let [a], [b] be two distinct equivalence classes. Suppose c [a] [b]. Then a R c and b R c. Hence by symmetry, c R b. nd so by transitivity, a R b. Let x [a], then x R c and by the above argument x R b (Why?), and so x [b]. Thus [a] [b]. Using a similar argument, we can show [b] [a]. Therefore [a] = [b], which contradicts the fact that [a] and [b] are distinct equivalence classes. For the second part of the theorem, suppose = { 1,..., n } is any partition of. Define R = {(a, b) a i and b i }. We must show that R is reflexive. Let a be any element. Then a i for some i, and hence by definition of R, a R a. Next we will show that R is symmetric. Suppose a R b. Then a i and b i for some i. Then clearly, b i and a i and hence b R a. We must show R is transitive. Suppose, a R b and b R c. Then a i and b i, and b j and c j for some i, j. Since b i j, i = j (since the elements of are pairwise disjoint). Therefore, a i and c i and hence a R c. qed Functions are a special type of relation: Definition 2.6 (Function) function f is a binary relation on and (i.e., f ) such that for all a, there exists a unique b such that (a, b) f. We write f : when f is a function, and if (a, b) f, then write f(a) = b. Suppose f : is a function. codomain. is said to be the domain and the Definition 2.7 (Image) The image of a set is the set: f( ) = {b b = f(a) for some a } Definition 2.8 (Range) The range of a function is the image of its domain. Suppose that f : is a function. 6
7 Definition 2.9 (Surjection) f is a surjection (or onto) if its range is equal to its codomain. I.e., f is surjective iff for each b, there exists an a such that f(a) = b Definition 2.10 (Injection) f is an injection (or 11) if distinct elements of the domain produce distinct elements of the codomain. I.e., f is 11 iff a a implies f(a) f(a ), or equivalently f(a) = f(a ) implies a = a. Definition 2.11 (ijection) f is a bijection if it is injective and surjective. In this case, f is often called a onetoone correspondence. Definition 2.12 (Inverse Image) Suppose that f : and that Y. The inverse image of Y is the set f 1 (Y ) = {x x and f(x) Y } 3 Proofs 3.1 Introduction Learning how to write mathematical proofs takes time and lots of practice. proof of a mathematical statement is simply an explanation of that statement written in the language of mathematics. It is very important that you become comfortable with the definitions. If you don t know or understand the formal definitions, then you will not be able to write down your explanations. It would be like trying to explain something to someone in Italian without actually knowing Italian. 3.2 Proving Equality and Subset How do you prove that two sets are equal? The answer to this question depends on who you are trying to convince. In this class, we will always err on the side of caution and given fairly detailed formal proofs. In turns out that proving two sets are equal reduces to proving the sets are subsets of each other. Fact 3.1 = if and only if and Why is this true? Well, if and are equal, then they both name the same collection of objects. I.e., is another name for the collection of objects that names and vice versa. So, if and are equal then of course since is always a subset of itself and is simply another name for. Similarly, we can show. Conversely, suppose and. We want to know that 7
8 and name the same collection of objects. Suppose they didn t, then there should be some object x that is not in or some object y that is not in. Well, we know such an object x cannot exist since, and so, every element of is an element of. Similarly, the element y cannot exist. Hence, and must name the same collection of objects. What about trying to prove that two sets are not equal? This turns out to be easier. In order to show that does not equal, you need only find an element in that is not in OR an element of that is not in. Showing two sets are equal reduces to proving that the sets are subsets of each other. ut, how to show that a set is a subset of another set? The general procedure to show is to show that each element of is also and element of. This is straightforward if and are both finite sets. For example, suppose = {2, 3, 4} and = {1, 2, 3, 4, 5, 6}. How do we show that. Since is finite, we simply notice that 2, 3 and 4. What if is the set of even numbers and is the set of all integers? We would get awfully tired (and bored) if we waited around to show that each and every element of is also an element of. Imagine and are two boxes, and you would like to know whether all the elements in s box are also in s box. Suppose you reach in box and select an element, say the number 10. fter inspecting 10, you notice that 10 is in fact an integer and so must also be an element of box. ut you are not satisfied, since you cannot be sure that the next element you choose from will also be an element of. In fact, even if you have shown that the first million even integers are all members of box, you cannot be sure that the next element you select from box will in fact be an integer. Instead, you should consider the property that all elements of have in common and show that any object satisfying that property must be an element of. What property does x satisfy if it is contained in s box? The answer is x = 2 n, where n is some integer. Then, you simply notice that if n is an integer, then 2 n is also an integer; and hence, any element of must also be an element of. 3.3 Examples Theorem 3.2 = Proof. We must show and. 8
