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.



Example :

Determine whether (Q+,*), where * is defined by a*b = a+b/2 has an identity.

Solution :

Let a be any element in Q+.

If e is the identity element of (Q+,*) then

e*a = a*e = a

Then

a*e = a+e/2 = a --- e = a

e*a = e+a/2 = a --- e = a

Since e = a represent any element of Q+, then (Q+,*) has no identity.