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Proof on functions of an intersection of sets

  1. Sep 3, 2008 #1
    I'm working out of Abbott's Understanding Analysis and I'm trying to show the following,

    For an arbitrary function [itex]g :\mathbb{R}\longrightarrow \mathbb{R}[/itex] it is always true that [itex]g(A\bigcap B) \subseteq g(A) \bigcap g(B)[/itex] for all sets [itex]A, B \subseteq \mathbb{R}[/itex].

    I'm confused on how to get going with this--any help or hints would be greatly appreciated. Thanks.
     
  2. jcsd
  3. Sep 3, 2008 #2

    statdad

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    Homework Helper

    If [tex] X, Y [/tex] are sets for which you know the definitions or other properties, the classical way to show [tex] X \subset Y [/tex] is this:

    1: Pick an arbitrary [tex] a \in X [/tex]

    2: Use the definitions of the sets to show that [tex] a \in Y [/tex]

    As a start, if you know that [tex] a \in g(A \cap B)[/tex], then you know that there is a value [tex] x_0 [/tex] such that [tex] a = g(x_0) [/tex] and that [tex] x_0 \in A \cap B [/tex]. What else do you know about [tex] x_0 [/tex], and how can you use that information?
     
    Last edited: Sep 3, 2008
  4. Sep 4, 2008 #3
    Thanks, that really helps, I think I've got it now.
     
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