1. ## well defined

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 $\displaystyle f$?

2. Originally Posted by heathrowjohnny
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 $\displaystyle f$?
I am not quite sure what you are asking.

The fibers of functions are the pre-images of singleton sets:
$\displaystyle f:A \mapsto B\quad ,\quad \overleftarrow f (\{ r\} ) = \left\{ {x \in A:f(x) = r} \right\}$.

If f is a surjection then the fibers form a partition of the set A.
Every partition of a set defines an equivalence on that set.
The cells of the partition, here the fibers, are the equivalence classes.