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Math Help - inverse image question, is it correct?

  1. #1
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    inverse image question, is it correct?

    Let X and Y be sets , let A \subseteq X and B \subseteq Y, and let f:X \times Y \longrightarrow X be a projection map.

    Show that f^{-1}(A) = A \times Y.


    I wonder if the question is correct. For example, let X = {1, 2} and Y = {a,b}. Then, X \times Y = {(1,a), (1,b), (2,a), (2,b)}.

    Suppose f(1,a)= f(1,b)=f(2,a)=2, and f(2,b)=1.

    Suppose A = {1} \subseteq {1,2} = X.

    Then f^{-1}(A) = {(2,b)}  \neq A \times Y = {(1,a), (1,b)}.

    Please check whether I am right that the proposition was wrong from the question.
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  2. #2
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    Quote Originally Posted by armeros View Post
    Let X and Y be sets , let A \subseteq X and B \subseteq Y, and let f:X \times Y \longrightarrow X be a projection map. Show that f^{-1}(A) = A \times Y.

    I wonder if the question is correct. For example, let X = {1, 2} and Y = {a,b}. Then, X \times Y = {(1,a), (1,b), (2,a), (2,b)}.
    You have the projection wrong.
    f_x(1,a)=f_x(1,b)=1~\&~f_x(2,a)=f_x(2,a)=2.
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  3. #3
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    Oh, how careless I am.

    I thought a projection is just another name for a function. The way I define in my inquiry was to define a function, not a projection.

    Sorry to bother you with such a careless question.

    Thanks a lot!
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