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Math Help - [SOLVED] Uncountable: A is a countable subset of an uncountable set X, prove X \ A un

  1. #1
    Member ilikedmath's Avatar
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    Question [SOLVED] Uncountable: A is a countable subset of an uncountable set X, prove X \ A un

    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.
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  2. #2
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    The union of two countable set is a countable set.
    What would happen if X \setminus A were countable?
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  3. #3
    Member ilikedmath's Avatar
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    Quote Originally Posted by Plato View Post
    The union of two countable set is a countable set.
    What would happen if X \setminus A were countable?
    *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.
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  4. #4
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    Quote Originally Posted by ilikedmath View Post
    *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.
    Well done.
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  5. #5
    Member ilikedmath's Avatar
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    Thanks!

    Quote Originally Posted by Plato View Post
    Well done.
    Thank you, and I definitely could not have done it without help.
    I tend to psych myself out even before I attempt a proof because I always think it has to be long and complex. But this proof is really "easy" once you see it and it's short too. Wow, math is amazing Thanks again!
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