Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook