Set Theory

Set Theory Curriculum

Set Theory Curriculum

This curriculum provides a structured overview of the study of Set Theory, from its foundational principles to advanced topics.

Core Concepts

Basic Set Theory:

  • What is a Set?
  • Elements and Subsets
  • Ways to Define Sets
  • Empty Set, Singleton Set
  • Equality of Sets

Set Operations:

  • Union of Sets
  • Intersection of Sets
  • Difference of Sets
  • Complement of a Set
  • Cartesian Product

Relations:

  • Definition of a Relation
  • Types of Relations (Reflexive, Symmetric, Transitive)
  • Equivalence Relations
  • Partial Orderings
  • Total Orderings

Functions:

  • Definition of a Function
  • Types of Functions (Injective, Surjective, Bijective)
  • Inverse Functions
  • Composition of Functions
  • Functions and Cardinality

Advanced Topics

Cardinality:

  • Finite and Infinite Sets
  • Countable and Uncountable Sets
  • Cardinal Numbers
  • Cantor's Theorem
  • Continuum Hypothesis

Axiomatic Set Theory:

  • Zermelo-Fraenkel Axioms (ZF)
  • Axiom of Choice (AC)
  • ZFC Set Theory
  • The Role of Axioms
  • Independence Results

Ordinal Numbers:

  • Well-ordered Sets
  • Ordinal Arithmetic
  • Transfinite Induction
  • Successor and Limit Ordinals
  • The Axiom of Replacement

Axiom of Choice:

  • Equivalent Forms of AC
  • The Well-Ordering Theorem
  • Zorn's Lemma
  • Applications and Consequences of AC
  • Independence of AC

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