Lecture on Set Theory

Definition of a Set

• Set: A collection of objects called elements.
  • Can include physical objects, thoughts, ideas, concepts, and especially mathematical objects.
• Purpose: Package objects with similar properties meaningfully.
  • Example: Set of triangles (clear membership rules).

Properties of Sets

• Clarity in Membership: Unambiguous determination of whether an object is in a set.
  • Example: Set of triangles (element has three sides).
• Notation:
  • Curly brackets {} used to denote sets.
  • Elements separated by commas.
  • Example: Set {1, 2, 3}.
• Symbolic Representation:
  • ∈ for membership, ∉ for non-membership.
  • Set builder notation for concise set descriptions.

Equality and Subsets

• Set Equality: Two sets are equal if they have the same elements.
• Subset: A set A is a subset of B if all elements of A are in B.
  • Symbolized by ⊆.
  • Proper subset: A subset that is not equal (⊂).

Cardinality and Types of Sets

• Cardinality: Number of elements in a set, denoted by |A|.
  • Example: {1, 2, 3} has cardinality 3.
• Infinite Sets: Use ∞ for cardinality.

Special Sets

• Empty Set: A set with no elements, symbolized by ∅.
  • Unique and a subset of every set.

Operations on Sets

Union and Intersection

• Union (∪): Combines all elements from both sets.
• Intersection (∩): Elements common to both sets.

Properties of Union and Intersection

• Union:
  • A union with empty set is A (A ∪ ∅ = A).
  • Commutative and associative properties apply.
• Intersection:
  • Intersection with empty set is empty.
  • Commutative and associative properties apply.

Set Difference

• Difference (A - B): Elements in A not in B.
• Complement: Elements not in the set, often relative to a universal set.

Theorems and Laws

• De Morgan's Laws:
  • ¬(A ∪ B) = ¬A ∩ ¬B
  • ¬(A ∩ B) = ¬A ∪ ¬B
• Distributive Properties:
  • A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Advanced Concepts

• Power Set: Set of all subsets of a set.
• Indexed Families: Sets indexed by numbers.
• Russell's Paradox: The paradox arising from set definitions that contain themselves.

Conclusion

• Naive vs. Axiomatic Set Theory: Axiomatic provides a rigorous foundation to avoid paradoxes.

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These notes capture the essential aspects of set theory from the lecture, focusing on definitions, operations, properties, and advanced concepts.