Use the results of exercise 28 and 29 to prove that a union of any two countable sets is countable.

To prove:

A union of any two countable sets is countable.

Given information:

1 A disjoint union of any finite set and any countably infinite set is countably infinite.

2. A union of any two countably infinite sets is countably infinite set.

Concept used:

Union of any two countably infinite sets is countably infinite.

Calculation:

Let A,B be two countable sets. Use the following cases to prove the given theorem.

Case 1:

Suppose both A,B are finite.

Then, A∪B=A∪(B−A)

Since A and B−A are finite and disjoint therefore, A∪B=A∪(B−A) is finite.

Hence A∪B is countable