Solve Binary Operations Question: Prove Isomorphism

In summary, the conversation discusses the definition of an isomorphism between two sets, Z and Z, using the function phi, where phi is defined as n+1 for n in Z. The conversation also includes a question about defining an operation, *, for the sets that will make phi an isomorphism. The conversation ends with the speaker realizing they have solved the problem.
  • #1
Jumblebee
12
0
So I have really been struggling with this question. The original question said: The map [tex]\varphi[/tex]:Z->Z defined by [tex]\varphi[/tex](n)=n+1 for n in Z is one to one and onto Z. For (Z, . ) onto (Z,*) (i am using . for usual multiplication) define * and show that * makes phi into an isomorphism.
I know that the operation must be m*n=mn-m-n+2. But I get stuck in proving that the operations are preserved. When I do [tex]\varphi[/tex](m.n) i get mn+1. and i can't get [tex]\varphi[/tex](m). [tex]\varphi[/tex](n) to work. I think I am doing something wrong. Can anyone help?
 
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  • #2
Never mind! i just got it to work!
 

1. What is a binary operation?

A binary operation is a mathematical operation that takes two operands and produces a single result. Some common examples of binary operations include addition, subtraction, multiplication, and division.

2. What does it mean to prove isomorphism in binary operations?

To prove isomorphism in binary operations means to show that two algebraic structures have the same structure and can be mapped onto each other in a way that preserves the operation between them. This means that the same operation performed on elements in one structure will yield the same result as performing the operation on their corresponding elements in the other structure.

3. How do you prove isomorphism in binary operations?

To prove isomorphism in binary operations, you must first define the operations and structures involved. Then, you must show that there is a one-to-one mapping between the elements of the two structures and that the operation is preserved under this mapping. This can be done through equations, tables, or other mathematical methods.

4. Why is proving isomorphism important in binary operations?

Proving isomorphism in binary operations is important because it allows us to understand and compare different algebraic structures in a meaningful way. It also helps us to identify patterns and similarities between structures, which can aid in solving complex problems and creating new mathematical concepts.

5. Can binary operations be isomorphic to themselves?

Yes, it is possible for a binary operation to be isomorphic to itself. In this case, the structure and elements remain the same, but the mapping between them may change. This can happen when the operation is commutative, meaning that the order of the operands does not affect the result.

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