18.090: The Threshold of Infinity sat in a plastic chair in Building 2, staring at a chalkboard covered in symbols that looked more like ancient runes than the math he knew from high school. For Leo, math had always been a series of recipes: plug
Week 13:
The MIT course 18.090: Introduction to Mathematical Reasoning is a foundational subject designed to bridge the gap between calculation-based mathematics (like standard calculus) and the abstract, proof-oriented world of higher mathematics. The Bridge to Advanced Mathematics Low-quality answer: "Assume rational, derive contradiction
Problem: Show that √2 is irrational.
Low-quality answer: "Assume rational, derive contradiction."
Extra Quality answer: Begins with "We use proof by contradiction. Step 1: Write √2 = a/b in lowest terms… Step 2: Square both sides → 2b² = a² → a is even… Step 3: Substitute a=2c → 2b² = 4c² → b² = 2c² → b even. Contradiction (a,b not coprime)."
Then adds: Common mistake: forgetting to state "lowest terms" – without that, no contradiction. Week 14: Wrap-up: proof strategies review
Preparatory Value: It is specifically recommended for students who want more experience with proofs before tackling advanced subjects like 18.100 Real Analysis, 18.701 Algebra I, or 18.901 Introduction to Topology. sample advanced proofs
Course structure & schedule (14 weeks)
Week 1: Logic, statements, connectives, truth tables, implication, quantified statements.
Week 2: Logical equivalences, predicate logic, negation of quantifiers, mathematical writing conventions.
Week 3: Proof techniques: direct proofs, contraposition, contradiction; examples with integers and parity.
Week 4: Sets and set operations, Venn diagrams, De Morgan’s laws, indexed families, Cartesian products.
Week 5: Functions: definitions, injective/surjective/bijective, inverses, composition; images/preimages.
Week 6: Relations: properties (reflexive, symmetric, transitive), equivalence relations and partitions.
Week 7: Number theory basics: divisibility, gcd, Euclidean algorithm, fundamental theorem of arithmetic (statement and proof sketch).
Week 8: Mathematical induction and strong induction, well-ordering principle; applications to inequalities, divisibility, sequences. — Midterm around here.
Week 9: Sequences and limits (ε-N intuitive proofs for basic limits); monotone sequences and boundedness (intuitive proofs).
Week 10: Counting and combinatorics: basic rules, permutations/combinations, binomial theorem, combinatorial proofs.
Week 11: Elementary graph theory: definitions, trees, Eulerian and Hamiltonian paths, basic proofs and constructions.
Week 12: Relations revisited: partial orders, Hasse diagrams, minimal/maximal elements, Zorn’s Lemma statement (no proof).
Week 13: Cardinality: finite, countable, uncountable sets; Cantor’s diagonal argument; bijections and countability proofs.
Week 14: Wrap-up: proof strategies review, sample advanced proofs, final exam practice, student presentations/projects.
In 18.090, the questions change entirely. A problem might ask: Prove that the derivative of an even function is an odd function.