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**Mathematics I **

Basic Mathematics

Prepared by Prof. Jairus. Khalagai

**African Virtual university**

**Université Virtuelle Africaine**

**Universidade Virtual Africana**

African Virtual University

**Notice**

This document is published under the conditions of the Creative Commons

http://en.wikipedia.org/wiki/Creative_Commons

Attribution

http://creativecommons.org/licenses/by/2.5/

License (abbreviated “cc-by”), Version 2.5.

African Virtual University

**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

African Virtual University

**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)

African Virtual University 0

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) **