Basic Mathematics by Prof. Jairus. Khalagai - HTML preview

Introduction

a) A Story of Nursery School Children

Two Children brother and sister called John and Jane go to a Nursery school

called Little Friends.

One morning they woke up late and found themselves in a hurry to put on clothes

and run to school, Jane first put on socks then shoes. But her brother John first

put on shoes then socks. Jane looked at him and burst into laughter as she run to

school to be followed by her brother.

Question

Why did Jane burst into laughter?

b) Story of a visit to a beer brewing factory

A science Club in a secondary school called Nabumali High

School in Uganda, one Saturday made a trip to Jinja town

to observe different stages of brewing beer called Nile Beer.

It was noted that of special interest was the way some equi-

pment used in the process would enter some chamber and

emerge transformed. For example an empty bottle would

enter a chamber and emerge transformed full of Nile Beer

but without the bottle top. Then it would enter the next

chamber and emerge with the bottle top on.

empty

bottle

Chamber

Chamber

(1)

(2)

filled with

filled with

Nile

top on

Beerwithout

top

Question

Can you try to explain what happens in each chamber of the brewing factory?

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Activity

We note that in our story of the Nursery school Children what is at stake is the

order in which we should take instruction in real life situations. Jane laughed at

her brother because she saw the socks on top of the shoes. In other words her

brother had ended up with composite instruction or function which was untenable.

We can also look at other such cases through the following example.

Example 1

I think of a number, square it then add 3 or I think of a number add 3 then square

it. If we let the number to be x, then we will end up with two different results

namely 2

x + 3 and ( x + )2

3 respectively.

Example 2

Can you now come up with a number of examples similar to the one above?

If we now consider our story on the brewing of Uganda Warangi we note that

each Chamber has a specific instruction on the job to perform. This is why wha-

tever item passes through the chamber must emerge transformed in some way.

We can also look at an example where instructions are given in functional form

with explicit formulae as shown below.

Example 3

Consider the composition of the functions.

f : x → 2x and g : x → x + 5

Here if we are operating f followed by g then we double x before we add 5. But if operate g followed by f then we add 5 to x before we double the result.

For notation purposes

( f o g) ( x)= f ( g ( x))means g then f. While ( g o f ) ( x)= g ( f ( x)) means f then g

Thus we have:

4 8 11

g : x → 2x

g : x → x + 5

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Representing the composite function g ( f ( x)) = 2 x + 5

While:

4 9 18

g : x → x + 5

f : x → 2x

representing the composite function f ( g ( x)) = 2 ( x + 5) Exercise 4

Given f : x → 3x +1

g : x → x − 2

Determine the following functions:

(a) f o g

(b) g o f

(c) (

) 1

f g −

o

(d) (

) 1

g f −

o

Taking x = 3 draw a diagram for each of the composite functions above as is the

case in example 3 above.

Exercise 5

Sketch the graph for each of the following function: assuming the domain for

each one of them is the whole set ℜ of real numbers.

b)

f (x) = 2x − 3

c) g (x)

2

= 4x −12x

d) h(x)

3

= x − 3x +1

e) k (x) = 2 sin x

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Module 1: Basic Mathematics

Unit 2: Binary Operations

Specific Learning Objectives

By the end of this activity, the learner should be able to:

•

Give examples of binary operations on various operations

•

Determine properties of commutativity or associativity on some binary

operations.

•

Determine some equivalence relations on some algebraic structures

Overview

The concept of a binary operation is essential in the sense that it leads to the

creation of algebraic structures.

The well known binary operations like + (addition) and x (multiplication) do

constitute the set ℜ of real numbers as one of the most familiar algebraic structu-

res. Indeed the properties of commutativity or associativity can easily be verified

with respect to these operations on ℜ.

However, in this activity we define and deal with more general binary operations

which are usually denoted by*.

For example for any pair of points x, and y, in a given set say G, x * y could

even mean pick the larger of the two points. It is clear here that x * y = y * x.

Consequently we will exhibit examples of more general algebraic structures that

arise from such binary operations.

Key concepts

Algebraic structure: This is the collection of a given set G together with a binary operation * that satisfies a given set of axioms.

Binary operation: This is a mapping which assigns to each ordered pair of ele-

ments of a set G, exactly one element of G.

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Readings

All of the readings for the module come from Open Source text books. This

means that the authors have made them available for any reader to use them

without charge. We have provided complete copies of these texts on the CD

accompanying this course.

