Question

prove that (a,b) is uncountable set​

Answer 1

Answer:

Suppose that A \ B is countable. Then, A = (A \ B) ∪ B

is a union of two countable sets, hence A is countable, contrary to our

hypothesis.

(ii) As suggested in the hint, let C be a countably infinite subset of A \ B

(such C exists by Theorem 12 on page 33 in Pugh). Since B is countable

and C is countably infinite, their union B ∪ C is also countably infinite, so

there is a bijection φ : C → B ∪ C. Now define the map f : A \ B → A by

f(x) =

x if x ∈ (A \ B) \ C

φ(x) if x ∈ C.

It is clear that f is a bijection from A \ B to A.