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 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)

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}.