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Math Help - Family of subsets

  1. #1
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    Family of subsets

    Prove: If  f: A \rightarrow B is a function and {  {C_\lambda \mid \lambda \epsilon \Lambda}} is a family of subset of A, then  f  ( \bigcup_{\lambda \epsilon \Lambda} C_\lambda )= \bigcup_{\lambda \epsilon \Lambda} f(C_\lambda).

    I not getting family of subsets. How do I start this off. Assume x belongs to the left side???
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  2. #2
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    You are proving that the union of images is equal the image of a union.
    Because you are working with images you are using existential operators.
    Therefore it is best to prove each is a subset of the other.
    If t \in f\left( {\bigcup\limits_\Lambda  {C_\lambda  } } \right) then \left( {\exists x \in \bigcup\limits_\Lambda  {C_\lambda  } } \right)\left[ {f(x) = t} \right]. Then proceed to show that t \in \bigcup\limits_\Lambda  {f\left( {C_\lambda  } \right)} .

    Then reverse the process.
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