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Thread: Partitions and definitions

  1. #1
    Super Member Deadstar's Avatar
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    Partitions and definitions

    Am i doing this right? And in the case of (c) how do you prove it?

    Let B be a set.
    a) Define what it means to say that $\displaystyle B_1 , B_2$ give a partition of B, where $\displaystyle B_1 , B_2$ are subsets of B.

    b) Let f:A -> B be a function. Suppose that C is a subset of B. Write down the definition of $\displaystyle f^{-1}(C)$.

    c) Suppose that $\displaystyle B_1 , B_2$ is a partition of B. Prove, using your definitions, that $\displaystyle f^{-1}(B_1) , f^{-1}(B_2)$ is a partition of A.

    What i think the answers are...

    a) i) $\displaystyle B_1 , B_2$ are non-empty
    ii) since 1 is not equal to 2, $\displaystyle {B_1}\cap{B_2} = \phi$
    iii) $\displaystyle B = \bigcup B_i$ The U thing goes from i=1 to n but im not sure how to do that (does n=2?)

    b) $\displaystyle f^{-1}{C} = {(a \in | f(a) \in C)}$

    c) ?

    Am i right so far and if so how is c done?
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  2. #2
    MHF Contributor

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    You are correct in parts a & b.
    However, part c is not necessarily true.
    If the function is surjective (onto), then it is true.

    If $\displaystyle f$ is not surjective define $\displaystyle B_1 = f(A)\,\& \,B_2 = B\backslash f(A)$. That is a partition of $\displaystyle B$.
    It is counter-example to part c.
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