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Math Help - Help with Intro to Abstract Math Class...

  1. #1
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    Help with Intro to Abstract Math Class...

    Let A and B be sets and X and Y be subsets of A. Let f : A --> B be
    a function. Prove that f(X) - f(Y) is a subset of f(X - Y).

    I'm not very comfortable with sets to begin with, but now that I've got functions thrown in and a really bad textbook, I'm lost. I've only gotten to "Let a be a subset of f(X) - f(Y). Then a is in f(X) and a is not in f(Y)." but now I'm not sure where to go next. Any help at all would be greatly appreciated. Thanks. )
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  2. #2
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    Quote Originally Posted by erinthered
    Let A and B be sets and X and Y be subsets of A. Let f : A --> B be
    a function. Prove that f(X) - f(Y) is a subset of f(X - Y).

    I'm not very comfortable with sets to begin with, but now that I've got functions thrown in and a really bad textbook, I'm lost. I've only gotten to "Let a be a subset of f(X) - f(Y). Then a is in f(X) and a is not in f(Y)." but now I'm not sure where to go next. Any help at all would be greatly appreciated. Thanks. )
    I do not know what you mean by f(x) and f(x-y). Can you explain yourself?

    Let me try it like this:
    1) f:A\to B (function)
    2) X\subseteq A (subset)
    3)Then, f[X]=\{f(x)|x\in X\}
    4)Thus, f[X]\subseteq B
    Because, since f(x),x\in A we have f(x)\in B because this function maps an element from A into B. But, x\in X implies x\in A thus, for all x\in X we have, f(x)\in B. Thus, for all e\in f[X] we have e\in B thus, f[X]\subseteq B
    But I do not know if you were asking this?
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  3. #3
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    f(X) is the image of X under f according to my textbook, same with f(Y). I'm thinking that f(X-Y) is supposed to be the image of the complement of Y in X and I have to prove that it's a subset of the complement of the image of Y in the image of X.

    I was trying to pick a random arbitrary element in the left hand side, then prove that it's also in the right hand side. I'm just not sure how to go about doing that.
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  4. #4
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    My proof so far (after corrected a stupid mistake.....) is:

    Let a be an element of f(X) - f(Y). Then a is in f(X) and is not in f(Y). Since a is in f(X), then a = f(x) for some x in X. Since x is in X, x is in X-Y. Hence a is an element of f(X-Y) and f(X) - f(Y) is a subset of f(X-Y).

    I think I'm skipping something or doing something wrong, because it just looks simpler than it should. Bleh. I don't know.
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  5. #5
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    Quote Originally Posted by erinthered
    f(X) is the image of X under f according to my textbook, same with f(Y). I'm thinking that f(X-Y) is supposed to be the image of the complement of Y in X and I have to prove that it's a subset of the complement of the image of Y in the image of X.

    I was trying to pick a random arbitrary element in the left hand side, then prove that it's also in the right hand side. I'm just not sure how to go about doing that.
    Okay I get you know, I was close though.

    Proofs on set theory I like to write out in works, or in easily understandable sets.
    You have f:A\to B
    X,Y\subseteq A
    Thus, X-Y=\{x\in X, x\not \in Y\}
    Thus, F[X-Y]=\{f(x)|x\in X,x\not \in Y\}
    Thus, F[X]=\{f(x)|x\in X\}
    Thus, F[Y]=\{f(x)|x\in Y\}
    Thus, F[X]-F[Y]=\{f(x)|f(x)\in F[X],f(x)\not \in F[Y]\}
    You need to show that,
    \{f(x)|f(x)\in F[X],f(x)\not \in F[Y]\}\subseteq \{f(x)|x\in X,x\not \in Y\}

    Which is true, you see why?
    (I just cannot complete the proof. SO tired now- got a bussiness law test soon. Got to go bye.)
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  6. #6
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    yay! I see it clearly now instead of the vague fuzz that I was seeing. Thanks. Luck on the Business Law test.
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  7. #7
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    Quote Originally Posted by erinthered
    yay! I see it clearly now instead of the vague fuzz that I was seeing. Thanks. Luck on the Business Law test.
    Thanks, let me add something.

    Consider two cases:
    1) F[X]-F[Y] empty.
    2) F[X]-F[Y] non-empty.

    If 1 then there is nothing to prove, because we defined the empty set to be the subset of any set.

    If 2 you need to show if x\in F[X]-F[Y] then x\in F[X-Y] then by definition of subset we have F[X]-F[Y]\subseteq F[X-Y]
    -------
    Finally, always know the difference between \subset and \subseteq. They mean different things.
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