Sets: Definitions and Operations

Sets

• Definition: A set is a collection of distinct objects, considered as a whole. The objects are called elements or members.

• Notation:

  • Sets are usually denoted by uppercase letters (e.g., A, B, C).
  • Elements are listed within curly braces, e.g., A = {1, 2, 3}.

• Types of Sets:

  • Empty Set: A set with no elements, denoted as Ø or {}.
  • Finite Set: A set with a limited number of elements (e.g., A = {1, 2, 3} has 3 elements).
  • Infinite Set: A set with an unlimited number of elements (e.g., B = {1, 2, 3, ...}).
  • Subset: A set A is a subset of B if all elements of A are also in B (A ⊆ B).
  • Proper Subset: A set A is a proper subset of B if A ⊆ B and A ≠ B.
  • Universal Set: The set that contains all possible elements in a particular context, usually denoted by U.

• Set Operations:

  • Union: The set containing all elements from both sets A and B. Denoted as A ∪ B.
  • Intersection: The set containing elements common to both sets A and B. Denoted as A ∩ B.
  • Difference: The set of elements in A that are not in B. Denoted as A - B.
  • Complement: The set of elements in the universal set U that are not in A. Denoted as A'.

• Venn Diagrams: Visual representations of sets and their relationships, where circles represent sets and their overlaps represent intersections.

• Power Set: The set of all subsets of a set A, including the empty set and A itself. Denoted as P(A).

• Cardinality: The number of elements in a set, often denoted as |A|.

• Examples:

  • A = {2, 4, 6}, B = {4, 5, 6}
    • A ∪ B = {2, 4, 5, 6}
    • A ∩ B = {4, 6}
    • A - B = {2}

• Important Properties:

  • Sets are unordered: {1, 2} is the same as {2, 1}.
  • No duplicates are allowed in sets: {1, 2, 2} = {1, 2}.

Sets

• A set is a collection of distinct objects considered as one unit.
• Sets are typically denoted by uppercase letters (e.g., A, B, C).
• Elements within a set are listed within curly braces (e.g., A = {1, 2, 3}).
• Empty Set: A set with no elements, represented as Ø or {}.
• Finite Set: A set with a specific, finite number of elements.
• Infinite Set: A set with an unlimited number of elements.
• Subset: A set A is a subset of B if all elements of A are also present in B (A ⊆ B).
• Proper Subset: A set A is a proper subset of B if A is a subset of B but not equal to B (A ⊆ B and A ≠ B).
• Universal Set: Encompasses all possible elements in a given context, symbolized as U.
• Set Operations:
  • Union (∪): Combining all elements from sets A and B into a single set.
  • Intersection (∩): Creates a set containing common elements shared by both sets A and B.
  • Difference (-): Produces a set including elements present in A but not in B.
  • Complement ('): Generates a set containing elements from the universal set U that are not in set A.
• Venn Diagrams: Visual representations of sets and their relationships, using circles to represent sets and overlaps for intersections.
• Power Set (P(A)): The collection of all possible subsets of set A, including the empty set and A itself.
• Cardinality (|A|): Represents the number of elements within a set.
• Key Properties:
  • Sets are unordered, meaning {1, 2} is the same as {2, 1}.
  • Sets do not allow duplicate elements, so {1, 2, 2} is equivalent to {1, 2}.