1. Introduction to Measure Theory
MTH 426
Introduction to Measure Theory
By Dr. Saqib Hussain

2. MTH 426
Lecture # 5
Countable Sets

3. Previous Lecture’s Review
Equivalent sets
Infinite sets

6. Lecture’s Outline
Countable sets
Uncountable sets

7. Denumerable sets
Set D is said to be denumerable if it is equivalent to the set of natural number N.
Examples:

9. Example:
Show that the set of integers is denumerable.
Solution:

11. Example:
Show that an infinite sequence of distinct elements is denumerable.
Solution:

12. Theorem:
If A and B are denumerable sets then A x B is denumerable.
Proof:

13. Theorem:
Every infinite set contains a subset which is denumerable
Proof:

15. Theorem:
A subset of a denumerable set is either finite or denumerable
Proof:

17. Countable set
A set is said to be countable if it is either finite or denumerable.
Examples:

18. Remark:
A subset of a countable set is countable

19. Theorem:
Proof:

21. Example:
Show that the set of rational numbers is denumerable.
Solution:

23. Example:
Show that the set [0, 1] is non-denumerable.
Solution:

25. Example:
Show that the set [a, b] is non-denumerable.
Solution:

26. Example:
Show that the set of irrational numbers is non-denumerable.
Solution:

27. References:
Set Theory and Related Topics by Seymour Lipschutz.
Elements of Set Theory by Herbert B. Enderton