Question

Show that the relation R on the set of differentiable functions from R to R consisting of all pairs of functions f, g such that f(x) -() is an equivalence relation. What functions are in the equivalence class of f(x) ??

Answer 1

For the first part of proving that f'(x) = g'(x) is an equivalence relation:

1.) Reflexivity:

A differentiable function will always be equal to its differentiation. So, f'(x) = f;(x) always holds true proving that the relation is reflexive.

2.) Symmetric:

If, there are two functions f and g, such that f'(x) = g'(x), then since = operation is commutative, it implies that g'(x) = f'(x), which proves the symmetric nature of the relation.

3.) Transitive:

If, for functions f, g and h, we have f'(x) = g'(x) and g'(x) = h'(x), we always have f'(x) = h'(x). So, if fRg and gRh, we have fRh, which proves transitivity.

Since all three holds true, it proves that the given relation is an equivalence relation.

For the equivalence class:

All the functions holding equivalence relation to the given function are in the equivalence class. Since, the relation is f'(x) = g'(x), it means that every function whose derivative is equal f'(x) is in the equivalence class.

This simply means that any function whose derivative equals 2*x is in the equivalence class.

Formally, any differentiable function g(x), having g'(x) = 2*x is in the equivalence class.

Please ask in comments if you have any doubt.