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**Subject Mathematics Abstract Algebra**

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1. We assume by contradiction that (R, +) ~ (R*, x).

Then neutral element from the left side must be mapped to neutral element from the right side.

This means f(0, +)=f(1,x) (0 is the neutral element for (R,+) and 1 is the neutral element for (R*,x).

Also, since the two are isomorphic => f is bijection => f is both one-to-one and onto....

Then neutral element from the left side must be mapped to neutral element from the right side.

This means f(0, +)=f(1,x) (0 is the neutral element for (R,+) and 1 is the neutral element for (R*,x).

Also, since the two are isomorphic => f is bijection => f is both one-to-one and onto....

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