18.090 transitions quickly from basic logic to set theory, which forms the fabric of modern mathematics. Set Operations Students must become comfortable with intersections ( ∩intersection ), unions ( ), complements ( Accap A to the c-th power ), and power sets (
: The course operates on clear true/false principles, training students to produce arguments that are logically sound.
Leo’s first "Problem Set" (pset) felt like a trap. It didn't ask him to calculate anything. It asked him to prove that there are infinitely many prime numbers. Leo knew it was true—he’d read it in a book—but proving it felt like trying to catch smoke with his bare hands. He spent three hours in the Barker Library
In standard calculus or linear algebra, success is often measured by finding the correct numerical answer. In 18.090, the "answer" is the itself. Students are introduced to the rigorous language of set theory, logic, and functions. The goal is to move away from intuition—which can be deceptive—and toward deductive certainty . This requires a high level of "extra quality" in thought, as a single logical gap can invalidate an entire argument. Mastering the Tools of the Trade It didn't ask him to calculate anything
: A deep dive into abstract algebraic structures like groups, rings, and vector spaces.
Building a conclusion step-by-step from known axioms.
Focuses on understanding and constructing mathematical arguments. He spent three hours in the Barker Library
These provide a concise summary of proof techniques.
: Both injective and surjective. This implies the function is invertible. How to Write "Extra Quality" Mathematical Proofs
Its "gateway" status is so significant that MIT advises students to take , as these later courses require solid proof experience. Additionally, because 18.090 requires calculus only as a corequisite, it can be taken concurrently with MIT’s multivariate calculus sequence, allowing students to build reasoning skills in tandem with computational ones. and vector fields. They are
: The primary goal is teaching students how to write clear, logical, and rigorous mathematical proofs. Mathematical Language
The initial portion of the course focuses on the bedrock of all mathematics: logic and set theory.
There is a quiet crisis that happens in mathematics departments around the world. A student breezes through Calculus I, II, and III, mastering integrals, derivatives, and vector fields. They are, by all standard metrics, good at math. Then, they walk into their first upper-level proof-based course—Real Analysis or Abstract Algebra—and hit a wall.
Start by defining the shift in perspective. Most early math is about "finding the answer" through algorithms. In 18.090, the goal shifts to —proving why an answer must be true using logical principles. Mention that this course is particularly suitable for students before they tackle high-level proof-heavy subjects like 18.100 (Real Analysis) or 18.701 (Algebra I) . 2. The Core Pillars of Reasoning Discuss the specific technical toolkit the course provides: Logic and Quantifiers : Understanding how to use "for all" ( ∀for all ) and "there exists" ( ∃there exists ) to define mathematical statements precisely.