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Thread: well defined

  1. #1
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    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 $?
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    Quote Originally Posted by heathrowjohnny View Post
    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.
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