Isomorphisms

Date: 08/16/99 at 02:20:37
From: Brendan Bates
Subject: Isomorphism

Hi,

I am a high school student currently studying math. For my assignment, 
I have to give an oral presentation on a topic in mathematics. I was 
given the topic of isomorphism. I have looked everywhere on the 
Internet and cannot find an explanation I could give to the class 
(because I myself can't understand them!) It would be a big help if 
you could give me an explanation and a nice example that someone like 
me could understand, and then interpret it so I can explain it to the 
rest of the class.

Thanking you in advance,

David Sessna :)

Date: 08/16/99 at 12:46:00
From: Doctor Rob
Subject: Re: Isomorphism

Thanks for writing to Ask Dr. Math!

Hmmmmmm...  This isn't so easy!

Let's start with two examples. Consider the multiplication table for 
odd and even numbers:

       *  | EVEN  ODD
     -----------------
     EVEN | EVEN  EVEN
     ODD  | EVEN  ODD

Now consider the multiplication table for the numbers {0,1}:

       *  |  0    1
     ---------------
       0  |  0    0
       1  |  0    1

These are very similar! In fact, if you replace the word "EVEN" with 
"0" and the word "ODD" with "1", they will be identical.

If you take the set {EVEN,ODD} with the operation defined by the first 
multiplication table, and you also take the set {0,1} with the 
operation defined by the second multiplication table, we express the 
similarity between these by saying that they are "isomorphic."

That means that the two sets can be put into one-to-one 
correspondence, and this correspondence respects the two operations on 
the two sets, in the following sense. Start with two elements of the 
first set. First multiply them together, and find the element in the 
second set that corresponds to this product. Then find the two 
elements in the second set corresponding to the two given elements of 
the first set, and multiply those together. These two results will be 
the same.

In symbols, we can call the first set A and the second set B, and the 
one-to-one correspondence is then expressed as a function from A to B, 
f:A --> B, which is one-to-one and onto. Then the fact that f respects 
the operations is given by the statement

     f(x*y) = f(x)*f(y), for every x and y in A.

Isomorphisms between objects tell us that the two set-with-operation 
objects are very similar. If no such isomorphism exists, then the two 
objects are different from each other in some important way.

I hope this helps. If not, write again.

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