(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let F be the set of all functions f mapping R into R that have derivatives of all orders. Determine whether p is an isomorphism of the first binary structure with the second.

1. [tex]<F, +>[/tex] with [tex]<R, +>[/tex] where p(f) = f'(0)

2. [tex]<F, +>[/tex] with [tex]<F, +>[/tex] where p(f)(x) = [tex]\int^{x}_{0} f(t)dt[/tex]

3. [tex]<F, +>[/tex] with [tex]<F, +>[/tex] where p(f)(x) = [tex]d/dx \int^{x}_{0} f(t)dt[/tex]

4. [tex]<F, \cdot>[/tex] with [tex]<F, \cdot>[/tex] where p(f)(x) = [tex]x \cdot f(x)[/tex]

2. Relevant equations

3. The attempt at a solution

Some ideas I have:

I think #1 is false, since f=x and f=x+1 can have the same derivative at 0. Isomorphism requires that p be bijective.

#3 is true. Simplifying gives p(f)(x)=f(t) for all f in F and x in R.

I'm not so sure about #2, and #4... The book says they are both false, but I don't really understand it. I'd appreciate any hints/suggestions.

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# Homework Help: Determining whether the map is an isomorphism

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