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Contents
About the Author vi
Preface vii
- 1 ELEMENTARY SET THEORY Goals of a Basic Mathematics Course ix
- 1.1 Introduction
- 1.2 Rudiments of Set Theory
- 1.2.1 Notation and Terminology
- 1.2.2 Fundamental Operations on Sets
- 1.3 Laws of Set Theory
- 1.3.1 Venn Diagrams
- 1.3.2 Elements Argument Method
- 1.4 Fundamental Counting Principle
- 1.4.1 Counting and Venn Diagrams
- 1.5 Real Number Systems
- 1.5.1 Some Useful subsets of Real Numbers
- 1.5.2 Intervals
- 1.6 Application of Laws of Set Theory
- 1.7 Exercises
- 1.8 Complex Numbers
- 1.8.1 Arithmetic Operations of Complex Numbers
- 1.8.2 The Complex Plane or The Argand Diagram or The Gauss Plane
- 1.8.3 Conjugates, Absolute Values and Arguments of Complex Numbers
- 1.8.4 Polar Form of a Complex Number
- 1.8.5 De Movrie’s Theorem
- of Complex Numbers 1.8.6 Application of Polar Form in Computing Products and Quotients
- 1.9 Exercises
- 2 ELEMENTARY LOGIC
- 2.1 Introduction
- 2.2 Mathematical Reasoning and Creativity
- 2.2.1 Inductive and Deductive Reasoning
- 2.3 propositional Logic
- 2.3.1 Propositions and Truth Values
- 2.3.2 Logical Connectives and Truth Tables
- 2.3.3 Tautologies and Contradictions
- 2.3.4 Logical Equivalence and Logical Implication
- 2.3.5 The Algebra of Logical Equivalence of Propositions
- a Conditional Proposition 2.3.6 Relationship between Converse, Inverse and the Contrapositive of
- 2.4 Predicate Calculus
- 2.4.1 Universal Quantifier
- 2.4.2 Existential Quantifier
- 2.4.3 Negation of a Quantified Statement
- 2.5 Application of Logic In Mathematical Proof
- 2.5.1 Proof by a Counter-example
- 2.5.2 Direct Proof
- 2.5.3 Proof by Cases
- 2.5.4 Proof by the Contrapositive
- 2.5.5 Proof by Contradiction
- 2.5.6 Proof by Mathematical Induction
- 2.6 Applications of Logic
- 2.7 Exercises
- 3 PERMUTATIONS AND COMBINATIONS
- 3.1 Introduction
- 3.2 Basic Counting Principle
- 3.3 Permutations
- 3.3.1 General Formula for P (n, r)
- 3.3.2 Permutations of Repeated Objects
- 3.4 Combinations
- 3.4.1 Comparing Combinations and Permutations
- 3.4.2 Combinations with Repetition
- 3.5 Problems involving Both Permutations and Combinations
- 3.6 Applications of Combinations
- 3.7 Exercises
- 4 RELATIONS AND FUNCTIONS
- 4.1 Introduction
- 4.2 Cartesian Products and Relations
- 4.3 Properties of Relations
- 4.4 Combining Relations
- 4.5 Functions
- 4.6 Types of Functions
- 4.6.1 One-to-one Functions and Many-to-one Functions
- 4.6.2 Onto Functions
- 4.6.3 Bijective Functions
- 4.7 Composition and Inverses of Functions
- 4.7.1 Composition of Functions
- 4.7.2 Inverse of a Function
- 4.7.3 The Vertical Line Test for a Function
- 4.8 Some Special Real Valued Functions
- 4.9 Some Classification of Real Valued Functions
- 4.9.1 Even Functions
- 4.9.2 Odd Functions
- 4.9.3 Functions which are Neither Even nor Odd
- 4.9.4 Algebraic Functions
- 4.9.5 Irrational Functions and Rational Functions
- 4.10 Solved Problems
- 4.11 Exercises
- 5 TRIGONOMETRY
- 5.1 Radian and Degree Measure of an Angle
- 5.2 Trigonometric Ratios
- 5.3 Trigonometric Identities
- 5.3.1 Cofunction Identities
- 5.3.2 Pythagorean Identities
- 5.3.3 Sum and Difference Identities
- 5.3.4 Double-Angle Identities
- 5.3.5 Half-Angle Identities
- 5.4 Proving Identities
- 5.5 Solving Trigonometric Equations
- 5.6 Exercises
- Bibliography
About the Author
