What is the definition for ten years old of isomorphism...? What the definition of morphism?
I don't like Wikipedia.
Can you write in a single paragraph on the definition of them.
You have asked an almost impossible question to answer for even high school. The reason being that there are several things going on at once here. Having two sets each has its own operation. If it is possible to match the elements of each one-to-one in such a way that operations preserved. That is very vary loose description.
You asking what I know...
OK.
I knew that all the "creators" you give me is for establishing theories in number theory. [Field and group, and etc.] and also my student know.
I write that my explanation is give to smart [or my Enlglish is bad by the word "gifted", it can be an error in using of the English language.]
My student know what is field and what is ring.
Can This help me to explain them what is isomorphism?!
An isomorphism from one algebraic structure to another is a "one-to-one", "onto" mapping, f, that "preserves" operations. That is, if x+ y is defined in the first algebraic structure then f(x)+ f(y) must be defined in the second algebraic structure and f(x+ y)= f(x)+ f(y). If x o y is an operation defined in the first algebraic structure then f(x) o f(y) must be defined in the second structure and f(x o y)= f(x) o f(y).
Another way of looking at it is that two algebraic structures are isomorphic if and only if the second is just the first with things "renamed".
Here is what you posted,
Thus I posted a greatly dumb-down version of the Wikipedia piece.
Now you post this hard to understand request.
What does Axioms of E mean? Look at Wikipedia again.
There is a discussion of the mapping $\exp:\mathbb{R}\to\mathbb{R}^+$ given by $x\mapsto e^x$.
You see that $\exp(xy)=\exp(x)\exp(y)$
What the different between isomorphism to homomorphism?
If you can write in a little line.
Thank you for everybody, you are very helpful.
If it can be in a little line, so please help me to understand.
I need to give only a conclusion of the matter. So, it is not matter so much.
I ask for this example only.
An isomorphism between algebraic structures of the same type is commonly defined as a bijective homomorphism.
READ this article on homomorphisms
Isomorphism is a very general concept that appears in several areas of mathematics. The word derives from the Greek iso, meaning "equal," and morphosis, meaning "to form" or "to shape."
Formally, an isomorphism is bijective morphism. Informally, an isomorphism is a map that preserves sets and relations among elements. "A is isomorphic to B" is written A=B. Unfortunately, this symbol is also used to denote geometric congruence.
An isomorphism from a set of elements onto itself is called an automorphism.