prove that (a,b) is uncountable set
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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.
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