18090 Introduction To Mathematical Reasoning Mit Extra Quality ~repack~ 〈UPDATED — STRATEGY〉
What truly distinguishes 18.090 is its pedagogical execution, which maximizes student engagement and deep conceptual processing. The course bypasses passive lecture listening in favor of active, collaborative synthesis.
(False: there is no single number that yields zero when added to every other number). Essential Proof Techniques Covered in 18.090
Excellent for students who want a step-by-step breakdown with ample practice problems. MIT OpenCourseWare (OCW) What truly distinguishes 18
You assume the opposite of what you want to prove. Then, you show this assumption leads to a logical impossibility. Example: Proving 2the square root of 2 end-root
Do you need specific on any of these topics? Share public link Essential Proof Techniques Covered in 18
Always signal the end of your argument. Use the traditional ( quod erat demonstrandum ) or a solid square tombstone symbol ( Self-Study Resources for 18.090 Success
The fundamental language of all modern mathematics. Quantifiers: Mastering the nuance between "for all" ( ∀for all ) and "there exists" ( ∃there exists 2. The Core Pillars of Proof Writing Example: Proving 2the square root of 2 end-root
| Week | MIT Topic | Extra Quality Action | | :--- | :--- | :--- | | 1-2 | Propositional Logic, Truth Tables | Read Velleman Ch. 1-2. Do 10 truth-table problems without the table (use algebraic simplification). | | 3-4 | Quantifiers, Predicate Logic | Watch TrevTutor’s "Negating Quantifiers." Write the negation of every statement in your lecture notes. | | 5-6 | Direct & Contrapositive Proofs | Read Hammack Ch. 5. For each proof, write the contrapositive statement before starting. | | 7-8 | Proof by Contradiction & Induction | The "(\sqrt2) is irrational" proof is classic. Then attempt a double induction (induction on two variables). | | 9-10 | Set Theory, Russell’s Paradox | Watch VSauce’s "The Banach-Tarski Paradox" (not directly in 18.090, but builds intuition for weird sets). | | 11-12 | Relations & Functions (Injective/Surjective) | Prove that if ( f ) and ( g ) are injective, then ( g \circ f ) is injective. Do it three ways: direct, contrapositive, contradiction. | | 13-14 | Cardinality, Cantor’s Theorem | Read the "Hilbert’s Hotel" essay by George Gamow. Then attempt a proof that the power set of ( \mathbbN ) is uncountable. |
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
Example: Euclid's proof of the infinitude of prime numbers, or proving that 2the square root of 2 end-root is irrational. 4. Mathematical Induction Used to prove a statement is true for all natural numbers (
Exploring the Fundamental Theorem of Arithmetic. 5. Cardinality and Infinite Sets