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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
African Virtual University 36
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
African Virtual University 37
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.
African Virtual University 39
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
African Virtual University 45
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.