IsomorphismsDate: 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/ |
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