Mathematics I
Basic Mathematics
Prepared by Prof. Jairus. Khalagai
African Virtual university
Université Virtuelle Africaine
Universidade Virtual Africana
African Virtual University
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Table of ConTenTs
I.
Mathematics 1, Basic Mathematics _____________________________ 3
II. Prerequisite Course or Knowledge _____________________________ 3
III. Time ____________________________________________________ 3
IV. Materials _________________________________________________ 3
V. Module Rationale __________________________________________ 4
VI. Content __________________________________________________ 5
6.1 Overview ____________________________________________ 5
6.2 Outline _____________________________________________ 6
VII. General Objective(s) ________________________________________ 8
VIII. Specific Learning Objectives __________________________________ 8
IX. Teaching and Learning Activities ______________________________ 10
X. Key Concepts (Glossary) ____________________________________ 16
XI. Compulsory Readings ______________________________________ 18
XII. Compulsory Resources _____________________________________ 19
XIII. Useful Links _____________________________________________ 20
XIV. Learning Activities _________________________________________ 23
XV. Synthesis Of The Module ___________________________________ 48
XVI. Summative Evaluation ______________________________________ 49
XVII. References ______________________________________________ 66
XVIII. Main Author of the Module _________________________________ 67
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I.
Mathematics 1, basic Mathematics
By Prof. Jairus. Khalagai, University of Nairobi
II.
Prerequisite Courses or Knowledge
Unit 1: (i) Sets and Functions (ii) Composite Functions
Secondary school mathematics is prerequisite.
This is a level 1 course.
Unit 2: Binary Operations
Basic Mathematics 1 is prerequisite.
This is a level 1 course.
Unit 3: Groups, Subgroups and Homomorphism
Basic Mathematics 2 is prerequisite.
This is a level 2 course.
III.
Time
120 hours
IV.
Material
The course materials for this module consist of:
Study materials (print, CD, on-line)
(pre-assessment materials contained within the study materials)
Two formative assessment activities per unit (always available but with spe-
cified submission date). (CD, on-line)
References and Readings from open-source sources (CD, on-line)
ICT Activity files
Those which rely on copyright software
Those which rely on open source software
Those which stand alone
Video files
Audio files (with tape version)
Open source software installation files
Graphical calculators and licenced software where available
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V.
Module Rationale
The rationale of teaching Basic mathematics is that it plays the role of filling up
gaps that the student teacher could be having from secondary school mathematics.
For instance, a lack of a proper grasp of the real number system and elementary
functions etc. It also serves as the launching pad to University Mathematics by
introducing the learner to the science of reasoning called logic and other related
topics.
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VI.
Content
6.1 Overview
This module consists of three units which are as follows:
Unit 1: (i) Sets and Functions (ii) Composite Functions
This unit starts with the concept of a set. It then intoroduces logic which gives
the learner techniques for distinguishing between correct and incorrect argu-
ments using propositions and their connectives. A grasp of sets of real numbers
on which we define elementary functions is essential. The need to have pictorial
representations of a function necessitates the study of its graph. Note that the
concept of a function can also be viewed as an instruction to be carried out on a
set of objects. This necessitates the study of arrangements of objects in a certain
order, called permutations and combinations.
Unit 2: Binary Operations
In this unit we look at the concept of binary operations. This leads to the study
of elementary properties of integers such as congruence. The introduction to
algebraic structures is simply what we require to pave the way for unit 3.
Unit 3: Groups, Subgroups and Homomorphism
This unit is devoted to the study of groups and rings. These are essentially sets of
numbers or objects which satisfy some given axioms. The concepts of subgroup
and subring are also important to study here. For the sake of looking at cases of
fewer axiomatic demands we will also study the concepts of homomorphisms
and isomorphisms. Here we will be reflecting on the concept of a mapping or a
function from either one group to the other or from one ring to the other in order
to find out what properties such a function has.
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6.2 Outline
Unit 1: (i) Sets and Functions (ii) Composite Functions (50 hours)
Level 1. Priority A. No prerequisite.
Sets (4)
Elementary logic (8)
Number systems (6)
Complex numbers (4)
Relations and functions (8)
Elementary functions and their graphs (8)
Permutations (7)
Combinations (5)
Unit 2: Binary Operations (35 hours)
Level 1. Priority A. Basic Mathematics 1 is prerequisite.
Binary operations. (7)
Elementary properties of integers. (7)
Congruence. (7)
Introduction to Algebraic structures. (7)
Applications (7)
Unit 3: Groups, Subgroups and Homomorphism (35 hours)
Level 2. Priority B. Basic Mathematics 2 is prerequisite.
