Without 18.090, students often struggle in these upper-level courses because they understand the computations but fail to construct the necessary proofs.
Defining functions strictly as relations, and proving whether a function is injective (one-to-one), surjective (onto), or bijective (invertible).
: Collaboration is central to the MIT experience. Discussing problem sets with your peers helps expose holes in your logical reasoning before the grading teaching assistants find them.
Working with congruence classes, which form the bedrock of modern cryptography and abstract algebra. 18.090 introduction to mathematical reasoning mit
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Before you can build a proof, you must understand the building blocks. Students learn about sentential logic (and, or, implies), quantifiers (for all, there exists), and the basic properties of sets. This provides the syntax needed to write clear, unambiguous mathematical statements. 2. Proof Techniques Without 18
, which contradicts the initial assumption that the fraction was in simplest form. Thus, the square root of 2 end-root must be irrational. Which specific mathematical topic are you planning to cover in your paper? Course 18: Mathematics IAP/Spring 2026
By taking 18.090, students can expect to develop the following skills and takeaways:
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 . Discussing problem sets with your peers helps expose
Assuming the opposite of what you want to prove and showing it leads to a logical impossibility.
: Assuming the statement is false and finding a logical flaw in that assumption.
Proving that if the conclusion is false, the hypothesis must also be false. 3. Basic Structures
Unlike calculus, where you apply formulas, this course teaches you . You will learn the language of mathematics.
Many MIT students find that transitioning to 18.090 is where they actually start "loving" math because they stop memorizing formulas and start understanding the underlying structures. It's often the class that helps students decide if they want to double-major in Course 18 (Mathematics) 18.0x - MIT Mathematics