B.M. Nzimbi is a Lecturer in Pure Mathematics at the University of Nairobi. Dr. Nzimbi received his B.sc in Mathematics and Computer Science from the University of Nairobi (1995), Msc(Pure Mathematics) from University of Nairobi (1999), Msc(Mathematics) from Syracuse University (New York, USA) (2004), and his Ph.D in Pure Mathematics from University of Nairobi (2009), where he wrote his thesis in the area of Operator Theory under the direction of Prof. J.M. Khalagai. Before joining the University of Nairobi, he held a position at Catholic University of Eastern Africa (CUEA), where he was a part-time lecturer. Dr. Nzimbi has authored, co-authored and published numerous articles in professional journals in the areas of Operator Theory and differential geometry. He is the author of the textbooks ”Linear Algebra I ” and ”Linear Algebra II ”, which are extensively used in the ODL programme at the University of Nairobi.
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other descriptive material, followed by several exercises of varying difficulty. I finally wish to record my appreciation to my former students for their invaluable sug- gestions and critical review of the manuscript that made the writing of this book easy.
Bernard Mutuku Nzimbi Nairobi, 2011
viii
Goals of a Basic Mathematics Course
A Basic Mathematics course will enable students to learn a particular set of mathemat- ical facts and how to apply them and how to think mathematically. To achieve this goal, this text stresses set theory and mathematical reasoning and the different ways problems are solved.
Special Features of this Book
Accessibility: There are no mathematical prerequisites beyond high school algebra for this text. Each chapter begins at an easily understood and accessible level. Once basic mathematical concepts have been developed, more difficult material and applica- tions are presented.
Accessibility: This text has been carefully designed for flexible use. The depen- dence of chapters has been minimized. Each chapter is divided into sections and each section is divided into subsections that form natural blocks of material for teaching. Instructors can easily pace their lectures using these blocks.
Writing Style: The writing style of this book is direct and pragmatic. Precise mathematical language is used without excessive formalism and abstraction. Notations are introduced and used when appropriate. Care has been taken to balance the mix of notation and words in mathematical statements.
Mathematical Rigour and Precision: All definitions and theorems in this book are stated extremely carefully so that students will appreciate the precision of language and rigour need in mathematics. Proofs are motivated and developed and their steps are carefully justified.
Figures and Tables: Figures and tables in this book are carefully presented and illustrated.
Exercises: There is an ample supply of exercises in this book that develop basic
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Chapter 1
ELEMENTARY SET THEORY
1.1 Introduction
Set theory is a natural choice of a field where students can first become acquainted with an axiomatic development of a mathematical discipline. The central concept in this chapter revolves around a set, which is simply a collection, group, conglomerate, aggregate of objects. All fundamental tools of elementary set theory as needed in mathematics and elsewhere in the sciences and social sciences are included in detailed exposition in this chapter.
Objectives At the end of this lecture, you should be able to:
- Define a set.
- Carry out some set operations.
- Apply set theory in solving some practical problems.
1.2 Rudiments of Set Theory
Definition 1.1 A set is any well-defined collection, group, aggregate, class or conglomerate of objects.
These objects (which may be cities, years, numbers, letters, or anything else ) are called
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elements of the set, and are often said to be members of the set. A set is often specified by ⊙ listing its elements inside a pair of braces or curly brackets or parentheses ⊙ means of a property of its elements.
Example 1.
The set whose elements are the first six letters of the alphabet is written
{a, b, c, d, e, f }
Example 1.
