Proof Writing & Problem Solving· Study notes
Proof
How to read, find and write a proof: direct proof, contrapositive and contradiction, induction, pigeonhole and extremal arguments, quantifiers and counterexamples, problem-solving strategy, and the craft of writing it up.
What you'll be able to do
- Write a correct, complete proof in full sentences
- Choose the right proof technique for a claim
- Find proofs with small cases, working backwards and invariants
- Audit a proof for gaps
Read the notes as a guided lesson
Walk the notes one box at a time — prose, worked examples, quick checks — with a timed practice round after every section. For the full course with interactive lessons, go back to the course.
Every test ends with a score by topic and a list of what to revisit. Test out of a topic with 80% or better to mark it mastered.
Topics
Direct Proof
4 lessonsFrom hypothesis to conclusion, one justified step at a time
Contrapositive and Contradiction
5 lessonsProving by assuming the opposite
Induction
5 lessonsProving a statement for every natural number
Pigeonhole and Extremal Arguments
5 lessonsProving that something must exist without finding it
Existence, Uniqueness and Counterexamples
5 lessonsThe four kinds of claims and what each needs
Problem-Solving Strategies
5 lessonsHow to find a proof when you do not know where to start
Writing It Up
4 lessonsHow a proof should read on the page