Associated Topics || Dr. Math Home || Search Dr. Math

### Proof of Only One Identity Properity for Binary Operations

```
Date: 10/31/2001 at 12:15:42
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/
```
Associated Topics:
College Modern Algebra

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search