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Lecture notes in Set Theory

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BBA120 Business MathematicsThe course aims to provide an introduction to mathematical concepts and lay down a foundation for applications of basic tools and techniques for various areas of business such as economics, accountancy and the life and social sciences. It begins with non calculations topics as Basic mathematics, equations, functions, matrix algebra, mathematics of finance etc. Then it progress through both single-variable and multi-variable calculus. An abundance and variety of applications appear throughout the course. Students continually see how the mathematics they are learning can be applied to practical business problems. These applications over such diverse areas as business, economics, sociology, finance etc. Moses Mwale. School of Business and Economics Department of Business Administration Email:moses.mwaleictar.ac.zm BBA120 Business Mathematics Lecture Notes 2013 Preface These lecture notes are for the course BBA120 “Business Mathematics” for first semester 2013 intake at the University of Lusaka. The author wishes to acknowledge that these lecture notes are collected from the references listed in Bibliography, and from many other sources the author has forgotten. The author claims no originality, and hopes not to be sued for plagiarizing or for violating the sacred copyright laws. Moses, August 8, 2013 BBA 120 Business Mathematics Contents Unit 1: Mathematical Preliminaries 3 1.1 Set Theory ................................................................................................................... 3 1.1.1 Sets and elements ........................................................................................... 3 1.1.2. Specification of sets ....................................................................................... 4 (a) List notation. ............................................................................................. 4 (b) Set builder notation. ................................................................................. 4 (c) Recursive rules. ........................................................................................ 4 1.1.3. Identity and cardinality .................................................................................. 5 1.1.4. Subsets ........................................................................................................... 5 1.1.5. Power sets ...................................................................................................... 5 1.1.6. Operations on sets: union, intersection. ......................................................... 5 Venn Diagrams ...................................................................................... 6 1.1.7 More operations on sets: difference, complement .......................................... 7 1.1.8 De Morgan's laws ........................................................................................... 8 1.1.9 Associative and Distributive laws of Set Operations ..................................... 9 Exercise 1.1 : Sets and Subsets ............................................................................... 9 Exercise 1.2 : Set Operations ................................................................................ 10 1.2.0 Common Number Sets ........................................................................................... 13 1.2.1 Intervals ........................................................................................................ 15 Inequalities ................................................................................................... 16 Interval Notation .......................................................................................... 16 Number Line ................................................................................................ 17 Open or Closed Intervals ............................................................................. 18 Intervals To Infinity (but not beyond) ........................................................ 18 Exercises 2.1 .......................................................................................................... 19 BBA 120 Business Mathematics Unit 1: Mathematical Preliminaries Many models and problems in modern economics and finance can be expressed using the language of mathematics and analyzed using mathematical techniques. The next few sections are a review of important material that you studied in high school algebra, trigonometry and pre-calculus courses ant that is essential prerequisite for studying calculus. The aim throughout this Unit is to show how a range of important mathematical techniques work and how they can be used to explore and understand the structure of economic models. 1.1 Set Theory Mathematics has its own terminologies and notation which is clear and concise and enables us to carry out calculations. Set theory is the milieu in which mathematics takes place today. 1.1.1 Sets and elements Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Description: a set is a collection of objects which are called the members Definition or elements of that set. If we have a set we say that some objects belong sets consist of their elements. A Set is a collection Examples: the set of students in this room; the English alphabet may be of objects viewed as the set of letters of the English language; the set of natural numbers1; etc. So sets can consist of elements of various natures: people, physical objects, numbers, signs, other sets, etc. (We will use the words object or entity in a very broad way to include all these different kinds of things.) The membership criteria for a set must in principle be well-defined, and not vague. Sets can be finite or infinite. There is exactly one set, the empty set, or null set, which has no members at all. A set with only one member is called a singleton or a singleton set. (“Singleton of a”) 3 4 Unit 1: Mathematical Preliminaries Notation: A, B, C, … for sets; a, b, c, … or x, y, z, … for members. b ∈ A if b belongs to A (B ∈ A if both A and B are sets and B is a member of A) and c ∉ A, if c doesn’t belong to A. ∅ is used for the empty set. 1.1.2. Specification of sets There are three main ways to specify a set: a) by listing all its members (list notation); b) by stating a property of its elements (Set builder notation); c) by defining a set of rules which generates (defines) its members (recursive rules). (a) List notation. The first way is suitable only for finite sets. In this case we list names of elements of a set, separate them by commas and enclose them in braces: Examples: 1, 12, 45, George Washington, Bill Clinton, a,b,d,m. Note that we do not care about the order of elements of the list, and elements can be listed several times. 1, 12, 45, 12, 1, 45,1 and 45,12, 45,1 are different representations of the same set (see below the notion of identity of sets). (b) Set builder notation. Example: xx is a natural number and x 8 Reading: “the set of all x such that x is a natural number and is less than 8” So the second part of this notation is a property the members of the set share (a condition or a predicate which holds for members of this set). Other examples: xx is a letter of Russian alphabet y y is a student of UMass and y is older than 25 General form: xP(x), where P is some predicate (condition, property). (c) Recursive rules. Example – the set E of even numbers greater than 3: a) 4 ∈ E b) if x ∈ E, then x + 2 ∈ E c) nothing else belongs to E. The first rule is the basis of recursion, the second one generates new elements from the elements defined before and the third rule restricts the defined set to the elements generated by rules a and b. (The third rule BBA 120 Business Mathematics should always be there; sometimes in practice it is left implicit. It’s best when you’re a beginner to make it explicit.) 