An isomorphism is a type of function that preserves the structure and operations of mathematical objects, such as groups, rings, and fields. It is a one-to-one and onto mapping between two algebraic structures that preserves the properties and relationships between their elements.
Homomorphism is a more general type of function that preserves the operations of mathematical structures, but not necessarily the structure itself. Isomorphism is a stricter form of homomorphism in which the function preserves both the operations and the structure of the objects.
Yes, two isomorphic structures can be considered equivalent in the sense that they have the same abstract structure and operations, even if the elements themselves are different. For example, two groups may have different elements, but if they are isomorphic, they have the same group structure and operations.
To prove that two algebraic structures are isomorphic, you must show that there exists a bijective homomorphism between them. This can be done by explicitly constructing the function and showing that it preserves the operations and structure of the objects, or by demonstrating that the structures have the same properties and can be mapped onto each other.
Isomorphism is a powerful tool in abstract algebra that allows us to study and understand complex mathematical structures by relating them to simpler or more familiar ones. It also helps us to identify important properties and relationships between different algebraic structures, and can be used to prove theorems and solve problems in various areas of mathematics.