that answer must be true. It transforms math from a set of rules you follow into a logical structure you build from the ground up. Proof as a Tool
For students aiming to succeed in MIT's Pure Mathematics or Applied Mathematics tracks, 18.090 provides the essential "mathematical maturity" required for the rigorous proof-heavy courses that follow. 18.0x - MIT Mathematics
A proof is a piece of expository writing. It should read smoothly from top to bottom. Mathematical symbols like ∈is an element of act as verbs and connectives within complete sentences. 2. State Your Strategy Up Front that answer must be true
is true, use definitions and axioms, and logically deduce that conclusion must be true.
is irrational, or that there are infinitely many prime numbers. Mathematical Induction Used to prove statements indexed by natural numbers ( Nthe natural numbers : Prove the statement holds for Inductive Step : Assume it holds for (Inductive Hypothesis) and prove it must therefore hold for Visual Analogy : Knocking down an infinite line of dominoes. Set Theory and Functions: The Language of Higher Math and power sets.
A masterclass in proof by contradiction. 🛠️ The 4 Proof Techniques You Must Master
: Direct proof, contrapositive, contradiction, and mathematical induction. Number Theory Basics : Properties of integers, divisibility, and prime numbers. Department of Mathematics | University of Washington Recommended Resources & "Extra Quality" Content and mathematical induction.
The course typically covers the foundational "alphabet" of higher mathematics: Understanding quantifiers ( ) and logical connectives.
The language of modern mathematics, including unions, intersections, and power sets.
University of Washington's Introduction to Mathematical Reasoning notes cover nearly identical topics to MIT's 18.090. Department of Mathematics | University of Washington sample proof problem
To succeed in 18.090, students are generally expected to have a strong background in: 18.03 (Differential Equations) or 18.06 (Linear Algebra)