1

Sets of relations and functions by Ivo Duntsch and Gunther Gediga,

Methodos Publishers, UK, 2000 pp 30-34 (File name on CD: Sets_Rela-

tions_Functions_Duntsch)

Internet Resources

Binary Color Device (visited 06.11.06)

http://www.cut-the-knot.org/Curriculum/Algebra/BinaryColorDevice.shtml

•

This is a puzzle involving binary operations and group tables. Use the

puzzle to develop your understanding.

Wolfram MathWorld (visited 06.11.06)

http://mathworld.wolfram.com/BinaryOperation.html

•

Read this entry for Binary Operations.

•

Follow links to explain specific concepts as you need to.

Wikipedia (visited 06.11.06)

http://www.wikipedia.org/

•

Type ‘Binary Operations’ into the search box and press ENTER.

•

Follow links to explain specific concepts as you need to.

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Introduction: The Story of the Reproductive System

In a real life situation among human beings, you will find that an in-

dividual gets into a relation with another individual of opposite sex.

They then reproduce other individuals who constitute a family. We

then have that families with a common relationship will constitute a

clan and different clans will give rise to a tribe etc…

We note that even in ecology the same story can be told. For example

we can start with an individual like an organism which is able to re-

produce other organisms of the same species that will later constitute

a population. If different populations stay together then they will

constitute a community etc…

Question:

What is the mechanism that can bring together two individuals (human beings

or organisms of ecology) in order to start reproduction?

Activity

We note that in the case of human beings in our story above we could say that

it is marriage that brings together a man and woman to later constitute a family

after reproduction. In mathematics the concept of marriage could be looked at as

a binary operation between the two individuals. If we can reflect on our mapping

diagram we have the following:

A

B

x

*

y

Where A = set of men wedding in a given time

B = set of women getting marriage at the same time

* = operation which says x weds y

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Clearly x * y = y * x

In this case this particular binary operation is commutative. If we denote the

relation implied by the binary operation * by R then we write. x R y to mean x is related to y or y R x to mean y is related to x.

If x R y ⇒ y R x then the relation is said to be symmetric.

Question

Can you try to define some relations on sets of your choice and check whether

they are symmetric?

In general we note that if a binary operation * gives rise to a relation R then:

a) R is reflexive if x R x

b) R is symmetric if x R y ⇒ y R x

c) R is transitive if x R y and y R z ⇒ x R z

For all elements x, y, z in a given set

We also note that a relation R satisfying all the three properties of reflexive, symmetric and transitive above is said to be an equivalence relation.

Example 1

Let U be the set of all people in a community.

Which of the following is an equivalence relation among them?

i. is an uncle of

ii. is a brother of

We note that in part (i) if R is the relation “is an uncle of” then x R y does not imply y R x.

Thus R is not symmetric in particular. Hence R is not an equivalence relation.

However in part (ii) if R is the relation “is a brother of” then x R x is valid.

Also x R y ⇒ y R x and finally x R y and y R z ⇒ x R z.

Hence R is an equivalence relation.

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Exercise 2

Which of the following is an equivalence relation on the set of all human

beings?

i.

is a friend of

ii. is a relative of

Exercise 3

a) Determine whether the binary operation * on the set ℜ of real numbers is

commutative or associative in each of the following cases

i.

x * y = y 2 x

ii. x * y = xy + x

b) Define a relation ~ on the set of integers as follows

a ~ b if and only if a + b is even.

Determine whether ~ is an equivalence relation on ℜ.

c) Give an example of an equivalence relation on the set ℜ of real numbers.

If you are working in a group each member of the group should give one such

example.

d) Complete exercise 2.4.1 p 34 in Sets, Relations and Functions by Duntsch

and Gediga (solutions on pp. 48 – 49)

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Module 1: Basic Mathematics

Unit 3: Groups, Subgroups and Homomorphism

Specific Objectives

By the end of this activity, the learner should be able to:

•

State axioms for both a group and a ring.

•

Give examples of groups and subgroups.

•

Give examples of rings and subrings.

•

Give examples of homomorphisms between groups and isomorphisms

between rings.

•

Prove some results on properties of groups and rings.

Overview

Recall that in our Unit 2 activity 2 we looked at the case of an individual organism

being able to reproduce and give rise to a population. Note that a population here

refers to a group of individuals from the same species. In this activity we are

going to demonstrate that a general algebraic structure can give rise to a specific

one with well stated specific axioms.

