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Math Help - Image and Inverse Images for Projection Functions

  1. #1
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    Question Image and Inverse Images for Projection Functions

    Let A and B be nonempty sets and  \pi_1 : A \times B \rightarrow A be the projection function  \pi_1 (x,y) = x for  (x,y) \epsilon A \times B .

    (a) For  C \subset A , determine the inverse image set  \pi^{-1} (C) .

    (b) For  C \subset A, D \subset B and  D \neq \emptyset , determine the image  \pi_1 (C \times D) .
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  2. #2
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    Quote Originally Posted by onemore View Post
    Let A and B be nonempty sets and  \pi_1 : A \times B \rightarrow A be the projection function  \pi_1 (x,y) = x for  (x,y) \epsilon A \times B .

    (a) For  C \subset A , determine the inverse image set  \pi^{-1} (C) .

    (b) For  C \subset A, D \subset B and  D \neq \emptyset , determine the image  \pi_1 (C \times D) .
    You probably mean  \pi_1^{-1} (C) , not  \pi^{-1} (C) . Have you at least tried to solve this problem?

    I thinkt the if you understand the definitions and if you draw a picture or think for a while, you see immediately that:
    \pi_1^{-1} (C) = C\times Band
    \pi_1 (C\times D) = C.

    One question, to see whether you understand the problem - can you explaine why D \ne \emptyset is needed in (b)?
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