Use Theorem 536 to prove that the union of a finite pairwise

Use Theorem 5.3.6 to prove that the union of a finite pairwise disjoint family of denumerable sets {A_i: i=1,2,...,n} is denumerable.

(HOW DO YOU USE THE THM? ON WHICH STEP?)

Theorem 5.3.6 If A and B are disjoint denumerable sets, then AUB is denumerable. 1-1 Proof B. Define h: N Let f: N A and g: N AUB via onto onto n 1 if n is odd h(n) if n is even

Solution

Let us define the finite disjoint family of denumerable sets.

the family of odd numbers O, the family even numbers,E

now,

O={1,3,5,7,...}

E={2,4,6,8,...}

O and E are denumerable since they are infinite.

now o union e={1,2,3,4,...}

O U E ={SET OF NATURAL NUMBERS}

set of natural numbers is also denumerable.

thus the theorem is proved.

use of the theorem is clearly brought out in the example .