Lecture 2 Set Theory PDF

NOTES PREPARED BY : CPA CHARLES KATUA KITHANDI [email protected] LECTURE TWO: SET THEORY 2.1. INTRODUCTION This lecture offers foundational knowledge in mathematical logic. Set theory is the branch of mathematics that deals with formal propertie...

NOTES PREPARED BY : CPA CHARLES KATUA KITHANDI [email protected] LECTURE TWO: SET THEORY 2.1. INTRODUCTION This lecture offers foundational knowledge in mathematical logic. Set theory is the branch of mathematics that deals with formal properties of well-defined collection of objects under study 2.2. LECTURE OBJECTIVES By the end of this lecture the learner should be: 1. Understand the definition and types of sets 2. Understand set operations and analysis 3. Methods of representation 4. Evaluate Venn diagram and apply in set analysis NOTES PREPARED BY : CPA CHARLES KATUA KITHANDI [email protected] 2.3. DEFINITION OF SET A set is a collection of distinct objects. For example, we can consider all the oceans in the world to be a set. If A is a set; A = {4,6,8,13} , the objects in the set, the integers 4,6,8,and 13 are referred to as the members or elements of the set. The elements can be listed in any order. The notation  is used to indicate membership of a set. For example, in the set A we can write 4  A or 4  {4,6,8,13} A set S is said to be a subset of another set T if every element in S is a member of T. a subset is denoted by . If B = {4,8} then since the integers 4 and 8 are members of set A we can write B  A or {4,8}  {4,6,8,13} Other definitions 1. Finite and infinite set A set can be classified as finite set or infinite set depending on the number of elements that it has. a finite set has a finite number of elements while an infinite set has an infinite number of elements. If P and S are sets: P = {2,4,6,....20}; S = {1,3,5,............} P is a finite set while S is an infinite set. 2. Universal set A universal set refers to a set that contains all the elements that an analyst wishes to study. if a researcher wishes to study the fresh water lakes in a country X then the universal set will be all the fresh water lakes in that country. The various classifications of the fresh water lakes in the country NOTES PREPARED BY : CPA CHARLES KATUA KITHANDI [email protected] would be sub sets of the universal set. The notation U is generally used to denote universal sets. Given a universal set we can derive its subsets. for example if U = {1,2,3} the subsets are { }, {1}, {2}, {3}, {1,2}, {1,3}, {2,3} and {1,2,3}. 3. The Null set It is a set that contains no element hence is also referred to as the empty set. It is designated by a Greek letter  or { }.Note that the sets { } and {0} are not the same. The former has no element in it while the latter has one element in it. This means that both the Null set, {}, and the Universal set, U, are subsets of U 4. Equal sets Two sets A and B are said to be equal if every member of a set A belongs to B and every element of set B belongs to A. That is the two sets contain the same elements. For example if A = {a, d. c. b} and B = {d, c a, b} are equal sets. 2.4. METHODS OF SET REPRESENTATION Sets are represented using capital letters such as A, B, C…… The objects in a set are said to be members or elements of a set. For instance A= {the set of all prime numbers≤ 5} has the following members {2, 3, 5}. It’s read as: A is the set of all prime numbers less than or equal to 5. The sign ∈ is used to show that an element is a member of a certain set. For example one may write that 2∈A. Read as 2 is a member/ an element in set A. The sign ∉ shows that an object is not an element of a certain set. For instance one may write 4∉A, read as 4 is not an element of set A. There are two different methods of representing members of a set. i) Descriptive method ii) Enumerative method NOTES PREPARED BY : CPA CHARLES KATUA KITHANDI [email protected] In the descriptive method, members of a set are presented in such a way that one can determine the elements of the set without difficulty. For example, P = x x = 0,1,2,...,7 or P = x 0  x  7 The enumerative method requires that one writes all the members of the set within the curly brackets. For example the set P could be written as follows P = {0,1,2,3,4,5,6,7} Example 1: Given that B= {1, 2, 3, 4, 5} list all (enumerate) the members of the following sets a) The set of all x2 such that x∈B b) The set of all 4x +1 such that x∈B Solution a) C={1, 4, 9, 16, 25} b) D={5, 9, 13, 17, 21} Using description the sets in example 1 are written as follows: a) C={x2 : x∈B} read as C is the set of all x2 such that x∈B. b) D={4x+1 : x∈B } Subset of a set, Consider the following sets B={x: 1≤ x≤5} and E={x: 1≤x≤10}. Using enumeration set B can be written as B= {1, 2, 3, 4, 5} whereas E can be written as E= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Notice that all the elements of set B are also contained in set E; we say that Set B is a subset of set E written as B⊂E. In general A⊂B if all the elements of Aare contained in B. The sign ⊄ shows that a set is not a subset of the other. 2.5. VENN DIAGRAMS AND SET OPERATION The Venn diagram provides a simple way of representing sets and relations between sets. It consists of a rectangle that represents the universal set. Subsets of the universal set are represented by circles drawn within the rectangle, or the universe. Consider the Venn diagram below NOTES PREPARED BY : CPA CHARLES KATUA KITHANDI [email protected] 1. Universal Set E.g. A universal set U with subsets A, B and C. U B A C 1. Complement of a set The complement of Set A denoted by A′ is the subset of U containing elements that do not belong to A. That is all the elements in the universal set but not in A. Example 4, Let U = {1, 2, 3, 4, 5….20} and A = {3, 4, 8, 12, 20}. Find A′. Solution A′ = { 1,2,5,6,7,9,10,11,13,14,15,16,17,18,19} i.e. the set of all the elements in the universal set but not in A. using Venn diagrams it can represented as shown in figure 2.2 follows: U A A′ A′ is the shaded region. 2. Intersection of Sets The intersection of Sets A and B denoted by ∩ is the set containing elements which belong to both A and B. Example 5; Let A= {3, 5, 7} and B= {2, 3, 4, 7, 8}. Find A∩B. Solution: A∩B={X: X∈ A and X∈ B} = {3, 7}. That is X is both in A and B. using Venn diagrams it can be represented as in figure 2.3 A B NOTES PREPARED BY : CPA CHARLES KATUA KITHANDI [email protected] A∩B is the shaded area. Example 6; given the following sets: T = {0, 2, 4, 6, 8, 10, 12} C = {4, 8} D = {10, 2, 0} E = {0} Find: i. D∩ E ii. C∩D∩E Solution: i. D∩ E = {10, 2, 0} ∩ {0} = {0} T E D ii. C ∩ D ∩ E = { 4,8} ∩ { 10,2,0} ∩ {0} = {} = Ø 3. Mutually exclusive or disjoint sets Two sets A and B are said to be disjoint or mutually exclusive if they have no elements in common. The intersection of any two mutually exclusive sets is empty. Example 6: Given the following sets M={2, 4, 6} and N={1, 3, 5}. Find i) M ∩N ii) Based on your answer in (i) above what can you conclude about sets M and N. iii) Suppose that 2∈N, would you be willing to revise your answer in (ii) above? Solution i) M ∩N={}= Ø since there are no elements common to both sets ii) Sets M and N are disjoint or mutually exclusive sets iii) Yes; sets M and N would end up with a common element 2 implying that M∩N= {2} meaning that the sets would seize to be disjoint 4. The union of sets NOTES PREPARED BY : CPA CHARLES KATUA KITHANDI [email protected] The union of two sets A and B i.e. A U B is the set of elements that belong to either A or B. It’s denoted as A∪ B Example 7; Given that A = {1, 3, 5}, B = {3, 5, 7, 9}. Find A∪B. Solution A∪B={x: x∈A or x∈B} = {1, 3, 5, 7, 9}. That is x ix in set A or B. 2.6. LAWS OF SET ALGEBRA 1. Commutative Law (of unions and intersections) Given a universal set U and subsets A, C and D as shown below: A U B= B U A A ∩ B = B ∩ A that is the order does not matter. 2. Associative Law It applies in case of three sets. A U (BUC) = (A U B) U C A ∩ (B∩ C) = (A∩ B) ∩ C 3. Distributive Law It applies when unions and intersections are used in combination A U (B∩C) = (A U B) ∩ (AUC) A ∩ (BU C) = (A∩ B) U (A ∩C) that is we are opening the brackets. 4. Further Laws of set algebra Given a universal set U and subsets A, C and D as shown below: U A D U AU=A NOTES PREPARED BY : CPA CHARLES KATUA KITHANDI [email protected] A=U AUA=A A∩A=A A∩U=A A U A′ = U A ∩ A′ =  CUD=DUC C∩D=D∩C 2.7. SUMMARY Set theory is important in capturing logical relationship between collections of distinct well defined objects. Venn diagrams are important in representing sets pictorially, and to show relationships and logical relationships between sets. NOTE The complement of a universal set is the null set. A set with a single element zero {0} is not a null or empty set  NOTES PREPARED BY : CPA CHARLES KATUA KITHANDI [email protected] 2.8. ACTIVITIES Activity Attempt exercises on the following website http://www.mathgoodies.com/lessons/sets/challenge_unit15.html 2.9. FURTHER READING References Anthony, M. & N. Biggs (1996), Mathematics for Economics and Finance. Cambridge University Press. Jacques, I. (1999), Mathematics for Economics and Business, 3rd ed. Addison Wesley Longman. Mukras M.S. (1986). Elements of Mathematical economics, Kenya Literature bureau. Pemberton, M. & N. Rau (2001), Mathematics for Economists. Manchester University Press. Simon, C. P. & L. Blume (1994), Mathematics for Economists. W. W. Norton & Co.