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### 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.

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!

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/
```
Associated Topics:
High School Definitions
High School Sequences, Series

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