The union of two countable set is a countable set.
What would happen if were countable?
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.
*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.
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!