9 We will show. Suppose x. Then x or x. Suppose x then x. Then x (if x is not in then x is certainly not in both and ). Hence x. Suppose x. For similar reason, x. Hence in either case, x. Therefore,. We must show. Suppose x. Then x, and so x or 3 x. Hence either x or x. In either case, x. qed Theorem 3.3 iff =. Proof. We must show implies = and = implies. ssume that. We must show =. I.e. we must show (1) and (2). The first statement is trivial, it is always the case that. For the second statement, assume x. We must show x. Since, x. Hence x. ssume =. We must show. Let x. Then x since =. Hence x. qed ttempt to answer these questions before looking at the answers. Exercise 3.4 Suppose that f :. Prove or disprove the following: 1. If X and Y, then f(x Y ) = f(x) f(y ). 2. If X, Y and f is 11,then f(x Y ) = f(x) f(y ). 3. If X and Y, then f 1 (X Y ) = f 1 (X) f 1 (Y ). Claim 3.5 It not the case that if X and Y, then f(x Y ) f(x) f(y ). Proof of Claim 3.5. To prove this, we must find counterexample. Let = {1, 2, 3} and = {a, b, c}. nd f : be defined as follows: f(1) = c, f(2) = b and f(3) = c. Let X = {1, 2} and Y = {2, 3}. Then X Y = {2} and f(x Y ) = f({2}) = {f(2)} = {b}. ut, f(x) f(y ) = {f(1), f(2)} {f(2), f(3)} = {c, b} {b, c} = {b, c}. Hence, f(x Y ) f(x) f(y ). qed (of Claim) 3 Notice that x does not imply that x and x. The and in italics is should be an or. Make sure you clearly understand the logic here, since this is often misunderstood by students. 9
10 It is true that for any function f : and all subsets X, Y, f(x Y ) f(x) f(y ) (for a proof see below). Claim 3.6 If X, Y and f is 11,then f(x Y ) = f(x) f(y ). Proof of Claim 3.6. Suppose that f : is a 11 function. Let X and Y. We must show (1) f(x Y ) f(x) f(y ) and (2) f(x) f(y ) f(x Y ). Notice that (1) is true even if f is not 11. Let y f(x Y ). Then there is an element x X Y such that f(x) = y. Since x X Y, x X and x Y. Therefore, y = f(x) f(x) and y = f(x) f(y ). Hence, y f(x) f(y ). We now prove (2). Let y f(x) f(y ). Then y f(x) and y f(y ). Since y f(x) there is x 1 X such that f(x 1 ) = y. Since y f(y), there is x 2 Y such that f(x 2 ) = y. Since f is 11, x 1 = x 2. Therefore x 1 = x 2 X Y ; and so, y = f(x 1 ) = f(x 2 ) f(x Y ). qed (of Claim) Claim 3.7 If X and Y, then f 1 (X Y ) = f 1 (X) f 1 (Y ). Proof of Claim 3.7. Let f : be any function and suppose X and Y. We must show f 1 (X Y ) f 1 (X) f 1 (Y ) and f 1 (X) f 1 (Y ) f 1 (X Y ). Suppose x f 1 (X Y ). Then f(x) X Y. Hence f(x) X and f(x) Y. Since f(x) X, x f 1 (X). Since f(x) Y, x f 1 (Y ). Therefore x f 1 (X) f 1 (Y ). Suppose x f 1 (X) f 1 (Y ). Then x f 1 (X) and x f 1 (Y ). Therefore, f(x) X and f(x) Y. Hence, f(x) X Y ; and so, x f 1 (X Y ). asdfasdfasdfasdfa qed (of Claim) 10
SETS, 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 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 informationMath/CSE 1019: Discrete Mathematics for Computer Science Fall Suprakash Datta
Math/CSE 1019: Discrete Mathematics for Computer Science Fall 2011 Suprakash Datta datta@cse.yorku.ca Office: CSEB 3043 Phone: 4167362100 ext 77875 Course page: http://www.cse.yorku.ca/course/1019 1
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 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 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 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 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 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 informationPOWER SETS AND RELATIONS
POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty
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 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 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 informationIntroducing 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 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 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 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 informationIntroduction to Relations
CHAPTER 7 Introduction to Relations 1. Relations and Their Properties 1.1. Definition of a Relation. Definition: A binary relation from a set A to a set B is a subset R A B. If (a, b) R we say a is related
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 informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 20
CS 70 Discrete Mathematics and Probability Theory Fall 009 Satish Rao, David Tse Note 0 Infinity and Countability Consider a function (or mapping) f that maps elements of a set A (called the domain of
More informationReview for Final Exam
Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters
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 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 informationSets. A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object.
Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: real numbers, complex numbers, C integers, All students in
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 informationCmSc 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
More informationRELATIONS AND FUNCTIONS
Chapter 1 RELATIONS AND FUNCTIONS 1.1 Overview 1.1.1 Relation A relation R from a nonempty set A to a non empty set B is a subset of the Cartesian product A B. The set of all first elements of the ordered
More informationClicker 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] 1.11.7 Exercises: Do before next class; not to hand in [J] pp. 1214:
More informationRELATIONS AND FUNCTIONS
Chapter 1 RELATIONS AND FUNCTIONS There is no permanent place in the world for ugly mathematics.... It may be very hard to define mathematical beauty but that is just as true of beauty of any kind, we
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 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 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 informationIntroduction Russell s Paradox Basic Set Theory Operations on Sets. 6. Sets. Terence Sim
6. Sets Terence Sim 6.1. Introduction A set is a Many that allows itself to be thought of as a One. Georg Cantor Reading Section 6.1 6.3 of Epp. Section 3.1 3.4 of Campbell. Familiar concepts Sets can
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 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 informationSome Definitions about Sets
Some Definitions about Sets Definition: Two sets are equal if they contain the same elements. I.e., sets A and B are equal if x[x A x B]. Notation: A = B. Recall: Sets are unordered and we do not distinguish
More informationWhat 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
More informationMath 320 Course Notes. Chapter 7: Countable and Uncountable Sets
Math 320 Course Notes Magnhild Lien Chapter 7: Countable and Uncountable Sets Hilbert s Motel: Imagine a motel with in nitely many rooms numbered 1; 2; 3; 4 ; n; : One evening an "in nite" bus full with
More information2.3. Relations. Arrow diagrams. Venn diagrams and arrows can be used for representing
2.3. RELATIONS 32 2.3. Relations 2.3.1. Relations. Assume that we have a set of men M and a set of women W, some of whom are married. We want to express which men in M are married to which women in W.
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 informationApplications of Methods of Proof
CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The settheoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are
More informationMath 421, Homework #5 Solutions
Math 421, Homework #5 Solutions (1) (8.3.6) Suppose that E R n and C is a subset of E. (a) Prove that if E is closed, then C is relatively closed in E if and only if C is a closed set (as defined in Definition
More informationf(x) is a singleton set for all x A. If f is a function and f(x) = {y}, we normally write
Math 525 Chapter 1 Stuff If A and B are sets, then A B = {(x,y) x A, y B} denotes the product set. If S A B, then S is called a relation from A to B or a relation between A and B. If B = A, S A A is called
More informationSections 2.1, 2.2 and 2.4
SETS Sections 2.1, 2.2 and 2.4 Chapter Summary Sets The Language of Sets Set Operations Set Identities Introduction Sets are one of the basic building blocks for the types of objects considered in discrete
More informationChap2: The Real Number System (See Royden pp40)
Chap2: The Real Number System (See Royden pp40) 1 Open and Closed Sets of Real Numbers The simplest sets of real numbers are the intervals. We define the open interval (a, b) to be the set (a, b) = {x
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 informationLECTURE NOTES ON RELATIONS AND FUNCTIONS
LECTURE NOTES ON RELATIONS AND FUNCTIONS PETE L. CLARK Contents 1. Relations 1 1.1. The idea of a relation 1 1.2. The formal definition of a relation 2 1.3. Basic terminology and further examples 2 1.4.