We will also reflect on the notion of relations between sets using mappings,

whereby we will define a mapping between any two given groups. It is at this

stage that the concept of a homomorphism will come into play. The situation

of looking at the properties of a mapping between two sets which are furnished

with algebraic structure as the groups are can be of great interest and indeed it is

the beginning of learning proper Abstract Algebra.

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Key Concepts

Abelian group: This is a group G , ∗ in which a ∗ b = b ∗ a for

a, ,

b ∈ G .

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

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 (xy) = f (x) f (y).

Isomorphism: This is a homorphism which is also a bijection.

Ring: This is a non-empty R set say with two binary operations + and * called

addition and multiplication respectively such that:

(i) R, + is an Abelian group.

(ii) R , ∗ is a multiplicative semigroup.

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 ∈ R we have:

a ∗ b + c = a ∗b + a ∗ c

(

)

and (a + b)∗ c = a ∗ c + b ∗ c

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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.

Readings

All of the readings for the module come from Open Source text books. This

means that the authors have made them available for any reader to use them

without charge. We have provided complete copies of these texts on the CD

accompanying this course.

Abstract Algebra: The Basic Graduate Year, by Robert B. Ash (Folder on CD:

Abstract_Algebra_Ash)

Internet Resources

Wolfram MathWorld (visited 06.11.06)

http://mathworld.wolfram.com/Group.html

•

Read this entry for Group Theory.

•

Follow links to explain specific concepts as you need to.

Wikipedia (visited 06.11.06)

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

•

Read this entry for Group Theory.

•

Follow links to explain specific concepts as you need to.

MacTutor History of Mathematics

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Development_group_

theory.html

•

Read for interest the history of Group Theory

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Introduction: Story of a Cooperative Society

In 1990, one hundred workers of a certain Institution in Kenya decided to form

a cooperative society called CHUNA in which they were contributing shares on

a monthly basis. They set up rules for the running of the society which included

terms for giving out loans. It was decided after they had run the society for some-

time that the officials should pay regular visits to other well established cooperative

societies in the country to see how they are run in comparison to there own.

It was noted after those visits to other societies that there was need to moderate

some of there rules of running the society in order to create consistency.

Questions

1. Why did they set up rules after forming the cooperative society?

2. What significance could you attach to their visits to other cooperative

societies?

Activity

In our story above we note that a cooperative society requires rules to create an

operating structure. This is equivalent to having axioms that are satisfied by

elements of a non-empty set as is the case with the group G.

Question

Can you now think of other situations where a group of people or objects could

have sets of rules among them, which resemble the axioms of a group?

Example 1

Consider the set ¢ of integers in order the operation of addition (+). We have

that

(i) a + b in ¢ for all a, b, ∈ ¢

(ii) a + (b + c) = (a + b) + c for all a, b, c ∈ ¢

(iii) there is 0 ∈ ¢ such that

a + o = a = o + a for all a ∈ ¢

(iv) for every a∈ ¢ there is – a such that

a + -a = o = -a + a

Hence, { ¢ ,+ } is a group

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Exercise 2

Verify that the set R of real numbers is also a group under addition.

Note that if for any group {G, ∗} we have that for any pair of points x, y ∈ G, x ∗ y = y ∗ x

Then G is called Abelian group.

In this case the group {R+, +} is Abelian.

The second question coming out of our story of the cooperative society above is

mainly for comparison purposes. This is to find out whether the structure set up

by CHUNA compares well with those of other societies. Similarly the structures

of groups are easily compared using mappings. Thus for any two given groups

say G and H a mapping can be defined between them in order to compare their

structures. In particular a homomorphism f: G ➝ H is a mapping that preserves

the structure. In other words G and its image under f (denoted by f(G) in H are

the same group structurally. Note that if a homomorphism is an onto mapping

then it is called an isomorphism.

Example 3

Let G and H be any two groups and e1 be the identity of H. Then the mapping

f:G ➝ H given by

f( xy) = e’

is a homomorphism.

Indeed for any pair x, y ∈ G,

f( xy) = e’ = e’ e’ = f( x)f( y)

Exercise 4

Let G be the group {R+, x} of positive real numbers under multiplication and let

H be the additive group{R+, +} of real numbers. Show that the mapping:

f:G➝H given by

f( xy) = log x

10

is a homomorphism.