Intro to Proofs

Truth, Vacuous Truth, and Implication

Explore how implication works and why vacuous truth matters in proofs.

The Quantifier Sandbox

Practice manipulating universal and existential quantifiers in statements.

The Counterexample Engine

Use counterexamples to disprove false statements systematically.

// Unit 2: Functions, Cardinality, and Infinity

Injections, Surjections, and Bijections

Compare function types and their role in counting infinite sets.

Composition and Inverses

Understand function composition and inverse functions with proofs.

Cantor’s Diagonalization

Explore infinity with Cantor’s diagonal argument and set cardinality.

// Unit 3: Relations and Abstract Structures

Equivalence Relations

Study equivalence classes and their role in abstract mathematics.

Partitions and Quotients

Learn how partitions and quotient sets arise from relations.

Modular Arithmetic: Building a Ring

Use proofs to understand modular arithmetic and ring structure.

// Unit 4: The Architecture of the Reals

Supremum and Infimum

Reason about bounds and completeness in the real numbers.

The Density of the Rationals

Prove and visualize why rationals are dense in the real line.

Cauchy Sequences

Explore the definition of Cauchy sequences and convergence rigorously.

// Unit 5: The Delta-Epsilon Limit Sandbox

Visualizing Epsilon Tubes

Build intuition for epsilon-delta definitions with geometric diagrams.

The Delta Response Algorithm

Use an algorithmic approach to find delta values for limit proofs.

Continuous vs Discontinuous

Compare continuous and discontinuous functions through epsilon-delta reasoning.