Question

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.

Answer 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).

Answer 2

so should i define f. that means should i specify f in order to find the classes?

Answer 3

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.

Answer 4

so can i say
x^2/R= X^2+C, C= is a constant

Answer 5

or
ax^2+bx+C , a,b,c are constant.. can i define the classes like that?

Answer 6

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...

Answer 7

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 \(a\;'=0\)

Answer 8

mmm.. sounds good.. i think ill never come up with that.. :( why?

Answer 9

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.

Answer 10

thank you very much

Answer 11

You're welcome.