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 informationLecture 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
More informationEquivalence relations
Equivalence relations A motivating example for equivalence relations is the problem of constructing the rational numbers. A rational number is the same thing as a fraction a/b, a, b Z and b 0, and hence
More informationMath 3000 Running Glossary
Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (
More informationDay 11. Wednesday June 6, Today, we begin by investigating properties of relations. We use a few toy relations as examples.
Day 11 Wednesday June 6, 2012 1 Relations Today, we begin by investigating properties of relations. We use a few toy relations as examples. 1.1 Examples of Relations There are a few relations that particularly
More informationNotes. Sets. Notes. Introduction II. Notes. Definition. Definition. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry.
Sets Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 1.6 1.7 of Rosen cse235@cse.unl.edu Introduction
More 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 informationThe Mathematics Driving License for Computer Science CS10410
The Mathematics Driving License for Computer Science CS10410 Venn Diagram, Union, Intersection, Difference, Complement, Disjoint, Subset and Power Set Nitin Naik Department of Computer Science VennEuler
More informationSolutions to InClass Problems Week 4, Mon.
Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics for Computer Science September 26 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised September 26, 2005, 1050 minutes Solutions
More 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 informationMATHEMATICS 117/E217, SPRING 2012 PROBABILITY AND RANDOM PROCESSES WITH ECONOMIC APPLICATIONS Module #5 (Borel Field and Measurable Functions )
MATHEMATICS 117/E217, SPRING 2012 PROBABILITY AND RANDOM PROCESSES WITH ECONOMIC APPLICATIONS Module #5 (Borel Field and Measurable Functions ) Last modified: May 1, 2012 Reading Dineen, pages 5767 Proof
More informationChapter Three. Functions. In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics.
Chapter Three Functions 3.1 INTRODUCTION In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics. Definition 3.1: Given sets X and Y, a function from X to
More informationSection 3.3 Equivalence Relations
1 Section 3.3 Purpose of Section To introduce the concept of an equivalence relation and show how it subdivides or partitions a set into distinct categories. Introduction Classifying objects and placing
More information4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.
Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,
More informationMathematics 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
More informationFinite and Infinite Sets
Chapter 9 Finite and Infinite Sets 9. Finite Sets Preview Activity (Equivalent Sets, Part ). Let A and B be sets and let f be a function from A to B..f W A! B/. Carefully complete each of the following
More informationAnnouncements. 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. 2.32.6 Homework 2 due Tuesday Recitation 3 on Friday
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 informationChapter 1. SigmaAlgebras. 1.1 Definition
Chapter 1 SigmaAlgebras 1.1 Definition Consider a set X. A σ algebra F of subsets of X is a collection F of subsets of X satisfying the following conditions: (a) F (b) if B F then its complement B c is
More information4.1. Sets. Introduction. Prerequisites. Learning Outcomes. Learning Style
ets 4.1 Introduction If we can identify a property which is common to several objects, it is often useful to group them together. uch a grouping is called a set. Engineers for example, may wish to study
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 informationChapter Prove or disprove: A (B C) = (A B) (A C). Ans: True, since
Chapter 2 1. Prove or disprove: A (B C) = (A B) (A C)., since A ( B C) = A B C = A ( B C) = ( A B) ( A C) = ( A B) ( A C). 2. Prove that A B= A B by giving a containment proof (that is, prove that the
More informationCHAPTER 3. Mapping Concepts and Mapping Problems for. Scalar Valued Functions of a Scalar Variable
A SERIES OF CLASS NOTES TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS REMEDIAL CLASS NOTES A COLLECTION OF HANDOUTS FOR REMEDIATION IN FUNDAMENTAL CONCEPTS
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
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 informationChapter 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
More informationIf f is a 11 correspondence between A and B then it has an inverse, and f 1 isa 11 correspondence between B and A.
