1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Uncountable: A is a countable subset of an uncountable set X, prove X \ A uncountable

  1. Oct 11, 2008 #1
    Statement to prove:
    If A is a countable subset of an uncountable set X, prove
    that X \ A (or "X remove A" or "X - A") is uncountable.

    My work so far:
    Let A be a countable subset of an uncountable set X.
    (N denotes the set of all naturals)
    So A is equivalent to N (or "A ~ N") by the definition of countable.
    X is not equivalent to N since X is uncountable.
    Assume X \ A is countable. That is (X \ A) ~ N. Thus there is a bijection from (X \ A) to N. However since X is uncountable it is not guaranteed there is an n in N such that there is an x in X that maps to n. And so X \ A is uncountable.

    I know that proof is incomplete and may even be completely off :|
    I am really struggling in this class (Intro to Real Analysis) and feel as if it will be my GPA-killer (but I don't want it to be!). I'm having a hard time with the countable/uncountable concepts we covered last week.

    Any help is greatly appreciated! Thank you for your time.
     
  2. jcsd
  3. Oct 11, 2008 #2
    Re: Uncountable: A is a countable subset of an uncountable set X, prove X \ A uncount

    If both A and X\A were countable, what would that imply about the countability of X = A union X\A?
     
  4. Oct 11, 2008 #3

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    Re: Uncountable: A is a countable subset of an uncountable set X, prove X \ A uncount

    "Thus there is a bijection from (X \ A) to N. However since X is uncountable it is not guaranteed there is an n in N such that there is an x in X that maps to n. And so X \ A is uncountable"
    Not bad at all, but you need to develop precision and what you need to prove.
    Here, I think your intuition was sort of correct (namely that IF X/A were countable, that would entail that all of X would be countable, something you know is forbidden)

    You are struggling, but in your case, you are moving upwards, whatever you might feel right now.
    :smile:
     
  5. Oct 11, 2008 #4
    Re: Uncountable: A is a countable subset of an uncountable set X, prove X \ A uncount

    *light goes on in head* I think I see it now:
    Okay, I think I got it (thanks to your help!):

    My proof would be:

    Let A be a countable subset of an uncountable set X. Assume X \ A is countable.
    A union X \ A = X which is countable because the union of two countable sets is countable. BUT, this is a contradiction to the assumption that X is uncountable. Therefore, X \ A is uncountable. QED.

    Did I get it? :D
     
  6. Oct 11, 2008 #5

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    Re: Uncountable: A is a countable subset of an uncountable set X, prove X \ A uncount

    Yes.
     
  7. Oct 11, 2008 #6
    Re: Uncountable: A is a countable subset of an uncountable set X, prove X \ A uncount


    Thanks for the encouraging note :) My professor did give a list of tips that to prove something uncountable we'd need a contradiction and so we would first assume it countable. Thanks to everyone's helps/tips, I'm slowly getting the hang of this.
    This forum rocks and is a tremendous lifesaver! :) Thanks, everyone!
     
  8. Oct 11, 2008 #7
    Re: Uncountable: A is a countable subset of an uncountable set X, prove X \ A uncount

    Awesome, thanks again! :cool:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Uncountable: A is a countable subset of an uncountable set X, prove X \ A uncountable
  1. Uncountable sets (Replies: 5)

Loading...