1.1.3. Identity and cardinality Two sets are identical if and only if 2 they have exactly the same members. So A = B iff for every x, x ∈ A ⇔ x ∈ B. For example, 0,2,4 = x x is an even natural number less than 5 From the definition of identity follows that there exists only one empty set; its identity is fully determined by its absence of members. Note that empty list notation is not usually used for the empty set, we have a special symbol ∅ for it. The number of elements in a set A is called the cardinality of A, written n(A) or A. The cardinality of a finite set is a natural number. Infinite sets also have cardinalities but they are not natural numbers. 1.1.4. Subsets A set A is a subset of a set B iff every element of A is also an element of B. Such a relation between sets is denoted by A ⊆ B. If A ⊆ B and A ≠ B we call A a proper subset of B and write A ⊂ B. (Caution: sometimes ⊂ is used the way we are using ⊆.) Both signs can be negated using the slash / through the sign. Examples: a,b ⊆ d,a,b,e and a,b ⊂ d,a,b,e, a,b ⊆ a,b, but a,b ⊄ a,b. Note that the empty set is a subset of every set. ∅ ⊆ A for every set A. Why? Be careful about the difference between “member of” and “subset of”; 1.1.5. Power sets The set of all subsets of a set A is called the power set of A and denoted as ℘(A). For example, if A = a,b, ℘(A) = ∅, a, b, a,b. From the example above: a ∈ A; a ⊆ A; a ∈ ℘(A) ∅ ⊆ A; ∅ ∉ A; ∅ ∈ ℘(A); ∅ ⊆ ℘(A) 1.1.6. Operations on sets: union, intersection. We define several operations on sets. Let A and B be arbitrary sets. The union of A and B, written A ∪ B, is the set whose elements are just the elements of A or B or of both. In the predicate notation the definition is A ∪ B = x x ∈ A or x ∈ B 5 6 Unit 1: Mathematical Preliminaries Examples. Let K = a,b, L = c,d and M = b,d, then K ∪ L = a,b,c,d (K ∪ L) ∪ M = K ∪ (L ∪ M) = a,b,c,d K ∪ M = a,b,d K ∪ K = K L ∪ M = b,c,d K ∪ ∅ = ∅ ∪ K = K = a,b. Venn Diagrams There is a nice method for visually representing sets and set-theoretic operations, called Venn diagrams. Each set is drawn as a circle and its members represented by points within it. The diagrams for two arbitrarily chosen sets are represented as partially intersecting – the most general case – as in Figure 1.1 below. The region designated ‘1’ contains elements which are members of A but not of B; region 2, those members in B but not in A; and region 3, A 1 2 3 4 B members of both B and A. Points in region 4 outside the diagram represent elements in neither set. Figure 1.1 The Venn diagram for the union of A and B is shown in Figure 1.2. The results of operations in this and other diagrams are shown by shading areas. A B Figure 1.2 The intersection of A and B, written A ∩ B, is the set whose elements are just the elements of both A and B. In the predicate notation the definition is A ∩ B = x x ∈ A and x ∈ B Examples: K ∩ L = ∅ (K ∩ L) ∩ M = K ∩ (L ∩ M) = ∅ BBA 120 Business Mathematics K ∩ M = b K ∩ K = K L ∩ M = d K ∩ ∅ = ∅ ∩ K = ∅. The general case of intersection of arbitrary sets A and B is represented by the Venn diagram of Figure 1.3. The intersection of three arbitrary sets A,B and C is shown in the Venn diagram of Figure 1.4. A A B C B Figure 1.3 Figure 1.4 1.1.7 More operations on sets: difference, complement Another binary operation on arbitrary sets is the difference “A minus B”, written A – B, which ‘subtracts’ from A all elements which are in B. Also called relative complement: the complement of B relative to A. The set builder’s notation defines this operation as follows: A – B = x x ∈ A and x ∉ B Examples: (using the previous K, L, M) K – L = a,b K – K = ∅ K – M = a K – ∅ = K L – M = c ∅ – K = ∅. The Venn diagram for the set-theoretic difference is shown in Figure 1–5. Figure1.5: A B A – B is also called the relative complement of B relative to A. This operation is to be distinguished from the complement of a set A, written 7 8 Unit 1: Mathematical Preliminaries A’, which is the set consisting of everything not in A. In predicate notation A’ = x x ∉ A It is natural to ask, where do these objects come from which do not belong to A? In this case it is presupposed that there exists a universe of discourse and all other sets are subsets of this set. The universe of discourse is conventionally denoted by the symbol U. Then we have A’ =U – A The Venn diagram with a shaded section for the complement of A is shown in Figure1.6 A U Figure 1.6 1.1.8 De Morgan's laws In set theory, de Morgan's laws relate