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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.
 2 years ago
 2 years ago
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.
 2 years ago
 2 years ago

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KingGeorgeBest ResponseYou've already chosen the best response.1
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).
 2 years ago

anilorapBest ResponseYou've already chosen the best response.0
so should i define f. that means should i specify f in order to find the classes?
 2 years ago

KingGeorgeBest ResponseYou've already chosen the best response.1
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.
 2 years ago

anilorapBest ResponseYou've already chosen the best response.0
so can i say x^2/R= X^2+C, C= is a constant
 2 years ago

anilorapBest ResponseYou've already chosen the best response.0
or ax^2+bx+C , a,b,c are constant.. can i define the classes like that?
 2 years ago

KingGeorgeBest ResponseYou've already chosen the best response.1
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...
 2 years ago

KingGeorgeBest ResponseYou've already chosen the best response.1
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 \(a\;'=0\)
 2 years ago

anilorapBest ResponseYou've already chosen the best response.0
mmm.. sounds good.. i think ill never come up with that.. :( why?
 2 years ago

KingGeorgeBest ResponseYou've already chosen the best response.1
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.
 2 years ago
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