Let α and β be cardinal numbers. The cardinal sum of α and β, denoted α + β, is the cardinal | A ∪ B |, where A and B are disjoint sets such that | A | = α and | B | = β.
(a) Prove that the sum is well-defined. That is, if | A | = | C |, | B | = | D |, A ∩ B = ∅, and C ∩ D = ∅, then | A ∪ B | = | C ∪ D |.
(b) Prove that the sum is commutative and associative. That is, for any cardinals α, β, and γ, we have α + β = β + α and α + ( β + γ ) = (α + β ) + γ.
(c) Show that n + ℵ0 = ℵ0 for any finite cardinal n.
(d) Show that ℵ0 + ℵ0 = ℵ0.
(e) Show that ℵ0 + c = c.
(f ) Show that c + c = c.