Chapter 5 Cardinality of sets 51 11 Correspondences A 11 correspondence between sets A and B is another name for a function f : A B that is 11 and onto If f is a 11 correspondence between A and B,
More informationTOPOLOGICAL PROOFS OF THE EXTREME AND INTERMEDIATE VALUE THEOREMS. Contents
TOPOLOGICAL PROOFS OF THE EXTREME AND INTERMEDIATE VALUE THEOREMS JAMES MURPHY Abstract. In this paper, I will present some elementary definitions in Topology. In particular, I will explain topological
More informationWeek 5: Binary Relations
1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all
More informationProblems on Discrete Mathematics 1
Problems on Discrete Mathematics 1 ChungChih Li 2 Kishan Mehrotra 3 Syracuse University, New York L A TEX at January 11, 2007 (Part I) 1 No part of this book can be reproduced without permission from
More informationCourse 421: Algebraic Topology Section 1: Topological Spaces
Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............
More informationSolutions to Homework Problems from Chapter 3
Solutions to Homework Problems from Chapter 3 31 311 The following subsets of Z (with ordinary addition and multiplication satisfy all but one of the axioms for a ring In each case, which axiom fails (a
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 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 informationStructure of Measurable Sets
Structure of Measurable Sets In these notes we discuss the structure of Lebesgue measurable subsets of R from several different points of view. Along the way, we will see several alternative characterizations
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 informationMA651 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
More informationTOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS
TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms
More informationSets and Logic. Chapter Sets
Chapter 2 Sets and Logic This chapter introduces sets. In it we study the structure on subsets of a set, operations on subsets, the relations of inclusion and equality on sets, and the close connection
More informationBasic 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
More informationDiscrete Mathematics, Chapter 5: Induction and Recursion
Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Wellfounded
More informationMethoδos Primers, Vol. 1
Methoδos Primers, Vol. 1 The aim of the Methoδos Primers series is to make available concise introductions to topics in Methodology, Evaluation, Psychometrics, Statistics, Data Analysis at an affordable
More informationAutomata 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
More informationp 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)
.9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other
More informationOh Yeah? Well, Prove It.
Oh Yeah? Well, Prove It. MT 43A  Abstract Algebra Fall 009 A large part of mathematics consists of building up a theoretical framework that allows us to solve problems. This theoretical framework is built
More informationSection 3.2 Partial Order
1 Section 3.2 Purpose of Section To introduce a partial order relation (or ordering) on a set. We also see how order relations can be illustrated graphically by means of Hasse diagrams and directed graphs.
More informationRelations Graphical View
Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary
More informationREAL ANALYSIS I HOMEWORK 2
REAL ANALYSIS I HOMEWORK 2 CİHAN BAHRAN The questions are from Stein and Shakarchi s text, Chapter 1. 1. Prove that the Cantor set C constructed in the text is totally disconnected and perfect. In other
More informationChapter 4: Binary Operations and Relations
c Dr Oksana Shatalov, Fall 2014 1 Chapter 4: Binary Operations and Relations 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition, subtraction,
More informationDiscrete Mathematics. Thomas Goller. July 2013
Discrete Mathematics Thomas Goller July 2013 Contents 1 Mathematics 1 1.1 Axioms..................................... 1 1.2 Definitions................................... 2 1.3 Theorems...................................
More informationaxiomatic vs naïve set theory
ED40 Discrete Structures in Computer Science 1: Sets Jörn W. Janneck, Dept. of Computer Science, Lund University axiomatic vs naïve set theory s ZermeloFraenkel Set Theory w/choice (ZFC) extensionality
More informationProblem Set 1 Solutions Math 109
Problem Set 1 Solutions Math 109 Exercise 1.6 Show that a regular tetrahedron has a total of twentyfour symmetries if reflections and products of reflections are allowed. Identify a symmetry which is
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 information