Problem: Determine whether the given map $\displaystyle \phi$ is an isomorphism of the first binary structure with the second.

$\displaystyle <F,+> with <F,+>$ where $\displaystyle \phi(f)(x) = \frac{d}{dx}[\int_0^x d(t) dt]$

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Well, $\displaystyle \phi(f)(x) = \frac{d}{dx}[\int_0^x d(t) dt]$ is simply $\displaystyle f(x)$, so does that fact make this problem trivial? Since $\displaystyle \phi(f(x)) = f(x)$, it appears that $\displaystyle \phi$ doesn't do anything at all, it just returns what you put into it. Since both binary structures are the same, and the mapping function just spits out what you put into it, is this trivially isomorphic?