Set Theory Fundamentals

Set Theory Basics

Definition of a Set

• A set is a collection of well-defined objects called elements.
• Elements can include numbers, words, functions, or other sets.

Properties of Sets

• Members of a set share common characteristics.
• Sets can be finite (having a specific number of elements) or infinite (having an undefined number of elements).
  • Example of a finite set: {1, 2, 3, 4, 5}.
  • Example of an infinite set: {2, 3, 5, 7, …}.

Methods of Defining Sets

1. Roster Method: List elements explicitly.
   • Example: Set A = {1, 2, 3, 4, 5}.
2. Set-builder Notation: Defines elements based on a property.
   • Example: Set A = {x | x > 0} for positive real numbers.
3. Descriptive Property: Highlights commonalities among elements.
   • E.g., Set of natural numbers or first 20 positive even numbers.

Universal Set

• Contains all elements relevant to a particular problem, often denoted by ( U ).

Empty Set

• A set with no elements, denoted by ( \emptyset ) or {}.