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.