Isomorphism between (R, +) and (R*, *)

To determine whether the additive group (R, +) is isomorphic to the multiplicative group (R*, *), we need to check if there exists a bijective function from (R, +) to (R*, *) that preserves the group structure.

Definition of Isomorphism:
An isomorphism between two groups G and H is a bijective function f: G -> H such that for any elements a, b in G, f(a * b) = f(a) * f(b), where * denotes the group operation.

Checking for Bijectivity:
To establish an isomorphism, we need to find a bijective function between (R, +) and (R*, *). However, we cannot find such a function because the cardinality (size) of (R, +) and (R*, *) is different. The set of real numbers R is uncountably infinite, while the set of non-zero real numbers R* is countably infinite. Therefore, it is not possible to establish a bijective correspondence between these two sets.

Checking for Group Structure Preservation:
Even if we ignore the issue of cardinality, we can see that the group operation in (R, +) is addition, while the group operation in (R*, *) is multiplication. These two operations are fundamentally different. Addition is commutative and has an identity element (0), while multiplication is not commutative and has an identity element (1). Therefore, it is not possible to find a function that preserves the group structure between (R, +) and (R*, *).

Conclusion:
In summary, (R, +) is not isomorphic to (R*, *) because there is no bijective function that preserves the group structure between these two groups. Additionally, the cardinality of the sets is different, further preventing the establishment of an isomorphism.