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

MTH 426 Lecture # 5 Countable Sets

Previous Lecture’s Review Equivalent sets Infinite sets

Lecture’s Outline Countable sets Uncountable sets

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

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

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

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

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

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

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

Remark: A subset of a countable set is countable

Theorem: Proof:

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

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

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

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

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