If you get a problem wrong on a homework assignment, rewrite the entire proof correctly from scratch.

: Instead of watching a Teaching Assistant (TA) write solutions on a chalkboard, students spend recitations working collaboratively in small teams. TAs act as facilitators, guiding student logic and helping teams refine their arguments in real time.

Before you prove anything, write down the exact definition of every term. Most mistakes in 18.090 stem from fuzzy definitions.

Solving congruences and understanding Fermat’s Little Theorem. 3. The 5 Essential Proof Techniques

is a premier undergraduate mathematics course specifically designed to bridge the gap between mechanical computational math and rigorous, abstract proof-based mathematics. The phrase "extra quality" highlights the exhaustive curriculum, uncompromising logical precision, and collaborative environment that defines this foundational course.

Students learn to communicate with absolute precision. This includes evaluating predicates, quantifiers (

An extra quality modern technique: Use a large language model (like GPT-4) not to solve the problem, but to critique your proof.

In high school and introductory college calculus (such as MIT 18.01 or 18.02), mathematics is predominantly computational. Students learn formulas, execute algorithms, and solve for specific numeric answers.

Do not just read the textbook; write alongside it. For every proof presented in lecture or in Eccles' book:

Before we add extra resources, let’s establish the foundational pillars of 18.090.

In standard high school or early university math—such as calculus and differential equations—the focus is largely on algorithms: plugging numbers into formulas, deriving functions, and finding numerical solutions. However, advanced mathematics requires an entirely different cognitive toolkit: creating, analyzing, and defending rigorous abstract 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.

He realized he didn't need to count every prime; he just needed a logical wall that nothing could jump over. He used Reductio ad Absurdum —assuming the primes