the three basic set operations to each other; the union, the intersection, and the complement. De Morgan's laws are named after the Indian-born British mathematician and logician Augustus De Morgan (1806-1871). If A and B are subsets of a set X , de Morgan's laws state that ′ ′ ′ ( ) 𝐴 ∪ 𝐵 = 𝐴 ∩ 𝐵 ′ ′ (𝐴 ∩ 𝐵 ) = 𝐴 ∪ 𝐵 ′ Here, ∪ denotes the union, ∩ denotes the intersection, and A’ denotes the set complement of A in X , i.e., A’=X-A . Above, de Morgan's laws are written for two sets. In this form, they are intuitively quite clear. For instance, the first claim states that an element that is not in A∪B is not in A and not in B . It also states that an elements not in A and not in B is not in A∪ B . BBA 120 Business Mathematics 1.1.9 Associative and Distributive laws of Set Operations Associative Law The associative laws establish the rules of taking unions and intersections of sets. They apply to all sets including the set of real numbers. 𝐴 𝑈 (𝐵 𝑈 𝐶 ) = (𝐴 𝑈 𝐵 ) 𝑈 𝐶 This law states that taking the union of a set to the union of two other sets is the same as taking the union of the original set and one of the other two sets, and then taking the union of the results with the last set. 𝐴 ∩ (𝐵 ∩ 𝐶 ) = (𝐴 ∩ 𝐵 ) ∩ 𝐶 This law states that taking the intersection of a set to the intersection of two other sets is the same as taking the intersection of the original set and one of the other two sets, and then taking the intersection of the results with the last set. Distributive Law The distributive laws also establish the rules of taking unions and intersections of sets. 𝐴 𝑈 (𝐵 ∩ 𝐶 ) = (𝐴 𝑈 𝐵 ) ∩ (𝐴 𝑈 𝐶 ) This law states that taking the union of a set to the intersection of two other sets is the same as taking the union of the original set and both the other two sets separately, and then taking the intersection of the results. 𝐴 ∩ (𝐵 𝑈 𝐶 ) = (𝐴 ∩ 𝐵 ) 𝑈 (𝐴 ∩ 𝐶 ) This law states that taking the intersection of a set to the union of two other sets is the same as taking the intersection of the original set and both the other two sets separately, and then taking the union of the results. Exercise 1.1 : Sets and Subsets 1. Let A=a, b, c, d, e. How many elements does A contain? 2. Let A = 2, 4, 5, 4. Which statement is correct? a) 5 is an element of A. b) 5 is an element of A. c) 4, 5 is an element of A. d) 5 is a subset of A. 3. Which of these sets is finite? 9 10 Unit 1: Mathematical Preliminaries a) x x is even b) x x 5 c) 1, 2, 3,... d) 1, 2, 3,...,999,1000 4. Which of these sets is not a null set? a) A = x 6x = 24 and 3x = 1 b) B = x x + 10 = 10 c) C = x x is a man older than 200 years d) D = x x x 5. Let S=1, 2, 3. How many subsets does S contain? 6. Let D E. Suppose a D and b E. Which of the following statements must be true? a) c D b) b D c) a E d) a D 7. Let A = x x is even, B = 1, 2, 3,..., 99, 100, C = 3, 5, 7, 9, D = 101, 102 and E = 101, 103, 105. Which of these sets can equal S if S A and S and B are disjoint? a) A b) B c) C d) D e) E S 8. Let S = a, b. How many elements does the power set 2 contain? S 9. Which set S does the power set 2 = , 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3 come from? a) 1,2,3 b) 1, 2, 3 c) 1, 2, 2, 3, 1, 3 d) 1, 2, 3 Exercise 1.2 : Set Operations 1. Let A = x, y, z, B = v, w, x. Which of the following statements is correct? a) A B = v, w, x, y, z b) A B = v, w, y, z c) A B = v, w, x, y BBA 120 Business Mathematics d) A B = x, w, x, y, z Set Theory 2. Let A = 1, 2, 3, ..., 8, 9 and B = 3, 5, 7, 9. Which of the following Applied to business statements is correct?. operations, set theory a) A B = 2, 4, 6 can assist in planning b) A B = 1, 2, 3, 4, 5, 6, 7, 8, 9 and operations. c) A B = 1, 2, 4, 6, 8 d) A B = 2, 4, 6, 8 Every element of business can be 3. Let C = 1, 2, 3, 4 and D = 1, 3, 5, 7, 9. How many elements does grouped into at least the set C D contain? one set such as accounting, How many elements does the set C D contain? management, operations, production and sales. 