18.090 — Introduction to Mathematical Reasoning (overview & key content)
📚 Core Topic Breakdown
Student Testimonials (Synthesized from MIT Course Evaluations)
"I came to MIT thinking I was bad at math. Turns out, I was bad at logic. 18.090 fixed that. It was the hardest 6 credits I've ever taken, and the most valuable."
— Anonymous, Course Evaluation 2022
"The first time I had to present a proof at the board, I forgot how to breathe. By week 10, I was arguing with the TA about the difference between 'there exists unique' and 'there exists at least one.' I grew more in 14 weeks than in 4 years of high school."
— Course Evaluation 2019
"If you are Course 18 (Math major), do not skip 18.090. I tried to go straight to 18.100 and got destroyed. I took 18.090 the next semester and got an A in 18.100. Correlation is not causation, but..."
— Reddit r/mit comment
1. Introduction
For many second-year undergraduates at MIT, the transition from problem sets involving derivatives and integrals to proving theorems about limits or number theory can be jarring. 18.090 – Introduction to Mathematical Reasoning is explicitly designed to ease this transition. Unlike standard “transition to proof” courses elsewhere, 18.090 leverages MIT’s problem-solving culture while emphasizing clarity, rigor, and creativity in logical argumentation.
The course is typically taken after single-variable calculus (18.01) and before real analysis (18.100) or abstract algebra (18.700). Its credit load is 3-0-9 (3 class hours, 0 lab hours, 9 expected study hours per week), reflecting MIT’s intensive unit system.
🛠️ MIT OpenCourseWare (OCW) Resources
While MIT often cycles through different variations of this course (sometimes combined with Discrete Math), the best resource on MIT OCW is:
- Course: Mathematics for Computer Science (6.042J / 18.062J)
- Note: This is often the "sister" course to 18.090. It covers Induction, Graph Theory, and Relations very thoroughly.
- Video Lectures: Look for Tom Leighton or Albert Meyer’s lectures on OCW. They are legendary for explaining Induction and Relations.
Study Strategies for Success
Transitioning to proof-based math is difficult. Here is how to succeed:
- Start Problem Sets Early: You cannot "cram" proofs. If you get stuck, you need time to let your subconscious work on it.
- Write in Sentences: A proof is an essay written in mathematical symbols.
- Bad: $x > 0 \implies x^2 > 0$.
- Good: "Let $x$ be a real number. If $x > 0$, then $x^2$ must also be greater than zero because the product of two positive numbers is positive."
- The "Grandma Test": Read your proof out loud. Does it sound like nonsense, or does it sound like a logical argument? If you can't follow your own logic, neither can the grader.
- Understand Definitions: In this course, definitions are your toolbox. You cannot prove a set is "countable" if you do not have the exact definition of "countable" memorized.
- Go to Office Hours: The TA and Professor can explain the subtle logic gaps in your writing that you might miss on your own.
How to Access Materials (Self-Study)
If you are not currently enrolled at MIT, you can take this course for free via MIT OpenCourseWare (OCW).
- Search: "MIT 18.090 OCW".
- Video Lectures: Look for the version taught by Prof. Munkres or Prof. Lian. The lectures are often recorded and available on YouTube.
- Assignments: OCW provides PDFs of problem sets and exams with solutions.
Recommended Textbooks for Self-Study:
- Book of Proof by Richard Hammack (Free, concise, excellent exercises).
- How to Prove It: A Structured Approach by Daniel Velleman (Excellent for logic and strategy).
- Journey into Mathematics: An Introduction to Proofs by Joseph Rotman.
1. Logic and Foundations
This is the grammar of mathematics. You cannot write a proof without understanding the syntax.
- Propositional Logic: Negation ($\neg$), Conjunction ($\land$), Disjunction ($\lor$), Implication ($\implies$).
- Truth Tables: The mechanical way to verify logical statements.
- Quantifiers: The difference between "There exists" ($\exists$) and "For all" ($\forall$). Crucial for negating statements.
- Sets: Subsets, unions, intersections, and power sets.
Representative theorems/problems (short list)
- Prove that √2 is irrational.
- Show there are infinitely many primes.
- Prove that if a function f has a left and right inverse then they are equal and f is bijective.
- Use induction to prove formulas for sums (e.g., sum of first n integers, sum of squares).
- Prove basic properties of divisibility and gcd using Euclidean algorithm.
- Prove that every equivalence relation partitions its set, and conversely.
