not really sure how to go about this problem...
define a relation R on , the set of functions from R to R by f R g if f(0) = g(0). prove that R is an equivalence relation on . let f(x) = x for all . describe [f].
Well, whoever gave you this problem expects you to know the definition of "equivalence relation"!
An equivalence relation, xRy, on a set is an equivalence relation if
1) It is "reflexive": xRx for every x in the set.
Suppose f is a function from R to R. Is fRf? That is, is f(0)= f(0)?
2) It is "symmetric ": if xRy then yRx.
Suppose f and g are such functions and fRg (that is, f(0)= g(0)). Is gRf (is g(0)= g(0))?
3) It is "transitive": if xRy and yRz then xRz.
Suppose f, g, and h are such functions, fRg (that is, f(0)= g(0)) and gRh (that is, g(0)= h(0)). Is fRh (is f(0)= h(0))?
[f] is the "equivalence class" of f, the set of all functions that are equivalent to f(x)= x. If g(x) is equivalent to that, if xRg, what must be true about g(x)?