4. Let A = 2, 3, 4, B = 3 and C = x x is even. Which statement Within those sets are is correct? other sets. In a) C A = B operations, for b) C B = A example, there are sets c) A C of warehouse operations, sales d) C / A = B (Alternate notation for C-A) operations and administrative 5. Let A B, B C and D A = C. Which statement is always false? operations. a) B D b) A C In some cases, sets c) A = B intersect as sales d) B D = and B A operations can intersect the operations set and the sales set 6. What is shaded in the Venn diagram below?. a) A B 11 12 Unit 1: Mathematical Preliminaries b) A B c) A d) B 7. What is shaded in the Venn diagram below?. a) A B b) A' c) A - B d) B - A 8. Let U = 1, 2, 3, ..., 8, 9 and A = 1, 3, 5, 7. Find A'. a) A' = 2, 4, 6, 8 b) A' = 2, 4, 6, 8, 9 c) A' = 2, 4, 6 d) A' = 9 9. Let U = 1, 2, 3,..., 8, 9, B = 1, 3, 5, 7 and C = 2, 3, 4, 5, 6. How many elements does the set (B C)' contain? How many elements does the set (C - B)' contain? BBA 120 Business Mathematics 1.2.0 Common Number Sets There are sets of numbers that are used so often that they have special names and symbols: Symbol Description Natural Numbers The whole numbers from 1 upwards. (Or from 0 upwards in some fields of mathematics). The set is 1,2,3,... or 0,1,2,3,... Integers The whole numbers, 1,2,3,... negative whole numbers ..., -3,-2,-1 and zero 0. So the set is ..., -3, -2, -1, 0, 1, 2, 3, ... (Z is for the German "Zahlen", meaning numbers, because I is used for the set of imaginary numbers). Rational Numbers A rational number is any number that can be written as a ratio of two integers. The set of rational numbers contains the set of integers since any integer can be written as a fraction with a denominator of 1. A rational number can have several different fractional representations. For example, 1/2 is equivalent to 2/4 or 132/264. In decimal representation, rational numbers take the form of repeating decimals. Some examples of rational numbers are: Irrational Numbers Any number that is not a Rational Number. These are numbers that can be written as decimals, but not as fractions. They are non-repeating, non- terminating decimals. Some examples of irrational numbers are: 13 14 Unit 1: Mathematical Preliminaries Algebraic Numbers Any number that is a solution to a polynomial equation with rational coefficients. Includes all Rational Numbers, and some Irrational Numbers. Transcendental Numbers Any number that is not an Algebraic Number Examples of transcendental numbers include π and e. Real Numbers All Rational and Irrational numbers. They can also be positive, negative or zero. They include the Algebraic Numbers and Transcendental Numbers. A simple way to think about the Real Numbers is: any point anywhere on the number line (not just the whole numbers). Examples: 1.5, -12.3, 99, √2, π They are called "Real" numbers because they are not Imaginary Numbers. Imaginary Numbers Numbers that when squared give a negative result. If you square a real number you always get a positive, or zero, result. For example 2×2=4, and (-2)×(-2)=4 also, so "imaginary" numbers can seem impossible, but they are still useful Examples: √(-9) (=3i), 6i, -5.2i The "unit" imaginary numbers is √(-1) (the square root of minus one), and its symbol is i, or sometimes j. 2 i = -1 Complex Numbers A combination of a real and an imaginary number in the form a + bi, where a and b are real, and i is imaginary. The values a and b can be zero, so the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers. Examples: 1 + i, 2 - 6i, -5.2i, 4 BBA 120 Business Mathematics Together, the rational numbers and the irrational numbers form the set of Real Numbers. Figure 2.1 The real number line Intervals 1.2.1 Intervals An Interval is all the real numbers between two given numbers. For example, all the numbers between 1 and 6 is an interval, i.e. the set of all numbers x satisfying 1 ≤ x ≤ 6 is an interval which contains 1 and 6, as well as all numbers between them. The interval 2 to 4 includes numbers such as: 15 16 Unit 1: Mathematical Preliminaries 7 2.1 2.1111 2.5 2.75 2.80001 π / 3.7937 2 And lots more Note that the boundary numbers may or may not be included in the interval. It isn't really clear. Example: "An economy class ticket allows baggage of up to 20 kg in mass" If your bag is exactly 20 kg ... will that be allowed or not? I will show you how to be precise about this in each of three popular methods:  Inequalities  The Number Line  Interval Notation Inequalities In mathematics, an inequality is a relation that holds between two values when they are different Real numbers can be compared in size.  