18.090 Introduction To Mathematical Reasoning Mit May 2026

This course is the bridge from computational calculus to rigorous proof-based mathematics. It covers logic, sets, functions, proof techniques (induction, contradiction), and basic number theory/analysis.

3. Learn to Read Proofs

When reading a proof in a textbook, do not just skim it. Cover the next step with a piece of paper and try to predict what comes next. Ask yourself: Why did they choose that specific variable? 18.090 introduction to mathematical reasoning mit

⚠️ Common Pitfalls

Assuming what you want to prove: Never start a proof by assuming the conclusion is true.
"It’s obvious": In this course, nothing is obvious. You must justify every step based on axioms or previous theorems.
Misunderstanding Implication: Remember that $P \implies Q$ is only False when $P$ is True and $Q$ is False. If $P$ is False, the implication is vacuously True.

Conclusion: Is 18.090 Right for You?

If you have typed "18.090 introduction to mathematical reasoning mit" into a search engine, you are probably standing at a crossroads. You have finished the computation-based math and are peering into the abstract unknown. This course is the bridge from computational calculus

The honest answer: 18.090 is hard. You will feel lost. You will erase entire proofs. You will question if you belong in a math major. Assuming what you want to prove: Never start

But you will also experience the unique thrill of constructing an ironclad argument from nothing but logic. You will learn to read a theorem and see its skeleton. And when you move on to analysis, topology, or number theory, you will realize that 18.090 gave you the only tool that matters: the ability to reason.

For MIT students, it’s a requirement. For anyone else reading this guide, it’s a blueprint. Mathematical reasoning is not a talent—it’s a craft. And 18.090 is the workshop where you learn the trade.

Are you an MIT student currently enrolled in 18.090? Check the MIT Student Information System (SIS) for current offerings and the Math Department’s undergraduate office for office hours. For self-learners, Richard Hammack's "Book of Proof" is available for free at people.vcu.edu/~rhammack/BookOfProof/ — that is the closest you can get to the MIT experience without the tuition.

Typical learning objectives

Parse and negate complex mathematical statements correctly.
Choose and apply appropriate proof strategies.
Prove simple theorems about numbers, sets, and functions.
Construct clear, rigorous written proofs and spot flaws in arguments.
Understand and use induction and basic combinatorial reasoning.

3. Relations and Functions

Relations: Equivalence relations (reflexive, symmetric, transitive) and partitions.
Functions: Definition of a function, injectivity (one-to-one), surjectivity (onto), and bijectivity.
Inverses and Composition: How functions interact.