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}.

Explore the fundamental concepts of sets, including definitions, notation, and types. This quiz covers empty sets, finite and infinite sets, as well as operations like union and intersection. Test your understanding of set theory in a concise format!