BBA120 Business Mathematics - University of Lusaka · introduction to mathematical ... areas as...

Lect

ure

No

tes

20

13

BB

A1

20

Bu

sin

ess

Mat

hem

atic

s

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:[email protected]

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


3

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

or elements of that set. If we have a set we say that some objects belong

sets consist of their elements.

Examples: the set of students in this room; the English alphabet may be

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”)

Definition

A Set is a collection

of objects

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:

{x|x 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:

{ x|x is a letter of Russian alphabet}

{y |y is a student of UMass and y is older than 25}

General form:

{ x|P(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


5

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}


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,

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.

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) = ∅

A

1 23

B 4

A

B


7

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.

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 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

A

B

A

B C

A

B


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

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 .

A

U


9

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?


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

8. Let S = {a, b}. How many elements does the power set 2S contain?

9. Which set S does the power set 2S = { , {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}


11

d) A B = {x, w, x, y, z}

2. Let A = {1, 2, 3, ..., 8, 9} and B = {3, 5, 7, 9}. Which of the following

statements is correct?.

a) A B = {2, 4, 6}

b) A B = {1, 2, 3, 4, 5, 6, 7, 8, 9}

c) A B = {1, 2, 4, 6, 8}

d) A B = {2, 4, 6, 8}

3. Let C = {1, 2, 3, 4} and D = {1, 3, 5, 7, 9}. How many elements does

the set C D contain?

How many elements does the set C D contain?

4. Let A = {2, 3, 4}, B = {3} and C = {x | x is even}. Which statement

is correct?

a) C A = B

b) C B = A

c) A C

d) C / A = B (Alternate notation for C-A)

5. Let A B, B C and D A = C. Which statement is always false?

a) B D

b) A C

c) A = B

d) B D = and B A

6. What is shaded in the Venn diagram below?.

a) A B

Set Theory Applied to business

operations, set theory

can assist in planning

and operations.

Every element of

business can be

grouped into at least

one set such as

accounting,

management,

operations, production

and sales.

Within those sets are

other sets. In

operations, for

example, there are sets

of warehouse

operations, sales

operations and

administrative

operations.

In some cases, sets

intersect -- as sales

operations can

intersect the

operations set and the

sales set


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?


13

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:


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.

i2 = -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


15

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:


2.1 2.1111 2.5 2.75 2.80001 π 7/2 3.7937

And lots more!

Note that the boundary numbers may or may not be included in the

interval.

It isn't really clear.

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

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?


17

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

Including 1 Not Including 1 Not Including

2 Including 2

Inequality:

x ≥ 1

"greater than

or equal to"

x > 1

"greater than"

x < 2

"less than"

x ≤ 2

"less than

or equal to"


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, +∞)


19

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.


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

7 is rational

3. −3 is a positive integer

4. 0 is not rational

5. √3 is rational

6. √25 is not a positive integer

7. 7

0 is a rational number

8. √2 is a real number

9. 0

0 is rational

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

BBA120 Business Mathematics - University of Lusaka · introduction to mathematical ... areas as...

Documents

Transcript of BBA120 Business Mathematics - University of Lusaka · introduction to mathematical ... areas as...