Prove that the set of all finite subsets of ω is countable. (Set theory)

Using exclusively induction, prove that, if X, X1, X2, . . . , Xn are sets,then X ∩(X1 ∪ X2 ∪ · · · ∪ Xn) = (X ∩ X1)∪(X ∩ X2)∪· · ·∪(X ∩ Xn).

Let {Fa}a€A be a collection of closed subsets of R. (a) Prove, from the definition of closed, that Naca Fa is closed. (b) Show by example that Uaca Fa need not be closed, and demonstrate that your example works.

Provide Cantor’s proof that the set (0, 1) is uncountable.