Set Theory

Set theory is a crucial theory that forms the foundation of modern mathematics, where undergraduate students first encounter the true abstraction of mathematics. The purpose of studying set theory is not to learn new concepts but to rigorously reconstruct vague concepts into precise and solid forms. Even if it seems useless at first, do not be complacent and study carefully.

Elementary Logic

Propositions

• Propositions, Connectives, Truth Tables
• Tautologies and Contradictions
  • Tautology
• De Morgan’s Laws

Indirect Proof Methods

• Mathematical Logic Proof of the Contrapositive
• Mathematical Logic Proof of Reductio ad Absurdum
• Mathematical Logic Proof of Syllogism
• Mathematical Induction

Sets and Axioms

Sets

• Definition of Sets and Propositional Functions
  • Subset Relation $\subset$
  • Quantification Rules $\forall, \exists$
  • Hat Notation in Element Listing Method $\hat{k}$
• Russell’s Paradox
• Families of Sets and Indices

Axiomatic Systems

• Zermelo-Fraenkel Set Theory with the Axiom of Choice
  • Axiom of Extensionality $=$
  • Axiom of the Empty Set $\emptyset$
  • Pairing Axiom
  • Schema of Specification $\cap$, $\setminus$, $X^{c}$
  • Union Axiom $\cup$
  • Power Set Axiom $\mathcal{P}(X) = 2^{X}$
  • Infinity Axiom
  • Axiom of Regularity
  • Schema of Replacement
  • Axiom of Choice

Relations and Functions

Ordered Pairs

• Cartesian Product $\times$
• Binary Relations
• Equivalence Relations $\sim$
• Partially Ordered Sets

Partitions

• Partition of a Set
  • Disjoint Union: Union of Mutually Exclusive Sets
  • Partition of Unity
• Equivalence Classes
• Partition of a Set by an Equivalence Relation

Mappings

• Functions, Images, and Sequences Rigorously Defined in Set Theory
• Preimage of a Function
  • Saturation, Fiber
• Injective, Surjective, Bijective Functions, and Inverses
  • Easy Ways to Remember Surjections, Injections, Codomain, and Range, with Explanations

Cardinality

Infinity

• Finite and Infinite Sets Rigorously Defined in Set Theory
• Countable and Uncountable Sets
• Cantor’s Diagonal Argument
• Cardinality of a Set $\text{card}$

Continuum Hypothesis

• Cantor-Bernstein Theorem
• Cantor’s Theorem
• Comparing the Cardinalities of Real and Rational Numbers
• Continuum Hypothesis

References

• 이흥천 역, You-Feng Lin. (2011). 집합론(Set Theory: An Intuitive Approach)