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.



Definition 2 (Closed Under)

Let * be a binary operation on S and let H be a subset of S. The subset H is closed under * if for all a, b element of H.

Example 2 :

On N, with operation subtraction (-).

Note that, 2, 5 element of N but 2 - 5 = -3 not element of N.

Then, condition 2 is not satisfied.
Thefore, subtraction is not binary operation on N.

Example 3 :

On Q, let a*b = a/b

Here, 2, 0 element of Q, but 2*0 = 2/0 ( no element)
Then, no rational number assigned by this rule to the pair (2,0).

• condition 1 is not satisfied.

Then, * is not a binary operation.

Note :
Referring to example 2, condition 2 is not satisfied. Then, N is not closed under (-).
Referring to example 3, condition 1 is not satisfied. Then * is not defined on Q.



sources : http://ofstednews.ofsted.gov.uk/resources/maths-pencils-graphic.jpg