The primary goal of 18.090 is to transform how students perceive mathematical truths. Instead of asking "What is the answer?", students learn to ask "Why is this true?".
A distinctive MIT feature is the use of LaTeX for final projects. Students write a short paper (3–5 pages) proving a non-trivial theorem of their choice, from Cantor’s diagonal argument to the infinitude of primes in arithmetic progressions (special case).
Before writing proofs, students must understand the language of logic. This section introduces logical connectives (AND, OR, NOT, IMPLIES) and truth tables. Students learn to manipulate quantifiers like "for all" ( ∀for all ) and "there exists" ( ∃there exists
Truth tables, logical connectives (AND, OR, NOT, implication), quantifiers (∀ "for all" and ∃ "there exists"), and the all-important concept of contrapositive. You learn that "If P then Q" is logically equivalent to "If not Q then not P"—a trick that will save your life on exams. 18.090 introduction to mathematical reasoning mit
Mastering injectivity (one-to-one functions), surjectivity (onto functions), and bijectivity (invertible functions).
Proving that if the conclusion is false, the hypothesis must also be false. 3. Basic Structures
18.090 Introduction to Mathematical Reasoning Prerequisites: Calculus I (18.01) is usually required; Calculus II (18.02) is recommended as a co-requisite. Goal: To transition students from solving computational problems (finding $x$) to constructing rigorous mathematical proofs and analyzing abstract structures. The primary goal of 18
Search for MIT OCW 18.090 – the archived site includes problem sets and exams.
Do not use advanced texts like Rudin's Principles of Mathematical Analysis or Munkres' Topology for this class – they assume you already know how to write proofs. 18.090 is where you learn that skill.
Why Hammack? It is exceptionally clear, conversational, and filled with graduated exercises. Chapters progress from simple truth tables to the mind-bending proof of the irrationality of ( \sqrt2 ) to the fact that the real numbers are uncountable. Students repeatedly praise the book for its "hand-holding without being condescending." Students write a short paper (3–5 pages) proving
Students practice "strong induction" (where you assume P(1) through P(k) to prove P(k+1)) and explore its connection to recursion. Classic problems include: proving the sum of the first n integers is n(n+1)/2, proving the Fundamental Theorem of Arithmetic, and analyzing the Tower of Hanoi.
You cannot memorize your way through 18.090. Focus on the underlying structure of a proof rather than the specific numbers used.