Set Theory

The Axiom of Choice is a fundamental axiom of set theory concerning the selection of elements from collections of non-empty sets. Learners will understand its equivalence to Zorn's Lemma and the Well-Ordering Theorem, and its implications for modern mathematics.

Cardinality measures the size of sets, distinguishing between different sizes of infinity. Learners will understand how to compare sets using bijections, define cardinal arithmetic, and apply Cantor's diagonal argument to prove the uncountability of the real numbers.

The Continuum Hypothesis proposes that there is no set whose cardinality is strictly between that of the integers and the real numbers. Learners will understand its independence from Zermelo-Fraenkel set theory and its historical significance in mathematical logic.

Functions are mathematical rules that map elements from a domain to a codomain. Learners will understand concepts of injectivity, surjectivity, and bijectivity, and be able to construct inverse functions, compose mappings, and analyze functional behavior in abstract contexts.

Ordinal numbers extend the natural numbers to describe the order types of well-ordered sets. Learners will understand the difference between cardinality and ordinality, perform ordinal arithmetic, and use ordinals to structure transfinite induction and recursion proofs.

Relations define connections between elements of sets, forming the basis for orderings and equivalence. Learners will understand properties like reflexivity, symmetry, and transitivity, and be able to analyze equivalence relations, equivalence classes, and partial orders.

This topic covers the foundational concepts of set theory, including subsets, unions, intersections, and complements. Learners will understand how to manipulate sets using Venn diagrams, prove basic set identities, and apply set operations to solve logical and algebraic problems.

Transfinite induction extends standard mathematical induction to well-ordered sets of any size. Learners will understand how to construct proofs over ordinal numbers, apply transfinite recursion to define mathematical objects, and solve advanced problems in set theory and topology.

The Well-Ordering Theorem states that every set can be well-ordered, meaning every non-empty subset has a least element. Learners will understand its equivalence to the Axiom of Choice and Zorn's Lemma, and its applications in constructing mathematical proofs.

The Zermelo-Fraenkel axioms form the standard foundation for modern set theory. Learners will understand each axiom's role in avoiding paradoxes, construct standard mathematical objects from empty sets, and evaluate the consistency and independence of mathematical statements.