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
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
Axiomatic Systems
Relations and Functions
Ordered Pairs
Partitions
Mappings
- Functions, Images, and Sequences Rigorously Defined in Set Theory
- Preimage of a Function
- Injective, Surjective, Bijective Functions, and Inverses
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)
All posts
- Propositions and Connectives, Truth Tables
- Contrapositive and Converse Propositions
- Proof of De Morgan's Laws
- Mathematical Proof of the Counterfactual
- Mathematical Logic Proof by Contradiction
- Mathematical Induction
- Mathematical Logic Proof of Syllogism
- Definition of Sets and Propositional Functions
- Quantifiers over Propositional Functions
- Set Inclusion
- Extensionality Axiom
- Empty Set Axiom
- Axiom of Pairs
- Disjoint Union: Disjoint Unions
- Classification Axiomatic Form
- Saturation and Definition of Fibers in Mathematics
- Union axiom
- Power Set Axiom
- Axiom of Infinity
- Axiom of Regularity
- Substitution Axiom Form
- Axiom of Choice
- Zermelo-Fraenkel Set Theory with the Axiom of Choice
- Sets and Indices
- Cartesian Product of Sets
- Mathematical Binary Relations
- Equivalence Relations in Mathematics
- Partition of a Set
- Homotopy Type
- Equivalence Relations and Set Partitions
- Functions and Mappings Rigorously Defined by Set Theory, Sequences
- The Original Image of a Function
- Injection, Surjection, Bijection, Inverse Function
- Easy Ways to Memorize Surjections, Injections, Ranges, and Domains, Explained
- What is Resonance?
- Finite Sets and Infinite Sets Strictly Defined by Set Theory
- Countable and Uncountable Sets
- Cantor's Diagonal Argument
- Cardinality of a Set
- Proof of the Cantor-Bernstein Theorem
- Proof of Cantor's Theorem
- Comparison of the Cardinality of Real Numbers and the Cardinality of Rational Numbers
- Partially Ordered Set
- Russell's Paradox
- Continuum Hypothesis
- Unit Partition in Mathematics
- Elementary Notation in Enumerating Elements