Fundamentals of Set Theory Explained

Nov 16, 2024

Lecture Notes on Set Theory

Introduction to Sets

  • A set is a collection of objects called elements.
  • Elements can be physical objects, thoughts, ideas, or mathematical objects.
  • Sets help in organizing objects with shared properties meaningfully.

Definition and Properties of Sets

  • Example of sets: Set of triangles.
    • Clear criteria to determine membership.
  • Claims about sets can be evaluated as true or false.
    • Example: Elements of triangles have three sides; the sum of internal angles is not always 360 degrees.
  • Sets are represented using curly brackets: {1, 2, 3}.
  • Sets can be named for easier reference (e.g., let A = {1, 2, 3}).
  • Membership notation:
    • 1 ∈ A (1 is in A)
    • 4 ∉ A (4 is not in A)

Set Builder Notation

  • Example: Set of prime numbers can be expressed as P = {p | p is prime}.
  • A variable must satisfy a certain criterion (predicate) to belong to the set.
  • Explicitly declare starting sets before the predicate (e.g., the natural numbers).

Equality and Subsets

  • Two sets are equal if they contain the same elements, regardless of order or repetition.
  • Cardinality: The size of a set, denoted by ||A||.
  • A set A is a subset of B if all elements of A are in B.
    • Notation: A ⊆ B.
    • Example: If A = {2, 4, 6} and B = {1, 2, 3, 4, 5, 6}, then A is a subset of B.
  • A proper subset (A ⊂ B) means A is a subset of B but not equal to B.
  • All sets are subsets of themselves.

The Empty Set

  • Definition: A set with no elements, denoted by {} or ∅.
  • The empty set is a subset of any set and is unique.

Venn Diagrams

  • Union of Sets: A ∪ B = {x | x ∈ A or x ∈ B}.
  • Intersection of Sets: A ∩ B = {x | x ∈ A and x ∈ B}.
    • Example: Let A = {0, 1} and B = {1, 2, 3}.
      • Union: A ∪ B = {0, 1, 2, 3}.
      • Intersection: A ∩ B = {1}.
  • Properties of unions and intersections:
    • A ∪ ∅ = A and A ∩ ∅ = ∅.
    • A ∪ A = A and A ∩ A = A.

Cardinality of Union and Intersection

  • Identity: |A ∪ B| = |A| + |B| - |A ∩ B|.
  • This leads to an inequality: |A ∪ B| ≤ |A| + |B|.

De Morgan's Laws

  • Complement of a union: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ.
  • Complement of an intersection: (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ.

Set Theoretic Difference

  • Definition: A \ B = {x | x ∈ A and x ∉ B}.
  • Complement: The complement of B with respect to A is A \ B.
  • Universal Set (U): The set containing all relevant elements for a specific topic.

Power Sets and Indexed Families of Sets

  • Power Set: Contains all subsets of a set A.
  • Example: If A = {0, 1}, then P(A) = {∅, {0}, {1}, {0, 1}}.
  • Indexed Families of Sets: Sets indexed by numbers (e.g., A_i for i in {1, 2, 3}).

Russell's Paradox

  • Discusses contradictions in set theory when considering sets that contain themselves.
  • Axiomatic set theory provides a rigorous framework to avoid such paradoxes.

Conclusion

  • Set theory provides a foundational structure for mathematics, yet it can lead to complex logical challenges.
  • Axiomatic approaches help clarify definitions and avoid paradoxes.