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Math Help - well defined

  1. #1
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    well defined

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

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

    3. Let  f: A \to B be a surjective map of sets. Prove that the relation  a \sim b \Longleftrightarrow f(a) = f(b) is an equivalence relation whose equivalence classes are the fibers of  f . So basically I have to prove reflexivity, symmetry, and transitivity? How about the fibers of  f ?
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  2. #2
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    Quote Originally Posted by heathrowjohnny View Post
    3. Let  f: A \to B be a surjective map of sets. Prove that the relation  a \sim b \Longleftrightarrow f(a) = f(b) is an equivalence relation whose equivalence classes are the fibers of  f . So basically I have to prove reflexivity, symmetry, and transitivity? How about the fibers of  f ?
    I am not quite sure what you are asking.

    The fibers of functions are the pre-images of singleton sets:
    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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