This course builds your toolkit for rigorous proof construction.
The official 18.090 problem sets are notoriously challenging. But to get extra quality, you need additional sources.
1. The "Gold Standard" Problems from the Harvard Math 23a Archive Harvard’s equivalent (Math 23a) offers problem sets that focus on writing quality. Try this one:
"Prove that ( \sqrt2 + \sqrt3 ) is irrational." (Hint: Square it, then use the rational root theorem—a connection to algebra often missed.)
2. The MIT PRIMES Problem-Solving Database MIT’s PRIMES (Program for Research in Mathematics, Engineering, and Science) has a public archive of "proof readiness" problems. These are short, elegant, and brutal. This course builds your toolkit for rigorous proof
3. Generating Your Own Proofs with AI (Ethically) An extra quality modern technique: Use a large language model (like GPT-4) not to solve the problem, but to critique your proof.
The Mistake: Proving ( P(k) \implies P(k+1) ) but forgetting the base case. Extra Quality Fix: Always check the smallest base case (often ( n=0 ) or ( n=1 )). Then check the next one manually. Induction without a base case is like building a ladder that doesn’t touch the ground.
You will stare at a blank page for 30 minutes. This is normal. This is "mathematical weightlifting." If you look up the solution immediately, you rob yourself of the neural pathway growth required for the exam.
After submission, the tool assigns scores (1–4) in: "Prove that ( \sqrt2 + \sqrt3 ) is irrational
Accompanied by specific, actionable comments (not just a score).
5.1. LaTeX Everywhere
5.2. Voice-to-Proof
5.3. Dark Mode for Theorem-Proving
Primary Recommendation: How to Prove It: A Structured Approach by Daniel J. Velleman (3rd Edition).
Secondary Recommendation (For the Ambitious): Book of Proof by Richard Hammack (Free online).
Tertiary (The MIT Culture Pick): How to Solve It by George Pólya.
You will master the standard architectures of mathematical proof: Accompanied by specific
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