Set Theory Lecture Notes

Definition of a Set

• A set is a collection of objects called elements.
• Examples of sets include:
  • Physical objects
  • Thoughts and ideas
  • Mathematical objects (main focus of lecture)
• A set packages objects sharing similar properties meaningfully.

Characteristics of Sets

• We can determine unambiguously whether an object belongs to a set.
  • Example: Set of triangles - can identify what is and isn’t a triangle.
• Claims about sets can be evaluated as true or false.
  • Example: Elements of triangles have three sides (true), sum of internal angles is 360 degrees (false).

Notation and Representation

• Sets are written using curly brackets, separating elements with commas.
  • Example: Set A = {1, 2, 3}
• To denote membership, we use the symbol ∈.
  • Example: If A = {1, 2, 3}, then 1 ∈ A, 2 ∈ A, but 4 ∉ A.
• Set Builder Notation: Describes sets without listing all elements.
  • Example: Set of prime numbers: P = {p | p is prime}
• Explicitly declare the starting set before the predicate.
  • Example: p in natural numbers such that p < 5.

Equality of Sets

• Two sets are equal if they contain the same elements.
  • Notation: A = B if
    • For all a in A, a ∈ B and vice versa.
• Order and repetition of elements do not matter in sets.
  • Example: A = {1, 2, 3}, B = {3, 2, 1} → A = B.

Cardinality of Sets

• The cardinality of a set is the number of elements it contains.
  • Notation: |A| denotes cardinality.
  • Example: If A = {1, 2, 3}, then |A| = 3.
• Infinite sets use the infinity symbol (∞) to denote cardinality.

Subsets

• A set A is a subset of B if all elements of A are also in B.
  • Notation: A ⊆ B.
  • Example: A = {2, 4, 6}, B = {1, 2, 3, 4, 5, 6} → A ⊆ B.
• Proper Subset: A is a proper subset of B if A ⊆ B but A ≠ B.
  • Notation: A ⊂ B (not always standard).
• All sets are subsets of themselves.

Properties of Sets

• If A ⊆ B and B ⊆ C, then A ⊆ C (transitive property).
• Empty Set: Contains no elements and is a subset of any set.
  • Unique: There’s only one empty set.

Venn Diagrams and Operations on Sets

• Union (A ∪ B): Set containing all elements in A or B.
• Intersection (A ∩ B): Set containing elements common to both A and B.
• Example Calculations:
  • Let A = {0, 1}, B = {1, 2, 3}.
    • Union: A ∪ B = {0, 1, 2, 3}.
    • Intersection: A ∩ B = {1}.

Properties of Union and Intersection

• Union with empty set: A ∪ ∅ = A.
• Intersection with empty set: A ∩ ∅ = ∅.
• Commutative property for union and intersection: A ∪ B = B ∪ A, A ∩ B = B ∩ A.
• Distributive property:
  • A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
  • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

Set Theoretic Difference

• The difference of two sets A and B (denoted A \ B) is the set of elements in A but not in B.
  • Example: A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8} → A \ B = {1, 3, 5}.

Complements and Universal Set

• The complement of a set A with respect to a universal set U consists of elements in U that are not in A.
  • Denoted as A^c.
• Universal Set (U): Contains all relevant elements for a discussion.

De Morgan's Laws

1. (A ∪ B)^c = A^c ∩ B^c
2. (A ∩ B)^c = A^c ∪ B^c

Power Set

• The power set of A contains all subsets of A.
  • Example: If A = {0, 1}, then power set P(A) = {∅, {0}, {1}, {0, 1}}.

Russell's Paradox

• Paradox arising from naive set theory regarding sets that contain themselves.
• Axiomatic set theory solves these paradoxes with a list of axioms defining a set rigorously.

Conclusion

• Set theory provides a foundational framework for understanding mathematical concepts.
• It's crucial in various fields of mathematics and logical reasoning.