Injective (one-to-one), surjective (onto), bijective, and inverse functions. Equivalence relations (reflexive, symmetric, transitive) and partitions.
Systematically evaluating the truth value of compound statements to identify tautologies and contradictions. 2. Methods of Mathematical Proof
: Recent offerings, such as in Spring 2025, have been taught by faculty like Semyon Dyatlov and Bjorn Poonen , often involving lecture notes and weekly problem sets designed to build analytical thinking.
MIT 18.090 is more than just a math class; it is a cognitive upgrade. It strips away the memorization of high school math and replaces it with the beauty of pure, unadulterated logic. By the end of the course, you will no longer look at math as a calculation tool, but as a playground of infinite structural possibilities. 18.090 introduction to mathematical reasoning mit
Mathematics is built on the language of sets. 18.090 covers the fundamental mechanics of how mathematical objects interact:
Mastering the syntax of mathematical statements, quantifiers, and logical connectives.
. This is often easier when the negation of a statement provides more concrete information to work with. Proof by Contradiction ( It strips away the memorization of high school
: Understanding infinite sets, cardinality (the "size" of infinity), and the structure of the real number system. Number Theory
Naïve set theory (with a warning about Russell's paradox). Union, intersection, complement, power sets, and Cartesian products. You learn to prove two sets are equal by showing mutual inclusion: ( A \subseteq B ) and ( B \subseteq A ).
For many incoming students at the Massachusetts Institute of Technology, the jump from high school calculus to upper-level theoretical mathematics feels like stepping off a firm dock into deep, murky water. In high school, math is often about calculation: find the derivative, solve for ( x ), compute the integral. But in college—especially at MIT—mathematics transforms into a discipline of . if ( n^2 ) is even
Confusion often arises because MIT has multiple courses that involve proofs. Here is the hierarchy:
Prove that for any integer ( n ), if ( n^2 ) is even, then ( n ) is even.
The heart of the course lies in writing proofs. In 18.090, you learn that a proof is not just a collection of symbols, but an essay written in prose that guides the reader inevitably to a conclusion. Here are the primary proof methods taught: Assuming a statement
If you are taking 18.090 at MIT, or self-studying the material via MIT OpenCourseWare, use these strategies to master the material: