Defining functions rigorously via injections (one-to-one), surjections (onto), and bijections (invertible).
This ritual is terrifying but transformative. It destroys the illusion that mathematics is about getting the right answer. It reveals that mathematics is about justification .
P-sets are released weekly and typically contain 6–8 problems. The first problem is usually a "warm-up" (build a truth table). The last problem is a "challenge" (a non-trivial proof from number theory or combinatorics). MIT students report spending 6–10 hours per week on the 18.090 p-set alone. The key rule: No collaboration on the final two problems. You must stand alone with your reasoning.
Number theory provides an excellent playground for practicing new proof techniques because the objects (integers) are familiar, but the properties require deep rigor. Topics include: Divisibility and the Euclidean Algorithm. Prime numbers and the Fundamental Theorem of Arithmetic. Modular arithmetic (often called "clock arithmetic"). 4. Functions and Relations
Getting stuck is a feature of advanced mathematics, not a bug. Spending hours on a single proof is normal and part of the learning process. 18.090 introduction to mathematical reasoning mit
18.090 Introduction to Mathematical Reasoning is more than just a course; it is a rite of passage for MIT students entering the world of abstract mathematics. By focusing on the creation of proofs and the language of logic, it provides the structural foundation necessary for success in everything from Real Analysis to Abstract Algebra. For any student seeking to see why a mathematical statement is true—not just that it is true—18.090 is an indispensable first step.
| Misconception | Reality (Taught in 18.090) | | :--- | :--- | | "A proof is just a sequence of equations." | A proof is a narrative. It requires words like "therefore," "assume," "note that," and "suppose." | | "One example proves a universal statement." | No. One example disproves a universal statement. To prove it, you need a general argument. | | "If you can't find a counterexample, the statement is true." | Absence of evidence is not evidence of absence. You must prove impossibility. | | "Proof by contradiction is the most powerful method." | Often, it's a crutch that obscures a constructive direct proof. Use it sparingly. |
Course focus and learning outcomes
Often cited as the first "true" proof course for many majors. 18.701 (Algebra I): It reveals that mathematics is about justification
Typical breakdown:
Assuming the opposite of what you want to prove and showing it leads to a logical impossibility.
The course covers the "alphabet" of higher mathematics, including: Foundational Logic : Mastering quantifiers like (for all) and there exists (there exists), and the mechanics of implication ( right arrow Set Theory
While some students enter MIT with extensive experience in math competitions or proof-based learning, many have only encountered computational math. 18.090 levels the playing field. It teaches students not just how to calculate an answer, but how to definitively prove why that answer must be true. Core Pillars of the Curriculum The last problem is a "challenge" (a non-trivial
While specific topics can vary by instructor (recent versions have been taught by faculty like Semyon Dyatlov Paul Seidel
A powerful tool for proving statements about integers.
Before writing a proof, you must understand the rules of logic. Students learn:
18.090 is a critical "gateway" course. It provides the crucial necessary for success in demanding upper-level classes. The Pure Mathematics Option explicitly recommends students gain proof experience in 18.090 before tackling 18.100 (Real Analysis) or 18.701 (Algebra I). It is also a Restricted Elective in Science and Technology (REST) , allowing students to fulfill a General Institute Requirement while building this essential skill.
For many students entering Course 18 (Mathematics) at MIT, hitting the "proof wall" in legendary classes like 18.100 (Real Analysis) or 18.701 (Algebra I) can be an intimidating transition [18.23]. This course acts as a vital incubator, training students to read, write, and think with the absolute precision required by modern mathematics. Course Overview & Strategic Placement