18.090 is an undergraduate subject focusing on understanding and constructing rigorous mathematical arguments. The curriculum covers foundational topics such as infinite sets, logical quantifiers, and various methods of proof. Simultaneously, it introduces selected concepts from algebra—including permutations, vector spaces, and fields—alongside key ideas from analysis, such as sequences of real numbers. The course is particularly suitable for students desiring additional experience with proofs before progressing to more advanced mathematics subjects or subjects in related areas with significant mathematical content.
The class explores the foundational landscape upon which all modern math is built.
: Assuming the opposite of what you want to prove and showing it leads to an impossibility. Mathematical Induction : Proving a statement is true for and that its truth for implies its truth for Department of Mathematics | University of Washington Prerequisites & Logistics Corequisite : You can take 18.090 concurrently with Multivariable Calculus (18.02) Self-Study Resource
Understanding mappings, injections, surjections, and equivalence relations. Cardinality: Exploring the different "sizes" of infinity. Why it Matters
Moving from high school mathematics to university-level mathematics is often a shock for students. In early schooling, math focuses heavily on computation, formulas, and algorithms. You are given an equation, and your job is to find the correct number. 18.090 introduction to mathematical reasoning mit
While the official course website for 18.090 does not always publish a specific textbook, the subject material aligns with standard resources such as "The Tools of Mathematical Reasoning" or "An Introduction to Mathematical Reasoning," which focus on numbers, sets, and functions.
Proving statements true for all natural numbers via base and inductive steps. 2. Set Theory and Cardinality
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
Solution outline (proof by contrapositive): Assume (n) is odd. Then (n = 2k+1) for some integer (k). Thus (n^2 = (2k+1)^2 = 4k^2+4k+1 = 2(2k^2+2k) + 1), which is odd. Therefore, if (n^2) is even, (n) cannot be odd, so (n) is even. ∎ The course is particularly suitable for students desiring
In abstract math, definitions are everything. If a problem asks you to prove a function is injective, your very first step should be writing down the exact mathematical definition of injectivity.
A key feature of 18.090 is its accessibility. The course has ; you can enroll regardless of your previous proof experience.
No textbook required; lecture notes provided. Recommended references:
Gaining the literacy required to read complex academic textbooks and math papers. Mathematical Induction : Proving a statement is true
Traditionally a 12-unit course (3-0-9) offered in the Spring term .
Working sequentially from accepted definitions to reach a logical conclusion. Proof by Contraposition: Proving that to establish that Proof by Contradiction (
: When studying mathematical definitions, always look for objects that fail to meet the definition. Understanding why a sequence fails to converge or why a set is not a field provides a sharper cognitive grasp than only looking at examples that work.