Results 1 to 3 of 3

Math Help - [SOLVED] Finally in LaTex: Pre-images and indexed set.

  1. #1
    Member ilikedmath's Avatar
    Joined
    Sep 2008
    Posts
    98

    Exclamation [SOLVED] Finally in LaTex: Pre-images and indexed set.

    Let f be a function from X into Y.
    If is a family of subsets of Y, prove that = .

    My thinking so far (which isn't a lot because I still don't have a grasp on this concept of indexed sets):

    So how I'm understanding this is that the question is asking me to prove that the pre-image (and not the inverse function!) of the intersection of a certain family of subsets of Y (the set {
    B_\alpha}) is the intersection of the pre-images of B_\alpha .

    The pre-image is in the domain. In this case the domain is X. So I have to show that the intersection of the pre-images of
    B_\alpha is the same as the pre-image of the intersection of all sets B_\alpha.

    Is that even close?
    Thanks for your time.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,712
    Thanks
    1641
    Awards
    1
    u \in \left( { \cap _\alpha  f^{ - 1} (B_\alpha  )} \right) \Leftrightarrow \left( {\forall \alpha  \in A} \right)\left[ {u \in f^{ - 1} (B_\alpha  )} \right]
    \begin{gathered}<br />
   \Leftrightarrow \left( {\forall \alpha  \in A} \right)\left[ {f(u) \in (B_\alpha  )} \right] \hfill \\<br />
   \Leftrightarrow \left[ {f(u) \in  \cap _\alpha  (B_\alpha  )} \right] \hfill \\<br />
   \Leftrightarrow u \in f^{ - 1} \left( { \cap _\alpha  (B_\alpha  )} \right) \hfill \\ <br />
\end{gathered}

    That is a start. Can you do the details?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member ilikedmath's Avatar
    Joined
    Sep 2008
    Posts
    98
    Quote Originally Posted by Plato View Post
    u \in \left( { \cap _\alpha f^{ - 1} (B_\alpha )} \right) \Leftrightarrow \left( {\forall \alpha \in A} \right)\left[ {u \in f^{ - 1} (B_\alpha )} \right]
    \begin{gathered}<br />
\Leftrightarrow \left( {\forall \alpha \in A} \right)\left[ {f(u) \in (B_\alpha )} \right] \hfill \\<br />
\Leftrightarrow \left[ {f(u) \in \cap _\alpha (B_\alpha )} \right] \hfill \\<br />
\Leftrightarrow u \in f^{ - 1} \left( { \cap _\alpha (B_\alpha )} \right) \hfill \\ <br />
\end{gathered}

    That is a start. Can you do the details?
    I think I can see it now. Thanks!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How to insert images in latex
    Posted in the LaTeX Help Forum
    Replies: 2
    Last Post: September 13th 2008, 01:28 PM
  2. How to Render Latex images
    Posted in the LaTeX Help Forum
    Replies: 1
    Last Post: May 12th 2008, 04:13 PM

Search Tags


/mathhelpforum @mathhelpforum