GROUPS 7

Example 5 :

Complete the following table so that * is commutative binary operation on the set S = {a,b,c,d}.


From the table, there is no identity and inverse element.(G1 and G2 are not satisfied).
Therefore, it is not a group.

GROUPS 6

Example 3 :

The following examples give groups with their respective tables.

1. Group V = {e,a,b,c} under multiplication (.). This group is the Klein 4-group. (notation V comes from the German word for four).

GROUPS 5

Finite Groups and Group Tables

Finite group - Group where the set G has finite element.

The rules :

1. List the identity first (use definition G2).
2. To find an inverse elements for each element e, a (use definition G3) a' must be either a or e.
   
3. All axioms are satisfied except possibly the associative property. Checking associativity on a case by case basis.

Note : each element of a group must appear once and only one in each row and column.

GROUPS 4

Theorem 3 ( The uniqueness of identity & inverse elements in G )

In a group G with binary operation *, there is only one element e in G such that

e*x = x*e = x

for all x in G.

Likewise, for each a in G, there is only one element a' in G that

a'*a = a*a' = e

In summary, the identity & inverse of each element are unique in a group.

GROUPS 3

Elementary Properties of Groups


Theorem 1 (Left & Right Cancellation Laws)

If G is a group with binary operation *, then the left and right cancellation laws hold in G, that is

a*b = a*c implies b = c, and
b*a = c*a implies b = c for all a,b,c in G

Proof:

(To prove the left cancellation law)
To show if a*b = a*c then b = c
Let a*b = a*c

Since G is a group, G3 is satisfied. Then, for all a' in G such that,

a'*(a*b) = a'*(a*c) --G3
(a'*a)*b = (a'*a)*c --G1,* associative
e*b = e*c --by def. of a' in G3
b = c

Similarly, we can prove the right cancellation law.

GROUPS 2

Example 3 :

Let G = {x element of G|x is not equal to 1}. On G, define * by a*b = a+b-ab. Determine whether the set G is a group under *.

G1 : (To show associativity of *)
Let a, b, c element of G & a,b, c is not equal to 1.

(a*b)*c = a*(b*c)

GROUPS

Definition 1 (Group)

A group is a binary operation structure (G,*) is a nonempty set and * is a binary operation on G such that the following axioms are satisfied.

G1 : For all a, b, c in G,

(a*b)*c = a*(b*c)

Associativity of *

G2 : There is an element e in G such that for all x in G,

e*x = x*e = x

Identity of *

G3 : Corresponding to each a in G, there ia an element a' in G such that

a*a' = a'*a = e

Inverse a' of a

Note : If any of these axioms is not satisfied, then a binary structure is not a group.

BINARY OPERATIONS 4

sources :http://www.youtube.com/user/videomathtutor#p/a


Theorem 1 ( Uniqueness of Identity Element )

A binary structure (S,*) has at most one identity. That is, if there is an identity, it is unique.

To prove "uniqueness"

- suppose two different elements to be the element that we want to prove unique.
- finally, to show these two elements are actually same.


(To show the identity element in (S,*) is unique).

Let e & f be identity element of S

Then,
Since e is the identity element

e*f = f*e = f ----1

Also, since f is the identity element

e*f = f*e = e ----2

Hence,
from 1 & 2 f = e

Therefore, the identity element in (S,*) must be unique.

BINARY OPERATIONS 3

sources : http://www.learningupgrade.com

Some important words:

Not everywhere define - Operation * is called not everywhere defined on S it - no element can be assigned to each possible ordered pairs.

Not well defined - Operation * is called not well defined on S it - several element of S are assigned to S (ambiguity).

BINARY OPERATIONS 2

Definition 3 (Commutative operation)

A binary operation * on a set S is commutative if a*b = b*a for all a, b in S.

Example 1 :

On Q, define a binary operation * by a*b = ab + 1. Show that the binary operation * is commutative on Q.
Let a, b element of Q,
(To check whether a*b = b*a)

LHS : a*b
= ab+1
= ba+1 since multiplication is commutative on Q.
= b*a = RHS
Therefore, * is commutative on Q.

Example 2 :
On Z, define a*b = a-b. Determine whether * is commutative on Z.

Let a, b element of Z
To check whether a*b = b*a

Note that,
by using counter example
let 1, 2 in Z.
LHS : 1 - 2 = -1
RHS : 2 - 1 = 1
but, since LHS is not equal to RHS, then * is not commutative on Z.

BINARY OPERATIONS

Definition 1 ( Binary Operations)

Let S a nonempty set. A binary operation on S is a function from S x S into S. Let * be a binary operation on S. For each a,b element of S, we denote.

the element *((a,b)) of S by (a*b).

An operation is called binary operation on S if :

1. Exactly one element is assigned to each possible ordered pair of element S.
2. For each ordered of element of S, the element assigned to again in S.

Example 1 :

On Z, with operation addition (+).

3 is assigned to (2, 1)

-1 is assigned to (0, -1)

Hence, condition 1 satisfied.
Also, for each ordered pair of elements of Z, the element assigned to it is again in Z.
Hence, condition 2 is satisfied.

Note : Since Z with operation + satisfied condition 2, we say Z is closed under addition.

RELATIONS

Definition 1 (Cartesian Product)

Let A & B be sets. The set A x B = {(a,b) | a element af A, b element of B} is the Cartesian product of A & B.


Definition 2 (Relation)
A relation between sets A & B is a subset R of A x B. We read aRb as "a is related to b".

Example (Equality Relation) :

The equality relation "=" defined on a set by = is the subset { (x,x) | x element of S} of S xS

Thus, for any X element of S, we have x = x

But, if x and y are different element of S, then (x,y) are not same.


Definition 3 (Partition)
A partition of a set is a collection of nonempty susets of S such that every element of S is in exactly one of the subsets.

The subsets are the cells of the partition.

SETS

Example 1:

{2,4,6,8} = {x|x is an even whole positive less than or equal 8}
= {2x|x = 1,2 ,3,4}


Example 2:

Let S = {1,2,3}

This set S has a total of 8 subsets :

{ }, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}.

LOGIC & PROOF

• Logic is the study of reasoning.

• Logic examines general forms which arguments may take, which forms are valid, and which are fallacies.

• Two parts of logic :

Inductive reasoning - give general conclusion from specific example.

Deductive reasoning - give logical conclusion from definitions and axioms.

• In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true.

• Also, proof is a sequence of inferences that starts with the axioms, and leads to the particular statement you wish to prove.

• A proof must demonstrate that a statement is true in all cases, without a single exception.

• Some terms :

Conjecture - An unproved that is believed to be true.

Theorem - The statement that is proved.

Lemma - The statement which use as a stepping stone in the proof of another theorem.

PRELIMINARIES

Algebra provides a generalization of arithmetic by using symbols, usually letters, to represent numbers.

For example, it is obviously true that

2 + 3 = 3 + 2

This arithmetic statement can be generalized using algebra to

x + y = y + x

where x and y can be any number.

*Algebra has been studied for many centuries.

HISTORY OF ALGEBRA

Egyption & Babylonian Algebra