18090 Introduction To Mathematical Reasoning Mit Extra Quality __hot__ Jun 2026

This is where students first apply their logical tools to concrete mathematical objects:

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

: Assigned every Tuesday and due the following Monday, these problem sets shift the focus from finding the right answer to the clarity and perfection of the written proof. Essential Textbooks and Learning Resources This is where students first apply their logical

| 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. |

Reading a proof and understanding why it is correct is significantly easier than creating that proof yourself. When practicing, always cover up the solution and try to reconstruct the logical path on your own. 2. Write in Complete Sentences | | 5-6 | Direct & Contrapositive Proofs | Read Hammack Ch

The "extra quality" of 18.090 lies in its pedagogical structure, which emphasizes and collaborative solving .

Mathematical reasoning involves the use of logical and systematic methods to solve problems. It requires: Cardinality and Infinite Sets

: Truth tables, quantifiers, and the structure of mathematical statements. Set Theory : Operations on sets, relations, and functions. Proof Techniques

A proof is a piece of explanatory prose, not a random jumble of symbols. Use capital letters, punctuation, and connective words (such as therefore , hence , suppose , and conversely ).

Exploring the Fundamental Theorem of Arithmetic. 5. Cardinality and Infinite Sets