The question is as follows:
Prove that if R is uncountable and T is a countable subset of R, then the cardinality of R\T is the same as the cardinality of R.
What i have:
I know that R is uncountable so it has a countable subset (this is a theorem of uncountable sets). Let T be this subset so T has the same cardinality as the set of natural numbers(by the definition of T being countable). My intution is telling me that we will have to use the cantor bernstein theorem to prove they have same cardinalities. So for that the first thing i did showed was that |R\T| <= |R| (pretty clear as its R\T, |R| means cardinality of R).i got a bit confused while trying to show that |R| <= |R\T|. Maybe we can show this by defining a bijection f from R --> R\T, such that f(x) = x when x is in R\T, but i dont know what to do if x is in R and not in R\T, if i can define that function then i can conclude |R| <= |R\T|, then use the cantor-bernstein theorem and then im done. Or maybe im doing this all wrong i cant think of any other way Any help is much appreciated!!