1.Determine whether the following functions $\displaystyle f $ are well defined: $\displaystyle f: \bold{Q} \to \bold{Z} $ defined by $\displaystyle f(a/b) = a $ and $\displaystyle f: \bold{Q} \to \bold{Q} $ defined by $\displaystyle f(a/b) = a^{2}/b^{2} $. I would say $\displaystyle f(a/b) = a $ is not well defined because, for example, $\displaystyle f(1/2) = 1 $, and $\displaystyle f(2/4) = 2 $.

2.Determine whether the function: $\displaystyle f: \bold{R}^{+} \to \bold{Z} $ defined by mapping a real number $\displaystyle r $ to the right of the decimal point in a decimal expansion of $\displaystyle r $ is well defined. I would say no, because $\displaystyle f(0.999) = 9 $ while $\displaystyle f(1) = 0 $.

3.Let $\displaystyle f: A \to B $ be a surjective map of sets. Prove that the relation $\displaystyle a \sim b \Longleftrightarrow f(a) = f(b) $ is an equivalence relation whose equivalence classes are the fibers of $\displaystyle f $. So basically I have to prove reflexivity, symmetry, and transitivity? How about the fibers of $\displaystyle f $?