18.090 Introduction To Mathematical Reasoning Mit Here
18.090 — Introduction to Mathematical Reasoning (overview & key content)
📚 Core Topic Breakdown
Student Testimonials (Synthesized from MIT Course Evaluations)
"I came to MIT thinking I was bad at math. Turns out, I was bad at logic. 18.090 fixed that. It was the hardest 6 credits I've ever taken, and the most valuable."
— Anonymous, Course Evaluation 2022
"The first time I had to present a proof at the board, I forgot how to breathe. By week 10, I was arguing with the TA about the difference between 'there exists unique' and 'there exists at least one.' I grew more in 14 weeks than in 4 years of high school."
— Course Evaluation 2019
"If you are Course 18 (Math major), do not skip 18.090. I tried to go straight to 18.100 and got destroyed. I took 18.090 the next semester and got an A in 18.100. Correlation is not causation, but..."
— Reddit r/mit comment 18.090 introduction to mathematical reasoning mit
1. Introduction
For many second-year undergraduates at MIT, the transition from problem sets involving derivatives and integrals to proving theorems about limits or number theory can be jarring. 18.090 – Introduction to Mathematical Reasoning is explicitly designed to ease this transition. Unlike standard “transition to proof” courses elsewhere, 18.090 leverages MIT’s problem-solving culture while emphasizing clarity, rigor, and creativity in logical argumentation.
The course is typically taken after single-variable calculus (18.01) and before real analysis (18.100) or abstract algebra (18.700). Its credit load is 3-0-9 (3 class hours, 0 lab hours, 9 expected study hours per week), reflecting MIT’s intensive unit system. "I came to MIT thinking I was bad at math
🛠️ MIT OpenCourseWare (OCW) Resources
While MIT often cycles through different variations of this course (sometimes combined with Discrete Math), the best resource on MIT OCW is:
- Course: Mathematics for Computer Science (6.042J / 18.062J)
- Note: This is often the "sister" course to 18.090. It covers Induction, Graph Theory, and Relations very thoroughly.
- Video Lectures: Look for Tom Leighton or Albert Meyer’s lectures on OCW. They are legendary for explaining Induction and Relations.
Study Strategies for Success
Transitioning to proof-based math is difficult. Here is how to succeed: "The first time I had to present a
- Start Problem Sets Early: You cannot "cram" proofs. If you get stuck, you need time to let your subconscious work on it.
- Write in Sentences: A proof is an essay written in mathematical symbols.
- Bad: $x > 0 \implies x^2 > 0$.
- Good: "Let $x$ be a real number. If $x > 0$, then $x^2$ must also be greater than zero because the product of two positive numbers is positive."
- The "Grandma Test": Read your proof out loud. Does it sound like nonsense, or does it sound like a logical argument? If you can't follow your own logic, neither can the grader.
- Understand Definitions: In this course, definitions are your toolbox. You cannot prove a set is "countable" if you do not have the exact definition of "countable" memorized.
- Go to Office Hours: The TA and Professor can explain the subtle logic gaps in your writing that you might miss on your own.
How to Access Materials (Self-Study)
If you are not currently enrolled at MIT, you can take this course for free via MIT OpenCourseWare (OCW).
- Search: "MIT 18.090 OCW".
- Video Lectures: Look for the version taught by Prof. Munkres or Prof. Lian. The lectures are often recorded and available on YouTube.
- Assignments: OCW provides PDFs of problem sets and exams with solutions.
Recommended Textbooks for Self-Study:
- Book of Proof by Richard Hammack (Free, concise, excellent exercises).
- How to Prove It: A Structured Approach by Daniel Velleman (Excellent for logic and strategy).
- Journey into Mathematics: An Introduction to Proofs by Joseph Rotman.
1. Logic and Foundations
This is the grammar of mathematics. You cannot write a proof without understanding the syntax.
- Propositional Logic: Negation ($\neg$), Conjunction ($\land$), Disjunction ($\lor$), Implication ($\implies$).
- Truth Tables: The mechanical way to verify logical statements.
- Quantifiers: The difference between "There exists" ($\exists$) and "For all" ($\forall$). Crucial for negating statements.
- Sets: Subsets, unions, intersections, and power sets.
Representative theorems/problems (short list)
- Prove that √2 is irrational.
- Show there are infinitely many primes.
- Prove that if a function f has a left and right inverse then they are equal and f is bijective.
- Use induction to prove formulas for sums (e.g., sum of first n integers, sum of squares).
- Prove basic properties of divisibility and gcd using Euclidean algorithm.
- Prove that every equivalence relation partitions its set, and conversely.
