Let m, n be positive integers with m | n, and let k be any integer. Show that f :
Zn-> Zm defined by f([x]n) = [x]m^k, for all [x]n belong to Zn
is a well-defined function.
Hint: We know that g : Z×
m ! Z×
m defined by g([x]m) = [x]k
m is a function if k is a
positive integer.
Do you mean that [x]n is the equivalence class of numbers that are equivalent mod n and [x]m^k is the equivalence class of numbers that are equivalent mod m^k? That is, [x]3 is an equivalence class mod 3 so that with n= 6, m= 3 (so that m|n) and k= 2, you are asked to show that the function f that maps any number, x, in to in is "well defined", that is that two numbers in the same equivalence class in are mapped into numbers in the same equivalence class in .