Let α and β be cardinal numbers. The cardinal product α β is defined to be the cardinal | A × B |, where | A | = α and | B | = β (a) Prove that the product is well-defined. That is, if | A | = | C |...

Let α and β be cardinal numbers. The cardinal product α β is defined to be the cardinal | A × B |, where | A | = α and | B | = β

(a) Prove that the product is well-defined. That is, if | A | = | C | and | B | = | D |, then | A × B | = | C × D |.

(b) Prove that the product is commutative and associative and that the distributive law holds. That is, for any cardinals α, β, and γ, we have

α β = β α, α ( β γ ) = (α β ) γ, and α ( β + γ ) = α β + α γ.

(c) Show that 0α = 0 for any cardinal α.

(d) Show that nℵ₀
= ℵ₀
for any finite cardinal n with n ≠ 0.

(e) Show that ℵ₀ℵ₀
= ℵ₀.

(f ) Show that cc = c.