Question

Prove that an uncountable set - countable set is uncountable.

Answer 1

Let A be uncountable
B be countable subset of A
Prove A - B is uncountable.

Assume (A - B) is countable
A = B U (A - B)
The union of 2 sets is countable
Therefore A must be countable,
but A is not countable.
Therefore our assumption is false.
Therefore (A-B) is uncountable.