18090 Introduction To: Mathematical Reasoning Mit Extra Quality

Book of Proof by Richard Hammack (Free online).

MIT's course 18090, Introduction to Mathematical Reasoning, is designed to introduce students to the basics of mathematical reasoning. This course focuses on teaching students how to read and understand mathematical proofs, how to construct their own proofs, and how to think mathematically. It's a course that lays the foundation for more advanced study in mathematics and related fields by ensuring that students have a solid grasp of mathematical language, logic, and proof techniques.

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Constructing logical matrices to verify the validity of arguments and tautologies. 2. Proof Techniques Book of Proof by Richard Hammack (Free online)

: Formally defining functions, domain, codomain, and composition.

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The logic and reasoning skills developed are highly valued not just in pure math, but in computer science, theoretical physics, economics, and quantitative finance . It's a course that lays the foundation for

: Exploring the properties of infinite sets and cardinality, which challenge basic intuition about "size". 3. Transitioning to Abstract Structures

Students move past casual definitions of "collections of objects" into rigorous axioms:

While 18.090's official MIT OCW page is not publicly listed, the department provides extensive support for students enrolled in the course. Key resources include: and bijectivity (invertible functions).

| 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. |

This is the core of the course. Students learn several foundational proof structures:

18.090: Introduction to Mathematical Reasoning (MIT) teaches students how to construct, write, and critique mathematical proofs. Students often struggle with logical flow, unjustified steps, quantifier errors, and proof structure.

The MIT course serves as a foundational bridge for students transitioning from computational mathematics to the rigorous world of formal proofs. Unlike standard calculus, this course focuses on the art of construction —how to build airtight mathematical arguments and interpret the complex writing of others. Essay: The Gateway to Formal Thought

Deep dives into injectivity (one-to-one), surjectivity (onto), and bijectivity (invertible functions).