. This is often easier when the negation of a statement provides more concrete information to work with. Proof by Contradiction (
Proving the Fundamental Theorem of Arithmetic and the infinitude of primes.
: Understanding infinite sets, cardinality (the "size" of infinity), and the structure of the real number system. Number Theory
To apply proof techniques, students are introduced to basic structures in abstract algebra. Studying arrangements and symmetric groups. 18.090 introduction to mathematical reasoning mit
Recent instructors include Semyon Dyatlov , Bjorn Poonen, and Paul Seidel. II. Educational Objectives
Before you can prove a theorem, you must understand the structure of a logical argument. Students learn:
"Book of Proof" by Richard Hammack (free online). This is more gentle than Velleman but excellent for drilling. : Understanding infinite sets, cardinality (the "size" of
The course syllabus typically covers foundational tools of logic and set theory, alongside specific concepts from algebra and analysis used to practice these tools: Methods of proof (Direct, Contradiction, Induction). Logical quantifiers ( ∀for all ∃there exists ) and conditional statements (Converse, Contrapositive). Set Theory: Operations on sets and properties of infinite sets. Functions, relations, and cardinality. Algebraic Concepts: Permutations and group-like structures. Introduction to vector spaces and fields. Analysis Concepts: Properties of sequences of real numbers. Introductory epsilon-delta arguments used in limits. Course Logistics Prerequisites: None, though Calculus II is a co-requisite.
Exploration of structures such as permutations , vector spaces , and fields .
The single greatest source of error in undergraduate proofs is the misuse of : "For all" (∀) and "There exists" (∃). 18.090 spends an unusual amount of time on the order of quantifiers. Recent instructors include Semyon Dyatlov , Bjorn Poonen,
: Methods of proof, logic, quantifiers, and set theory.
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