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Abstract Algebra Question: Let R be an equivalence relation on the set of real differentiable functions defined by fRg iff f and g has the same first derivative, i.e. f' = g'. Determine the equivalence classes of f for each f element of R.

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The equivalence class of f would just be the set of functions such that every function in the set is equal to f plus a constant term (The derivative of a constant term is 0).
so should i define f. that means should i specify f in order to find the classes?
f is just a real differentiable function. The most important thing is to define what a constant is. In this case, you should define the constant by saying it's a continuous differentiable function such that it's first derivative is 0.

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so can i say x^2/R= X^2+C, C= is a constant
or ax^2+bx+C , a,b,c are constant.. can i define the classes like that?
Partially. The only problem with defining classes like that, is you would have an infinite number of definitions, and you wouldn't include function like sin(x) and \(e^x\). Rather, I might recommend a definition similar to the following...
Let \(f, g, h, a\) be continuous differentiable functions on the set of real numbers, and define an equivalence relation R such that \(f\text{R} g \;\Leftrightarrow \; f'=g'\). Then \(h\) is in the equivalence class of \(f\) iff \(h=f+a\) where \(a\;'=0\)
mmm.. sounds good.. i think ill never come up with that.. :( why?
It just takes practice. When I first started doing this, it took me a while to get used to it. Just make sure to proofread your definitions.
thank you very much
You're welcome.

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