SET Theory Basics

SET Theory

Union

• The union of two sets A and B, denoted as A ∪ B, is the set of all elements that are in A, in B, or in both.
• The union of two sets can be represented using the symbol ∪.
• Example: A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}

Venn Diagrams

• A Venn diagram is a visual representation of sets and their relationships.
• It consists of overlapping circles, each representing a set.
• The regions of the diagram can be labeled to show the different sets and their intersections.
• Venn diagrams can be used to illustrate set operations such as union, intersection, and complement.

Intersection

• The intersection of two sets A and B, denoted as A ∩ B, is the set of all elements that are common to both A and B.
• The intersection of two sets can be represented using the symbol ∩.
• Example: A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}

Complement

• The complement of a set A, denoted as A', is the set of all elements that are not in A.
• The complement of a set can be represented using the symbol '.
• Example: A = {1, 2, 3}, then A' = {all elements except 1, 2, 3}
• The universal set, denoted as U, is the set of all possible elements.
• The complement of a set can also be represented as U \ A.

SET Theory

Union

• The union of two sets A and B, denoted as A ∪ B, contains all elements that are in A, in B, or in both.
• The symbol ∪ represents the union of two sets.
• Example: A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.

Venn Diagrams

• Venn diagrams visually represent sets and their relationships using overlapping circles.
• Each circle in a Venn diagram represents a set.
• The regions of a Venn diagram can be labeled to show different sets and their intersections.
• Venn diagrams illustrate set operations like union, intersection, and complement.

Intersection

• The intersection of two sets A and B, denoted as A ∩ B, contains all elements common to both A and B.
• The symbol ∩ represents the intersection of two sets.
• Example: A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}.

Complement

• The complement of a set A, denoted as A', contains all elements not in A.
• The symbol ' represents the complement of a set.
• Example: A = {1, 2, 3}, then A' = {all elements except 1, 2, 3}.
• The universal set U is the set of all possible elements.
• The complement of a set can also be represented as U \ A.

Set Operations

Union

• The union of two sets A and B, denoted as A ∪ B, contains all elements that are in A or in B or in both.
• Example: A = {1, 2, 3} and B = {3, 4, 5} → A ∪ B = {1, 2, 3, 4, 5}.
• Union is commutative: A ∪ B = B ∪ A, meaning the order of sets does not change the result.
• Union is associative: (A ∪ B) ∪ C = A ∪ (B ∪ C), allowing us to evaluate union operations in any order.

Intersection

• The intersection of two sets A and B, denoted as A ∩ B, contains all elements that are common to both A and B.
• Example: A = {1, 2, 3} and B = {3, 4, 5} → A ∩ B = {3}.
• Intersection is commutative: A ∩ B = B ∩ A, meaning the order of sets does not change the result.
• Intersection is associative: (A ∩ B) ∩ C = A ∩ (B ∩ C), allowing us to evaluate intersection operations in any order.

Difference

• The difference of two sets A and B, denoted as A \ B, contains all elements that are in A but not in B.
• Example: A = {1, 2, 3} and B = {3, 4, 5} → A \ B = {1, 2}.
• Difference is not commutative: A \ B ≠ B \ A, meaning the order of sets changes the result.
• Difference is not associative: (A \ B) \ C ≠ A \ (B \ C), requiring careful evaluation of difference operations.

Venn Diagrams

• Venn diagrams visually represent sets and their relationships using overlapping circles.
• Each circle represents a set, with the region inside the circle representing the elements of that set.
• Overlapping regions represent the intersection of the sets.

Complement

• The complement of a set A, denoted as A', contains all elements that are not in A.
• Example: A = {1, 2, 3} and universal set U = {1, 2, 3, 4, 5} → A' = {4, 5}.
• De Morgan's laws relate set operations to their complements:
  • (A ∪ B)' = A' ∩ B'
  • (A ∩ B)' = A' ∪ B'