Introduction:
R and R* are both sets of real numbers, but they have different structures and properties. In this response, we will explore whether R and R* are isomorphic, that is, if there exists a bijective function between them that preserves the algebraic operations.

Definition of R and R*:
R is the set of all real numbers, which includes both positive and negative numbers as well as zero. R* is the set of all positive real numbers, excluding zero.

Isomorphism:
To determine if R and R* are isomorphic, we need to find a function that satisfies the following conditions:
1. The function must be bijective, meaning it is both injective (one-to-one) and surjective (onto).
2. The function must preserve the algebraic operations, such as addition and multiplication.

Injectivity:
To establish injectivity, we need to show that no two distinct elements in R map to the same element in R*. However, it is not possible to find a function that maps positive and negative numbers to positive numbers only.

Surjectivity:
To establish surjectivity, we need to show that every element in R* has a pre-image in R. Since R* contains only positive numbers, it is not possible to find a pre-image for zero in R*.

Preservation of Algebraic Operations:
In order for R and R* to be isomorphic, the function between them must preserve the algebraic operations. However, it is not possible to find a function that preserves addition and multiplication between R and R*.

Conclusion:
Based on the above analysis, we can conclude that R and R* are not isomorphic. The lack of a bijective function between the two sets that preserves the algebraic operations prevents them from being isomorphic.