Understanding Sets and Their Definitions

Set Definition

• A set is a well-defined collection of objects.
• These objects are called elements or members.
• Sets are unordered, meaning the order of members doesn't change the set.
• A finite set has a limited or countable number of elements.
• An infinite set has unlimited or uncountable elements, making it impossible to identify the last element.
• A unit set, also called a singleton, has only one element.
• An empty set, or null set, contains no elements.
• The universal set (U) is a larger set holding all sets being investigated in a specific set theory application.
• Cardinality (N(A)) represents the number of elements in a set.

Set Writing Methods

Roster Method

• Lists all elements separated by commas.
• Examples:
  • {X, Y, Z}
  • P = {C, C++, JAVA}
  • E = {2, 4, 6, 8, 10, 12, 14, ...}
• Also known as Tabulation Method.

Rule Method

• Uses a descriptive phrase to explain the set's elements.
• Also called Set Builder Notation, written as {X|P(X)}.
• Examples:
  • 0 = { X | X IS AN ODD POSITIVE INTEGER LESS THAN 15}
  • R = { X | X IS A REAL NUMBER }
  • V = { X | X IS A VOWEL LETTER }

Venn Diagrams

• Created by John Venn.
• Uses diagrams to visually represent set theory relationships.
• Represents the universal set (U) as a containing box.

Set Operations

Union

• Denoted by $A∪B$, it contains all elements from sets A and B combined.

Intersection

• Denoted by $A∩B$, it contains only elements shared by both sets A and B.

Complement

• Denoted by $A'$, it contains all elements from the universal set (U) that are not in set A.

Difference

• Denoted by $A - B$, it contains elements in set A but not in set B.

Symmetric Difference

• Denoted by $A△B$ or $A⊕B$, it contains elements present in either set A or B, but not both.