Is the function f : S → {1,−1} f : [x]∼ ↦ (−1) well defined?

Question

a∼b  ⇔  a and b have the same remainder from division by n.

Let n=3, and consider the quotient set S = ℤ/∼ (we of course have that |S| = 3).

Is the function f : S → {1,−1}

f : [x]∼ ↦ (−1) well defined? Prove your answer.

Answer 1

Must be a typo, because any function that assigns the same value to every element is well-defined. In general, any function f: A --> {b} for some element b defined by f(a)=b for every a in A is well-defined: For if a=a', we always have f(a)=b and f(a')=b and hence a and a' map to the same element, namely b.