Question

Using the following theorem: If A and B are disjoint denumerable sets, then A ∪ B is denumerable, prove the union of a finite pairwise disjoint family of denumerable sets {A_i :1,2,3,....,n} is denumerable

Answer 1

To prove that if A and B are disjoint denumerable sets, then A U B is denumerable

? denumerable ⟹?: ℤ+→?
? denumerable ⟹?: ℤ+→?

We need to construct a bijective function, ℎ:ℤ+→?∪?

So, define h:

ℎ(?)={?(?+1)if ?=2?+1, for ?∈{0,1,2,…}
else
h(i)= {g(n) if ?=2?, for ?∈{1,2,3,…}

Then, 'h' is bijective because both f and g are bijective by definition.

Hence, ?∪? is denumerable, as required.

Now if there are many disjoint sets and we have to prove A₁ U A₂ U A₃ U........U An is denumerable , then since the sets
{A_i :1,2,3,....,n} are disjoint hence we can define
A₂ U A₃ U........U An= B set

now A₁ U B is still a disjoint set since A1 is disjoint with {Ai for i=2 to n} so A1 will be disjoint with their intersection as well.

Hence A₁ U B is denumerable , thus
A₁ U A₂ U A₃ U........U An is denumerable.

(Proved)