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
A_{1} U A_{2} U A_{3} U........U An is
denumerable , then since the sets

{A_{i} :1,2,3,....,n} are disjoint hence we can
define

A_{2} U A_{3} U........U An= B set

now A_{1} 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 A_{1} U B is denumerable , thus

A_{1} U A_{2} U A_{3} U........U An is
denumerable.

(Proved)

A countable union of disjoint countable sets is countable
Note: countable sets can be either finite or infinite
A countable union of countable sets is countable
Note: countable sets can be either finite or infinite

If A and B are denumerable sets, then A ∪ B is denumerable.
(This should be proven for disjoint and non-disjoint pairs of
sets.)

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