Learn - Enjoy

Learn - Enjoy
M.Velrajan

To learn we have to raise questions, understand, apply, explore and create.

For example, suppose we learn Group

theory using Topics in Algebra, by 
I. N. Herstein, second edition.
Only after understanding the

preliminary notions given in Chapter 1
and the hindsight through A(S), we have

to go through the definition of a group :

Unless we know what is a nonempty set

and binary operation we can not follow

the definition.
Even after knowing the preliminaries,

we have to understand the 4 conditions

to be satisfied by the binary operation.

 Before reading further, try to

understand the definition of a group.

If the natural question 
“Is the first condition ‘closed’ on the

binary operation necessary ?” arise in

our mind, we can appreciate ourselves. Otherwise, it means that we have not understood the definition of a group.
If we recollect the definition of binary

operation given in the  Chapter 1:

we can see that by the definition of

binary operation as a mapping from

G × G into G,
a.b ∈ G, for all a, b ∈ G.
Hence the first condition ‘closed’ is not

necessary, it is repeated.

So, we can conclude that the definition

of a group is : 


(After feeling proud) 
The next question “why the ‘closed’

condition is redundantly given ?”

have to arise within us. 

We can conclude that just to give an

importance on ‘closed’ condition it is 

redundantly given.
Of course, we can say that :  A nonempty

subset S of a set G with a binary

operation . is closed under . ,

if a.b ∈ S, for all a, b ∈ S.

This is what we mean by

understanding a concept.

After understanding the definition of a

group, we can find more examples of 

groups.
Now we consider the example :

The set G of symbols

{e=a0, a, a2, a3, …, an-1} is a group.
{ e, ,  , 
,  }
is a group, taking n = 5 and a = .

When we study the group of integers

modulo n, (Zn , ⊕n),
Zn = {0, 1, 2, . . . , n-1}, we have to note

the structure similarity of the cyclic

group G =

{ e = a0, a, a2, a3, …, an-1 } of order n and

Zn = {0, 1, 2, . . . , n-1},

the group of integers modulo n. 

This note helps us to understand

isomorphic groups and that any

cyclic group of order n is isomorphic

to the group of integers modulo n.

Think differently, innovatively, about

the cyclic group 
G = {e = a0, a, a2, a3, …, an-1} .

Can we consider any finite set

G = {a0, a1, a2, . . . , an-1} as a group ?

Yes, we can consider any finite set

G = {a0, a1, a2, . . . , an-1} as a

group under the binary operation

ai.aj = ai + j, if i + j ≤ n, where an = a0 and 
ai.aj = ai + j - n, if i + j > n, (or)ai.aj = ai ⊕n j and  e = a0. 

In fact, a multiplication can be defined on G so that G is a ring.

Think and define the multiplication.

Oh, we are able to create.
Note the beauty.
In the cyclic group

G = {e = a0, a, a2, a3, …, an-1}

the numbers 0, 1, 2, . . . , n-1 appear

as superscripts. 
In Zn = {0, 1, 2, . . . , n-1}

the numbers 0, 1, 2, . . . , n-1 are the

members.

In our finite set

G = {a0, a1, a2, . . . , an-1} as a group

the numbers 0, 1, 2, . . . , n-1 appear

as subscripts. 

After satisfying ourselves with more

examples of groups,we easily prove : 
The identity element of a group is unique; 
Inverse of each element of a group is unique; 
Cancellation laws hold in a group;  etc.

How do we prove all these preliminary

results of a group ?
Just by using the definition of a group.

We see that in order to be a group the

nonempty set G with an associative

binary operation should have an identity

element and each element of G should

have an inverse element. 
But using this we can easily prove all the

preliminary results of a group. 

This is the Uniqueness and the Beauty

of Pure Mathematics to be enjoyed.
And this is what is called Applying in

the process of learning.

Note that when we verify that G is a

group, to show e ∈ G is the identity

element of G we have to show ae = a = ea,

for all a ∈ G. 
But if G is a group, x ∈ G and ax = a

(or xa = a) for an element a ∈ G then

x = e, the identity element of G

(since ax = a = ae, by left cancellation,

x = e). 
In particular, if xx = x then x = e. 

Also note that when we verify that G is a

group, to show b ∈ G is the inverse

element of a, we have to show

ab = e = ba. 
But if G is a group, a ∈ G and ab = e

(or ba = e) then b = a - 1, the

inverse element of a

(since ab = e = aa - 1, by left cancellation,

b = a - 1). 

This observation or analysis is useful for

answering MCQ.
 
This is what is called Analysing in

the process of learning. 

So, whatever we learn we have to

understand, observe, analyse, apply

and create by innovation.

Simply, we have to learn as per the

Thirukkural
"கற்க கசடற கற்பவை கற்றபின் 
நிற்க அதற்குத் தக"
Read as :  "Karka kasadara karpavai katrapin 
                  Nirka atharku thaga” 
Meaning : “Learn thoroughly whatever to be

learn, after learning  
                 stand according it"

This is what is expected in any

competitive examinations.

Learn accordingly and Enjoy Learning.