## anilorap 3 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.

1. KingGeorge

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. anilorap

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

3. KingGeorge

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.

4. anilorap

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

5. anilorap

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

6. KingGeorge

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...

7. KingGeorge

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 $$a\;'=0$$

8. anilorap

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

9. KingGeorge

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.

10. anilorap

thank you very much

11. KingGeorge

You're welcome.