Proof of Only One Identity Properity for Binary Operations

Date: 10/31/2001 at 12:15:42
From: Tara Saddig
Subject: Proof of only one identity property for binary operations

Dear Dr. Math,

I am trying to prove for a college geometry class that there is one 
and only one identity property for every operation.

This is what I have so far:
(* is star, an unknown binary operation, not times)

Suppose 1 and 1' are both identities in a group

   a*1 = 1*a = a
   a*1' = 1'*a = a
   a*1 = a*1' = a

(I want to just divide by a and get that 1 = 1', but since we're 
working with * not times, I can't do this, and therein lies my 
problem.)

I also used the inverse properity to get some more stuff, but no 
proof:

   a*b = 1
   a'*b' = 1'
   a*1 = a*1'
   a*(a*b) = a*(a'*b')
   (a*1)*(a*b) = (a*1')*(a'*b')

I don't know where to go from here. I feel as if I have a lot of 
information, but don't know how to prove that there's only one 
identity for operation * without simply dividing both sides by a.  
Do you have any suggestions for finishing this proof? Thanks so much!

Date: 10/31/2001 at 13:49:36
From: Doctor Paul
Subject: Re: Proof of only one identity property for binary operations

You've done a lot of nice work, but it's not really going to lead you 
where you need to go.

The first portion of what you have above is good, but we need to take 
a different approach:

If 1 and 1' are both identity elements in a group, then the following 
hold:

   1 * 1' = 1' * 1 = 1 (since 1' is an identity element)

But notice also that:

   1' * 1 = 1 * 1' = 1' (since 1 is an identity element)

Then 1 * 1' = 1 and 1 * 1' = 1' so it follows that

   1 = 1'

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/