The set whose elements are the even integers between 1 and 11 is written
{ 2 , 4 , 6 , 8 , 10 }
We can also specify a set by giving a description of its elements (without actually listing the elements).
Example 1.
The set {a, b, c, d, e, f } can also be written
{T he f irst six letters of the alphabet}
Example 1.
The set { 2 , 4 , 6 , 8 , 10 } can also be written
{all even integers between 1 and 11 }
1.2.1 Notation and Terminology
For convenience, we usually denote sets by capital letters of the alphabet A, B, C, and so on. We use lowercase letters of the alphabet to represent elements of a set. For a set A, we write x ∈ A if x is a member of A or belongs to A. We write x ̸∈ A to mean that x is not a member of A or does not belong to A.
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Definition 1.5 If A ⊆ B and A ̸= B, we say that A is a proper subset of B, or A is properly contained in B, and write A ⊂ B.
We also write B ⊇ A instead of A ⊆ B and B ⊃ A instead of A ⊂ B.
Remark 1.
Note that since the empty set ∅ has no elements, every element in ∅ is also in any given set A. Hence ∅ ⊆ A. By the definition of subset, every set is a subset of itself. That is, for any set A we have A ⊆ A.
Lemma 1.2 (Uniqueness of the Empty Set). There exists only one set with no elements.
Proof. Assume A and B are sets with no elements. Then every element of A is an element of B (since A has no elements). Similarly, every element of B is an element of A (since B has no elements). Therefore, A = B, by the Principle of Extensionality.
Definition 1.6 (Cardinality of a Set). The number of elements in a set A is called the cardinality of A, and is denoted n(A) or |A|.
Note that cardinality of a set is always a non-negative integer or infinity. A set with one element is called a singleton set. A set A is said to be finite if n(A) < ∞. A set A is said to be infinite if n(A) = ∞. Note that n(∅) = 0.
Definition 1.7 (Universal Set) A Universal set U is a set which contains all el- ements under consideration. It is also called the universe of discourse or simply universe.
Example 1.
(a). If one considers the set of men and women, then a universal set is probably the set of human beings. (b). If one considers sets such as pigs, cows, chickens, or horses, the universal set is probably the set of animals.
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(c). If A = { 1 , 2 , 5 } and B = { 4 , 7 , 9 }, then a universal set is probably U = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 }.
Note that a universal set is not unique, unless specified.
1.2.2 Fundamental Operations on Sets
We introduce simple set-theoretic operations on sets and prove some of their properties. Given two or more sets, we can form a new set using these operations.
- Complement of a Set Let U be the universal set and let A be any set. The complement of A, written A{^ or sometimes A is defined as A{^ =
x ∈ U : x ̸∈ A
Example 1.
Let the universal set be U = { 0 , 1 , 2 , 3 , 5 , 6 } and A = { 3 , 5 }. Clearly, A{^ = { 0 , 1 , 2 , 6 }.
- Union of Sets Let A and B be sets. The union of A and B, denoted by A ∪ B is
A ∪ B =
x : x ∈ A or x ∈ B or both
More generally, if A 1 , A 2 , ..., An are sets, then their union is the set of all objects which belong to at least one of them, and is denoted by
A 1 ∪ A 2 ∪ · · · ∪ An
or by (^) n ∪ i=
Ai
This is the set of elements which belong to at least one Ai, i = 1, 2 , ..., n.
Example 1.
(a). If A = { 2 , 5 , 7 } and B = {T om, Bush, M ary}, then A∪B = { 2 , 5 , 7 , T om, Bush, M ary}. (b). If A 1 = {x, y, t, s}, A 2 = {q, r, f }, A 3 = { 0 , 1 , 3 , 4 , 5 , 6 , 7 , 8 , 20 }, then
A 1 ∪ A 2 ∪ A 3 = {x, y, t, s, q, r, f, 0 , 1 , 3 , 4 , 5 , 6 , 7 , 8 , 20 }
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Example 1.
(a). If A = { 1 , 2 , 3 , 5 , 6 , 7 } and B = { 3 , 5 , 9 }, then A − B = { 1 , 2 , 6 , 7 } and B − A = { 9 }. (b). If
A = {N ewY ork, Cairo, M umbai, Seoul, Beijing, M oscow, London}
and B = {N airobi, Kigali, P retoria, Beijing, Harare, P aris, London},
then A − B = {N ewY ork, Cairo, M umbai, Seoul, M oscow}
and B − A = {N airobi, Kigali, P retoria, Harare, P aris}
Clearly, if A − B = ∅ and B − A = ∅, then A = B. It is easy to verify that A − B = A ∩ B{. Note that A{^ = U − A.
- Symmetric Difference of Two Sets Let A and B be sets. The symmetric difference of A and B, denoted by A △ B is defined as A △ B =
x : x ∈ A or x ∈ B, but not both
Clearly, A △ B =
x : x ∈ A or x ∈ B, but not both
x : x ∈ A or x ∈ B, and x ̸∈ A ∩ B
x : x ∈ A ∪ B, and x ̸∈ A ∩ B
x : x ∈ (A ∪ B) − (A ∩ B)
= (A − B) ∪ (B − A).
= (A ∩ B{) ∪ (B ∩ A{)
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The symmetric difference of two sets is also called the Boolean sum of the two sets.
Example 1.
If A = { 2 , 1 , 3 , 5 } and B = {x, t, 7 , 1 }, then A ∪ B = { 1 , 2 , 3 , 5 , x, t, 7 } and A ∩ B = { 1 }. Therefore, A △ B =
2 , 3 , 5 , x, t, 7
- Cartesian Product of Sets Let A and B be sets. The Cartesian product of A and B, denoted by A × B is defined as A × B =
(a, b) : a ∈ A and b ∈ B
More generally, the Cartesian product of n sets A 1 , A 2 , ..., An is defined as
A 1 × A 2 × · · · × An =
(a 1 , a 2 , ..., an) : ai ∈ Ai, i = 1, 2 , 3 , ..., n
The expression (a 1 , a 2 , ..., an) is called an ordered n-tuple.
Example 1.
If A = { 0 , 1 , 2 } and B = {a, b}, then
A × B =
(0, a), (0, b), (1, a), (1, b), (2, 1), (2, b)
B × A =
(a, 0), (a, 1), (a, 2), (b, 0), (b, 1), (b, 2)
A × A =
Example 1.
Let R be the set of real numbers. Then the Cartesian product
R × R =
(x, y) : x, y ∈ R
= R^2
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- (A ∪ B){^ = A{^ ∩ B{^ De Morgan’s Laws (A ∩ B){^ = A{^ ∪ B{
- A ∪ B = B ∪ A Commutative Laws A ∩ B = B ∩ A
- A ∪ (B ∪ C) = (A ∪ B) ∪ C Associative Laws A ∩ (B ∩ c) = (A ∩ B) ∩ C
- A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) Distributive Laws A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
- A ∪ A = A Idempotence Laws A ∩ A = A
- A ∪ ∅ = A Identity Laws A ∩ U = A
- A ∪ A{^ = U Inverse Laws A ∩ A{^ = ∅
- A ∪ U = U Domination Laws A ∩ ∅ = ∅
- A ∪ (A ∩ B) = A Absorption Laws A ∩ (A ∪ B) = A
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1.3.1 Venn Diagrams
It is often useful a diagram called a Venn diagram(named after John Venn, a British Mathematician and philosopher (1834-1923))or sometimes Euler diagram (after Leonard Euler, who first introduced them) to visualize and prove some of the various properties of set operations. Venn diagrams are useful in many fields, including set theory, proba- bility, logic, statistics and computer science. In a Venn diagram, the universal set U is represented/depicted by the interior of a large rectangular area/region. Subsets within this universe are represented by interiors of circular areas/regions and wanted regions are to be shaded. For a set A, the region/area outside the cire for A represents A{. Set Operation Symbol
- Set B is contained in A B ⊆ A
Figure 1.1: Venn diagram for A ⊂ B
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