18.090 Introduction To Mathematical Reasoning Mit Online

MIT 18.090: Introduction to Mathematical Reasoning For many students arriving at MIT, mathematics has been a journey of calculation—solving for

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.

Core Learning Objectives

The primary goal is not to memorize facts, but to master the methodology of mathematics. By the end of the course, you should be able to: 18.090 introduction to mathematical reasoning mit

• 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."

• Problem sets: 30%
• Midterm exam: 25%
• Final exam: 30%
• Proof writing project: 15%

For the student standing at the threshold of advanced mathematics, 18.090 is the key that unlocks the door. Behind that door is a universe of infinite precision, elegant abstraction, and rigorous beauty. Turn the key. The proof awaits. MIT 18

: Familiarize yourself with basic set operations (union, intersection, complement), subsets, and power sets. Integer Properties Prove that √2 is irrational

• And (∧) and Or (∨) – including the crucial distinction between inclusive and exclusive or.
• Implication (⟹) – the most counter-intuitive concept for novices. Why is "If the moon is made of cheese, then 2+2=5" considered a true statement? (Answer: Vacuous truth).
• Contrapositive, Converse, and Inverse – Students drill until they can instinctively replace a difficult implication with its contrapositive.