Infinite sets

#matematik-discmath

Definition: Finite set, infinite set

A set is called finite if it is empty or for some . If is not finite it is called infinite.

Theorem

is infinite.

Proof:
We know is nonempty since . Now suppose is finite, so that . Then let . Then should be in , but we have for every , so , which is a contradiction.

Theorem

The set of all prime numbers is infinite.

Proof:
Suppose the set of all prime numbers is . Then let . Clearly is not a multiple of any since . This means is a prime not in , which is a contradiction.

Theorem

Let be a set and be a bijection. Then is infinite.

Proof:
If is finite, there is a bijection , so is a bijection between and , which is a contradiction.

Definition: Countable set

Let be a set. If there exists a bijection between and , we call countably infinite or just countable.