Final answer:
Sets are not isomorphic to their duals due to the difference in dimensions and properties.
Step-by-step explanation:
An isomorphism is a bijective and structure-preserving map between two mathematical objects. In this case, we are considering sets and their duals. The dual of a set is the set of all functions from the set to the field of scalars (usually the real numbers or complex numbers).
Now, let's suppose that a set and its dual are isomorphic. This would mean that there exists a bijective function that preserves the structure between the two sets.
However, sets and their duals have different dimensions. A set consists of elements, while its dual consists of functions. These two types of objects have different properties and cannot be mapped to each other bijectively. Therefore, sets are not isomorphic to their duals.