Answer to Question #149495 in Abstract Algebra for Sourav Mondal

Question

Let R[x] denote the set of all polynomials in x with real coefficients. On R[x], define a

relation ~ by f(x) ~g(x) if f'(x) = g'(x), where f'(x) is the derivative of f(x). Show that ~ is an equivalence relation on R[x]. For any f(x) E R[x], determine the equivalence class [f(x)].

Answer 1

R – is equivalence relation, by definition, iif it is symmetric, reflexive, transitive

• Symmetric

Let f and g be differentiable functions. If f'(x) = g’(x) then g’(x) = f’(x).

• Reflexive

Let f be a differentiable function. Then f'(x) = f’(x).

• Transitive

Let f, g, h be differentiable. Then if f’(x) = g’(x) and g’(x) = h’(x) then f’(x) = h’(x).

By the Mean Value Theorem if f’(x) = g’(x) the f(x) = g(x) + C, where C – is some real number.

Thus, the equivalence class [f(x)] is modulo addition by a constant

[f(x)] = {g(x), g(x) = f(x) + C, C in R}