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Math Help - Inverse Image Intersection Equality and Counterexample of a Function Proof

  1. #1
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    Inverse Image Intersection Equality and Counterexample of a Function Proof

    Suppose f is a function with sets A and B.
    1. Show that:
    I_{f} \left(A \cap B\right) = I_{f} \left(A\right) \cap I_{f} \left(B\right)

    2. Show by giving a counter example that:
    f\left(A \cap B\right) \neq f\left(A\right) \cap f \left(B\right)

    1.
    Let c be an element of I_{f} \left(A \cap B\right).
    By the definition of I_{f} \left(A \cap B\right) , there is a d\in(A \cap B) so that I_{f}(d)=c.
    Since, d\in(A \cap B), d \in A & d \in B. Since d\inA, I_{f}(d)\in I_{f}(A). This follows alongside d\inB, I_{f}(d)\inI_{f}(B).
    Since I_{f}(d)=c \in I_{f}(A) and I_{f}(d)=c \in I_{f}(B), c = I_{f}(A)\capI_{f}(B).

    Thoughts? Also would I need to show that the I_{f}(A)\capI_{f}(B) \in I_{f} \left(A \cap B\right) to show true equality?

    2.
    f\left(A \cap B\right) \neq f\left(A\right) \cap f \left(B\right)
    I'm thinking either the absolute value function or a square function of some sort would show that it is not equal. Though, I'm not sure how to proceed with depicting the counter example.
    Last edited by Icehuu; September 7th 2010 at 06:55 PM.
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  2. #2
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    ii.)

    f(x)=|x| {-3,...,-1} = A and {1,...,200} = B

    There is no intersection between A ∩ B. However, there is an intersection with f(A) ∩ f(B) that gives the set {1,3}.

    Thus, {null} != {1,3}.

    Any suggestions on i.)?
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  3. #3
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    Sorry, what is I_f? This is not a standard notation. It sounds like something inverse to f, but:
    Let c be an element of I_{f} \left(A \cap B\right).
    By the definition of I_{f} \left(A \cap B\right), there is a d\in(A \cap B) so that I_{f}(d)=c.
    Is c an element of the domain, and is d an element of the codomain of f? But the inverse of a function is in general a relation. One can think of it as a function that maps elements of the codomain into subsets of the domain. Maybe I am thinking in the wrong direction...

    Suppose f is a function with sets A and B.
    What does this mean?
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