Lecture Notes on Set Theory

Introduction to Sets

• Set: A collection of objects called elements.
  • Examples: Physical objects, thoughts, ideas, concepts, mathematical objects.
  • Sets package objects with similar properties meaningfully.
  • Example: Set of triangles (triangles vs. non-triangles).

Fundamental Principles of Sets

• Ambiguity: Clear whether an object is in a set or not.
  • Example: Elements of a triangle have 3 sides.
• Notation: Curly brackets {} and elements separated by commas.
  • Naming sets for convenience, e.g., A = {1, 2, 3}.
• Membership Symbol: ∈ (belongs to), ∉ (doesn't belong to).

Set Builder Notation

• General Format: {x | predicate}
  • Example: P = {p | p is prime}
  • Vertical line (|) means “such that”.

Explicit Set Declarations

• Example: P in natural numbers such that P < 5.
• Different sets: Real numbers < 5 vs. natural numbers < 5.

Equal Sets

• Definition: A and B are equal if they contain the same elements.
  • Example: A = {1, 2, 3}, B = {3, 2, 1} → A = B.

Cardinality of a Set

• Cardinality: Total number of elements in a set.
  • Denoted by vertical bars |A|.
  • Example: |A| = 3 for A = {1, 2, 3}.
  • Infinite sets: Example set of prime numbers → ∞.

Subsets

• Definition: A ⊆ B if all elements of A are in B.
  • Proper subset: A ⊂ B if A ⊆ B but A ≠ B.
  • Symbol: ⊆ (subset), ⊂ (proper subset).
  • Example: A = {2, 4, 6}, B = {1, 2, 3, 4, 5, 6} → A ⊆ B.

Venn Diagrams

• Union (∪): Combination of all elements in sets A and B.

  • Example: A = {0, 1}, B = {1, 2, 3} → A ∪ B = {0, 1, 2, 3}.

• Intersection (∩): Elements common to sets A and B.

  • Example: A = {0, 1}, B = {1, 2, 3} → A ∩ B = {1}.

Properties of Union and Intersection

• Union properties:

  • A ∪ ∅ = A
  • A ∪ A = A
  • A ⊆ B → A ∪ B = B
  • Commutative: A ∪ B = B ∪ A
  • Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)

• Intersection properties:

  • A ∩ ∅ = ∅
  • A ∩ A = A
  • A ⊆ B → A ∩ B = A
  • Commutative: A ∩ B = B ∩ A
  • Associative: (A ∩ B) ∩ C = A ∩ (B ∩ C)

Set Theoretic Difference

• Definition: A ∖ B is all elements in A not in B.
  • Example: A = {1, 2, 3, 4}, B = {2, 4} → A ∖ B = {1, 3}.
  • Complement: Universal set assumed for context: U ∖ A.

De Morgan’s Laws

• Laws:
  • (A ∪ B)' = A' ∩ B'
  • (A ∩ B)' = A' ∪ B'
• Examples: Explained with sets of animals, numbers.
• Duality Principle: Interchangeable union and intersection in identities.

Special Sets

• Empty Set (∅): Contains no elements.
  • Properties: ∅ ⊆ A, unique empty set.

Power Sets & Indexed Families of Sets

• Power Set: Contains all subsets of a set A.

  • Example: A = {0, 1} → power set of A = {∅, {0}, {1}, {0, 1}}

• Indexed Families: Sets indexed by numbers.

  • Example: A = {A1, A2, A3}, Ai are sets.

Russell's Paradox

• Russell's Paradox: Issue of sets containing themselves.
  • Solution in axiomatic set theory – rigorous definitions.

Concluding Remarks

• Set theory fundamentals crucial for advanced mathematics.
• Axiomatic set theory mitigates paradoxes.

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