Introduction to Sets

Sets

Definition of a Set

• A set is a collection of unique objects, known as elements or members, which can be anything (numbers, letters, objects, etc.)
• A set is denoted by a capital letter (e.g. A, B, C, etc.)

Representation of a Set

• There are two ways to represent a set:
  1. Roster Form: A set is represented by listing all its elements within curly braces {} Example: A = {1, 2, 3, 4, 5}
  2. Set Builder Form: A set is represented by describing the properties of its elements Example: A = {x: x is a natural number and x ≤ 5}

Types of Sets

• Empty Set (or Null Set): A set with no elements, denoted by ∅
• Singleton Set: A set with only one element
• Finite Set: A set with a finite number of elements
• Infinite Set: A set with an infinite number of elements

Operations on Sets

• Union of Sets: The union of two sets A and B, denoted by A ∪ B, is the set of all elements that are in A or in B or in both
• Intersection of Sets: The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that are common to both A and B
• Difference of Sets: The difference of two sets A and B, denoted by A - B, is the set of all elements that are in A but not in B
• Complement of a Set: The complement of a set A, denoted by A', is the set of all elements that are not in A

Sets

Definition of a Set

• A set is a collection of unique objects called elements or members
• Elements can be anything including numbers, letters, objects, etc.

Representation of a Set

• Sets can be represented in two ways: Roster Form and Set Builder Form
• Roster Form: lists all elements within curly braces {}
• Example of Roster Form: A = {1, 2, 3, 4, 5}
• Set Builder Form: describes properties of elements
• Example of Set Builder Form: A = {x: x is a natural number and x ≤ 5}

Types of Sets

• Empty Set (Null Set): has no elements, denoted by ∅
• Singleton Set: has only one element
• Finite Set: has a finite number of elements
• Infinite Set: has an infinite number of elements

Operations on Sets

• Union of Sets: A ∪ B is the set of all elements in A or B or both
• Intersection of Sets: A ∩ B is the set of all elements common to both A and B
• Difference of Sets: A - B is the set of all elements in A but not in B
• Complement of a Set: A' is the set of all elements not in A