Groups and subgroups. (7)
Cyclic groups. (2)
Permutation groups. (5)
Group homomorphisms. (4)
Factor groups. (3)
Automorphisms. (3)
Rings, sub-rings, ideals and quotient rings. (7)
Isomorphisms theorems for groups and rings. (4)
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This diagram shows how the different sections of this module relate to each
other.
The central or core concept is in the centre of the diagram. (Shown in red).
Concepts that depend on each other are shown by a line.
For example: Set is the central concept. The Real Number System depends on
the idea of a set. The Complex Number System depend on the Real Number
System.
Homomorphisms
and
Groups and
Propositional
Isomorphisms
Rings
Logic
Real Number
SE T
Algebraic
System
Structure
Complex
number
Functions and
Binary
system
their graphs
Operation
Permutations
Trigonometry
and
combinations
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VII. General objective(s)
You will be equipped with knowledge of elementary mathematical logic, sets,
numbers and algebraic structures required for effective teaching of mathematics
in secondary schools.
VIII. specific learning objectives
(Instructional objectives)
By the end of this module, the learner should be able to…”
•
Construct mathematical arguments.
•
make connections and communicate mathematical ideas effectively and
economically.
•
Examine patterns, make abstractions and generalize.
•
Understand various mathematical structures and the similarities and dif-
ferences among these structures.
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IX.
Teaching and learning activities
Module 1: Basic Mathematics, Pre-assessment
Unit 1: Sets and Functions
Assessments and Solutions
Pre-assessment Questions
1. Given the quadratic equation:
2
2x − x − 6 = 0
The roots are
a. {−4, }
3
b. {4, − }
3
⎧
3
c.
2,
⎫
⎨ − ⎬
⎩
2 ⎭
⎧
3
d.
2, ⎫
⎨−
⎬
⎩
2 ⎭
2. The value of the function f (x)
2
= 2 x + 3x + 1at x = 3 is
a. 19
b. 28
c. 46
d. 16
3. Which of the following diagrams below represents the graph of y=3x(2-x)
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b.
a.
c.
d
.
4. The solution of the equation
1
sin x = −
in the range 0 ≤ xo ≤ 360 is:
2
a. {150o,210o}
b. {30o,150o}
c. {210o,330o}
C
d. {30o,330o}
5. Given the triangle ABC below
√5
a o
A
B
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Which of the following statements is correct?
2
a) Cos a = 15
5
b) Sin a = 2
c) Tan a = 2
1
d) Sec a = 5
Unit 1: Pre-assessment Solutions
The following are the answers to the multiple choice questions.
Q 1 c
Q 2 b
Q 3 b
Q 4 c
Q 5 c
Unit 2: Binary Operations
1. The inverse of the function
1
f (x) =
is
x −1
(a)
1
f − (x) = x −1
(b)
1
1
−
− x
f
(x) = x
x + 1
(c) −1
f
(x) = x
1
(d) −1
f
(x) = − 1
x
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x
a
2. If sin
=
then
2
2
sin x in terms a is:
a
(a)
2
4 − a
(b)
2
a 4 − a
(c) a
2
4 − a
(d)
2
3. A girl has 3 skirts, 5 blouses and 4 scarves. The number of different outfits
consisting of skirt, blouse and scarf that she can make out of these is:
a. 220
b. 60
c. 12
d. 150
4. Given the complex number
z = 1 − i we have that Arg z is:
(a) 450
(b) 1350
(c) 2250
(d) 3150
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5. If
2
a ∗b = a + ab −1, then
5 ∗ 3 is
(a) 39
(b) 41
(c) 23
(d) 25
Unit 2: Pre-assessment Solutions
Q1. c
Q2. b
Q3. b
Q4. b
Q5. a
Unit 3: Groups, Subgroups and Homomorphism
1. Which of the following is a binary operation?
(a) Squaring a number.
(b) Taking the predecessor of a natural number.
(c) Taking the successor of a natural number.
(d) Finding the sum of two natural numbers
2. Recall the definition of a homormorphism and state which one of the following
is a homomorphism on a group G of real numbers under either multiplication
or addition?
