BINARY OPERATIONS 3

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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.