) : "A if and only if B." Requires proving the statement in both directions. 3. Quantifiers
One of the most mind-bending aspects of the course, cardinality explores the concept of infinite sets. Students learn to prove that some infinities are actually "larger" than others—such as the difference between the countable integers and the uncountable real numbers.
In computational math, you can check your answer against a textbook key. In proof-based math, a proof can look elegant and correct on the surface while containing fatal logical gaps. "Extra quality" self-study requires you to actively stress-test your own proofs. Ask yourself: Did I use every hypothesis given in the prompt? Did I accidentally assume what I was trying to prove? Write for an Audience ) : "A if and only if B
The MIT course serves as a critical bridge for students moving from the world of calculation to the world of formal abstraction. While many introductory math courses focus on "how" to solve a problem using established algorithms, 18.090 focuses on "why" a mathematical statement is true. It is, in essence, a bootcamp for mathematical literacy . The Shift from Computation to Proof
The course is ideally suited for:
MIT 18.090 is an introductory course designed to teach undergraduate students the language, structure, and aesthetics of rigorous mathematical proofs. The course shifts the focus from calculating an answer to proving why a mathematical statement must fundamentally be true. Core Course Objectives
MIT 18.090 is an introductory course designed to teach the language of higher mathematics. While courses like 18.01 (Calculus) and 18.02 (Multivariable Calculus) focus on calculations, derivatives, and integrals, 18.090 shifts the spotlight to mathematics works. Students learn to prove that some infinities are
You assume the entire statement you want to prove is false (i.e., the hypothesis is true, but the conclusion is false). You then reason logically until you reach a fundamental mathematical impossibility (a contradiction, like
The course dissects how simple statements combine to form complex theorems using logical operators: : True only if both Disjunction ( ) : True if at least one is true (inclusive "or"). Implication ( ) : "If ." This is the backbone of theorems. Crucially, if is false, the implication is vacuously true. Equivalence ( if is false
The curriculum is built to establish a solid foundation in the "language" of mathematics. Description