The notation a b means that a is less than b.  The notation a b means that a is greater than b. In either case, a is not equal to b. These relations are known as strict inequalities. The notation a b may also be read as "a is strictly less than b". In contrast to strict inequalities, there are two types of inequality relations that are not strict:  The notation a ≤ b means that a is less than or equal to b (or, equivalently, not greater than b, or at most b).  The notation a ≥ b means that a is greater than or equal to b (or, equivalently, not less than b, or at least b) Example: x≤ 20 means ‘up to and including 20’ Interval Notation In "Interval Notation" you just write the beginning and ending numbers of the interval, and use:  a square bracket if you want to include the end value, or  ( ) a round bracket if you don't BBA 120 Business Mathematics Like this: For example: (5, 12 Means from 5 to 12, do not include 5, but do include 12 Number Line With the Number Line you draw a thick line to show the values you are including, and:  a filled-in circle if you want to include the end value, or  an open circle if you don't Like this: Example: means all the numbers between 0 and 20, do not include 0, but do include 20 Here is a handy table showing you all 3 methods (the interval is 1 to 2): From 1 To 2 Not Including Including 1 Not Including 1 Including 2 2 x ≥ 1 x 1 x 2 x ≤ 2 Inequality: "greater than "greater than" "less than" "less than or equal to" or equal to" 17 18 Unit 1: Mathematical Preliminaries Number line: Interval notation: 1 (1 2) 2 Example 2: "Competitors must be between 14 and 18" So 14 is included, and "being 18" goes all the way up to (but not including) 19. As an inequality it looks like this: 14 ≤ Age 19 On the number line it looks like this: And using interval notation it is simply: 14, 19) Open or Closed Intervals The terms "Open" and "Closed" are sometimes used when the end value is included or not: (a, b) a x b an open interval a, b) a ≤ x b closed on left, open on right (a, b a x ≤ b open on left, closed on right a, b a ≤ x ≤ b a closed interval These are intervals of finite length. We also have intervals of infinite length. Intervals To Infinity (but not beyond) We often use Infinity in interval notation. Infinity is not a real number, in this case it just means "continuing on ..." Example: x greater than, or equal to, 3: 3, +∞) BBA 120 Business Mathematics Note that we use the round bracket with infinity, because we don't reach it There are 4 possible "infinite ends": Interval Inequality (a, +∞) x a "greater than a" a, +∞) x ≥ a "greater than or equal to a" (-∞, a) x a "less than a" (-∞, a x ≤ a "less than or equal to a" We could even show no limits by using this notation: (-∞, +∞) Exercises 2.1 I. Write the inequality 4 ≤ x 9 in interval notation II. Write the inequality 4 ≥ x -3 in interval notation III. What inequality is defined in interval notation by (-∞, -2) ∪ 3, +∞)? IV. What inequality is defined in interval notation by (-∞, 3 ∩ 1, +∞)? V. The interval shown on the number line can be expressed as which inequality: a. x ≤ -5 or x 4 b. x ≤ -5 and x 4 c. -5 ≤ x 4 d. -5 x ≤ 4 VI. 19 20 Unit 1: Mathematical Preliminaries Which of the following describes in interval notation the inequality shown in the diagram? a. (-∞, -2) ∪ 6, ∞) b. (-∞, -2 ∪ (6, ∞) c. (-∞, -2) ∩ 6, ∞) d. (-∞, -2 ∩ (6, ∞) VII. Determine the Truth of each statement. If the statement is false, give a reason why that is so 1. -13 is an integer −2 2. is rational 7 3. −3 is a positive integer 4. 0 is not rational 5. √3 is rational 6. √25 is not a positive integer 7 7. is a rational number 0 8. 2 is a real number √ 0 9. is rational 0 10. 𝜋 is a positive integer 11. -3 is to the right of -4 on the real- number line. 12. Every integer is positive or negative 13. 0 is a natural number
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