In computational math, you can get away with a vague understanding of a concept if you know the formula. In 18.090, you cannot prove anything about a "compact set" or an "injective function" unless you know its exact mathematical definition word-for-word.
Foundations: Infinite sets, quantifiers, and various methods of proof . Algebra: Permutations, vector spaces, and fields . Analysis: Sequences of real numbers . : Typically offered in the Spring semester . Why Take It?
is a specialized undergraduate course designed to bridge the gap between computational calculus and high-level abstract mathematical proofs. Offered by the MIT Department of Mathematics , this 3-0-9 unit course focuses explicitly on teaching students how to understand, construct, and write rigorous mathematical arguments. It serves as an essential preparatory pathway for undergraduates planning to transition into advanced, proof-heavy coursework like Real Analysis (18.100), Abstract Algebra (18.701), or Topology (18.901). 18.090 introduction to mathematical reasoning mit
For the student standing at the threshold of advanced mathematics, 18.090 is the key that unlocks the door. Behind that door is a universe of infinite precision, elegant abstraction, and rigorous beauty. Turn the key. The proof awaits.
| Week | Topic | |------|-------| | 1 | Logical connectives, truth tables, tautologies | | 2 | Quantifiers, negations, converse/inverse | | 3 | Proof techniques: direct, contrapositive, contradiction | | 4 | Mathematical induction (ordinary and strong) | | 5 | Sets: union, intersection, power sets, Cartesian products | | 6 | Functions: injective, surjective, bijective, inverses | | 7 | Relations: equivalence relations, partitions | | 8 | Midterm review & exam | | 9 | Number theory: divisibility, primes, GCD, Euclidean algorithm | | 10 | Modular arithmetic and proofs | | 11 | Real numbers: least upper bound property, sequences | | 12 | Countability: finite, countably infinite, uncountable sets | | 13 | Introduction to combinatorial proofs | | 14 | Final review and project presentations | In computational math, you can get away with
Starting from known axioms to reach a conclusion.
Are you a looking for open-source resources to study proof writing? Algebra: Permutations, vector spaces, and fields
: Your first draft of a proof is rarely the one you should turn in. Write out the rough logic first, and then carefully rewrite it to ensure every step follows logically from a definition, axiom, or previously proven theorem.
At MIT, higher-level courses like Real Analysis (18.100) and Abstract Algebra (18.703) assume a high level of mathematical fluency. Attempting those courses without a strong grasp of proofs often leads to academic struggle.
: It is listed as a Restricted Elective in Science and Technology (REST) subject. Core Topics