(a) f ( ) = 2x
x
(b) f (x) = 6x
(c) f ( )
2
x = x
(d) f (x) = x + 5
3. For a group G if a x a = b in G, then x is
(a) b
(b)
1
ba−
(c)
−1
a b
(d)
−1
−1
a b a
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4. If an element a is a ring R is such that 2
a = a then a is called
(a) nilpotent
(b) characteristic
(c) idempotent
(d) identity
5. Let R be a ring and x � R if there exists a unique element a ∈ R such that x a = x, then a x is:
(a) e
(b) a
(c) – x
(d) x
Unit 3: Pre-assessment Solutions
1. d
2. c
3. d
4. c
5. d
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Title of Pre-assessment : Pedagogical comment for learners
The questions in this pre-assessment are designed to test your readiness for stu-
dying the module.
The 5 questions preparing you for unit 1 require high school mathematics. If you
make any errors, this should suggest the need to re-visit the high school mathe-
matical topic referred to in the question.
The questions for unit 2 and unit 3 test your readiness after having completed
the learning activities for unit 1 and unit 2.
If you make errors in the unit 2 pre-assessment, you should check through your
work on unit 1 in this module. Likewise, If you make errors in the unit 3 pre-as-
sessment, you should check through your work on unit 2 in this module.
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X. Key concepts (glossary)
1. Abelian group: This is a group 〈 G, *〉 in which a * b = b * a for a,b, ∈ G.
2. Algebraic structure: This is the collection of a given set G together with a binary operation * that satisfies a given set of axioms.
3. Binary operation: This is a mapping which assigns to each ordered pair of
elements of a set G, exactly one element of G.
4. Composite Function: This is a function obtained by combing two or more
other simple functions in a given order.
5. Function: This is a special type of mapping where an object is mapped to a
unique image.
6. Group: This is a non-empty set say G with a binary operation * such that: (i) a ∗ b ∈ G for all a, b ∈ G .
(ii) a ∗ (b∗ c) =(a ∗ )
b ∗ c for all a, ,bc ∈ G.
(iii) There exists an element e in G such that e ∗ a = a = a ∗ e for all a ∈ G
where e is called identity.
(iv) For every a ∈ G there exists
−1
a
∈ G
−
−
such that
1
1
a ∗ a
= e = a
∗ a
Where
1
a− is called the inverse of a
7. Homomorphism: This is a mapping f from a group G into another group H
such that for any pair a,b, ∈ G. We have f (ab) = f (a) f (b).
8. Isomorphism: This is a homorphism which is also a bijection.
9. Mapping: This is simply a relationship between any two given sets.
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10. Proposition: This is a statement with truth value. Thus we can tell whether
it is true or false
11. Ring: This is a non-empty set say R with two binary operations + and *
called addition and multiplication respectively such that:
(i) R, + is an Abelian group.
(ii) 〈 R, * 〉 is a multiplicative semigroup.
12. Semigroup: This is a non-empty set S with a binary operation such * that:
(i) a ∗ b ∈ S for all a , b ∈ S.
(ii) a ∗ (b∗ c) = (a∗ )
b ∗ c for all a , ,bc ∈ S.
(iii) For all a , ,
b c ∈ S. we have:
a ∗(b+ c) = a ∗b+ a ∗ c and (a + b)∗ c = a ∗ c + b∗ c
13. Set: This is a collection of objects or items with same properties
14. Subgroup: This is a subset H of a group G such that H is also a group with respect to the binary operation in G.
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XI.
Compulsory Readings
Reading #1
A Textbook for High School Students Studying Maths by the Free High School
Science Texts authors, 2005, pg 38-47 (File name on CD: Secondary_School_
Maths)
Reading #2
Elements of Abstract and Linear Algebra by E. H. Connell, 1999, University of
Miami, pg. 1-13 (File name on CD: Abstract_and_linear_algebra_Connell)
Reading #3
Sets relations and functions by Ivo Duntsch and Gunther Gediga methodos pu-
blishers (UK) 2000. (File name on CD: Sets_Relations_Functions_Duntsch)
Reading #4
Abstract Algebra: The Basic Graduate Year, by Robert B. Ash (Folder on CD:
Abstract_Algebra_Ash)
General Abstract and Rationale
All of the compulsory readings are complete open source textbooks. Together
they provide more than enough material to support the course. However, the text
contains specific page references to activities, readings and exercises which are
referenced in the learning activities.
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XII. Compulsory Resources
Wolfram MathWorld (visited 29.08.06)
http://mathworld.wolfram.com/
•
A complete and comprehensive guide to all topics in mathematics. The
students is expected to become familiar with this web site and to follow
up key words and module topics at the site.
Wikipedia (visited 29.08.06)
http://www.wikipedia.org/
•
Wikipedia provides encyclopaedic coverage of all mathematical topics.
Students should follow up key words by searching at wikipedia.
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XIII. Useful links
Set Theory (visited 29.08.06)