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Math Help - Help with proving some facts

  1. #1
    Newbie
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    Sep 2009
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    Help with proving some facts

    Hi,
    I have to prove the following facts, but I am not sure if I figured them out correctly:

    f : X --> Y and F : P(X) --> P(Y )

    1) If A1 is a subset of A2, then F(A1) is a subset of F(A2)

    For this one, I have the following:
    For all x that are elements of A1 and A2
    (there is a y that is an element of F(A1) s.t. F(x)=y) and (y is an element of F(A2) s.t. F(X)=y)

    Therefore,
    the fact that x is an element of A1 implies that there is a y that is an element of F(A2) s.t. F(x) = y

    Hence, F(A1) is a subset of F(A2)

    (Here I feel that, I missed a step constructing the proof with quantifiers.)

    Also,

    2) For every A that is an element of P(X), A is a subset of the inverse function of F(A)

    For this one, I have:
    (For all x that are elements of A, there is a y that is an element of F(A) s.t. F(x)=y s.t. the inverse of F(y)=x) implies that (for all x belonging to A, x is an element of the inverse of F(F(A))) implies that (A belongs to the inverse of F(F(A))).

    I am also not sure about my logic here, do you guys agree with me?
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  2. #2
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    Quote Originally Posted by kaka87 View Post
    Hi,
    I have to prove the following facts, but I am not sure if I figured them out correctly:

    f : X --> Y and F : P(X) --> P(Y )

    1) If A1 is a subset of A2, then F(A1) is a subset of F(A2)

    For this one, I have the following:
    For all x that are elements of A1 and A2
    (there is a y that is an element of F(A1) s.t. F(x)=y) and (y is an element of F(A2) s.t. F(X)=y)

    Therefore,
    the fact that x is an element of A1 implies that there is a y that is an element of F(A2) s.t. F(x) = y

    Hence, F(A1) is a subset of F(A2)

    (Here I feel that, I missed a step constructing the proof with quantifiers.)

    Also,

    2) For every A that is an element of P(X), A is a subset of the inverse function of F(A)

    For this one, I have:
    (For all x that are elements of A, there is a y that is an element of F(A) s.t. F(x)=y s.t. the inverse of F(y)=x) implies that (for all x belonging to A, x is an element of the inverse of F(F(A))) implies that (A belongs to the inverse of F(F(A))).

    I am also not sure about my logic here, do you guys agree with me?
    1. I think you have it, or if not, you're on the right track.

    Let y\in F(A_1). Then \exists x\in A_1 such that F(x)=y. Because A_1\subset A_2, x\in A_2 also. Thus y=F(x)\in F(A_2), so F(A_1)\subset F(A_2).

    2. I'm not sure I understand the second question. It can't be asking you to prove that A\subset F^{-1}(F(A)), because that's trivial, but I don't know what else it could be asking.
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