Mathematical Proofs and Logic
Transition from calculation to proof. Build a rigorous foundation in propositional logic, set theory, direct and indirect proof techniques, induction, combinatorics, relations, functions, and the surprising hierarchy of infinite sets.
What This Course Covers
Mathematical Proofs and Logic is structured into 14 chapters that build on each other progressively:
Each chapter combines interactive AI tutoring with hands-on examples. After you learn the material, Lambdio's spaced repetition algorithm schedules review sessions at optimal intervals — so you retain concepts and techniques long-term.
How to Study Mathematical Proofs and Logic on Lambdio
Lambdio's AI-powered platform adapts to how Math courses are best learned. Here's our recommended approach:
Mathematical Proofs and Logic is the transition course from computation to theory, and it is best learned in Standard Mode because proof writing is a procedure that must be modeled and practiced rather than discovered. The AI tutor can introduce each definition precisely, walk through the strategy of a representative proof, explain why each step is licensed by a definition or a rule of inference, and then check comprehension with questions before the next technique is layered on top. Socratic Mode is a poor fit here: the subject is formal and formula-like, and asking a learner to rediscover De Morgan's laws, the structure of an induction proof, or Cantor's diagonal argument through open questioning alone would be slow and demoralizing. Quiz Mode is the ideal companion because the course rewards instant recall of exact statements — the negation rules, the contrapositive equivalence, the outlines of direct, contrapositive, and contradiction proofs, the inclusion-exclusion formula, the definitions of injective and surjective, the meaning of countable — and these are precisely the facts that later proofs depend on. Although the course is labeled Easy, its concepts are foundational and steeply cumulative: Chapters 1 and 2 supply the logic and set-theoretic vocabulary used by every later chapter, the proof techniques of Chapters 3 through 8 recur throughout, and the relations, functions, and cardinality material of Chapters 11 through 13 is assumed by more advanced courses. Set the priority to High so spaced repetition keeps the earliest definitions available when the later chapters rely on them. In practice, learn each chapter in Standard Mode, then write proofs from scratch before checking them, and use Quiz Mode between sessions to keep the named theorems and definitions sharp. Pair the course with Real Analysis for a rigorous treatment of calculus, Linear Algebra for the structures of linearity, or Measure-Theoretic Probability for graduate-level probability, and stagger your sessions so the proof habits transfer across subjects without overloading the review queue.
Interactive Quiz
Test your knowledge with these sample questions from the course. Tap an answer to see if you're right:
What You'll Be Able to Do After This Course
- ✓Translate statements into propositional and quantified form, build truth tables, and negate conditionals and quantified statements correctly
- ✓Combine quantifiers with logical connectives and apply rules of inference such as modus ponens and modus tollens
- ✓Work fluently with sets: set-builder notation, cardinality, subsets, power sets, Cartesian products, unions, intersections, and indexed families
- ✓Unpack precise mathematical definitions of even, odd, divisibility, prime, gcd, and congruence to drive a proof
- ✓Write direct proofs of conditional statements and organize arguments by exhaustive cases
- ✓Choose and execute contrapositive and contradiction proofs, and recognize when contradiction conceals a cleaner contrapositive argument
- ✓Prove biconditionals, chains of equivalent statements, and existence and uniqueness results, including constructive and non-constructive approaches
- ✓Disprove universal claims with carefully chosen counterexamples and disprove existence claims in full generality
- ✓Prove element membership, subset relations, and set equality, and apply set identities such as De Morgan's laws
- ✓Prove statements about the natural numbers using weak induction, strong induction, and proof by smallest counterexample
- ✓Count with the multiplication, addition, inclusion-exclusion, pigeonhole, and stars-and-bars principles, and construct combinatorial proofs
- ✓Analyze relations, equivalence classes, and partitions, and work with the integers modulo n
- ✓Classify functions as injective, surjective, or bijective, reason about composition and inverses, and use images and preimages
- ✓Compare cardinalities, distinguish countable from uncountable sets, and explain Cantor's theorem and the Continuum Hypothesis
- ✓Apply the triangle inequality and epsilon-delta definitions to limits, continuity, sequences, and series
Frequently Asked Questions
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