Session 2C
Proof by cases
45-75 min - work through lesson notes, practice, and the MCQ check.
In progress
Account sync
Sign in to keep this session synced across devices.
Account sync disabled
Add Supabase environment variables to enable account sync. Progress is saved locally in this browser for now.
Session tips
- Finish the MCQ before marking complete.
- Check the answer outlines after trying on paper.
- Mark complete to update your certificate progress.
Lesson notes
- Proof by cases is used when the universe naturally splits into a small number of exhaustive and non-overlapping possibilities. You prove the desired conclusion separately in each case, and because every object falls into one of the cases, the conclusion holds universally. Typical splits include: integer is even or odd, real number is >= 0 or < 0, and a set element is in A or not in A. The crucial requirement is exhaustiveness. If your cases do not cover every possibility, the proof is incomplete.
- A classic example is proving that n(n+1) is even for every integer n. The cases are: n is even, so the product is even, or n is odd, so n+1 is even, so the product is even. Case proofs are especially common when working with parity and modular arithmetic, because "mod m" arguments often reduce to checking a small set of residues. The main writing skill here is clarity: state the cases explicitly, finish each case cleanly, and then conclude that since the cases cover all possibilities, the claim is proven.
Worked examples
- For every integer n, n(n+1) is even (case on n even or odd).
Practice
- 1Prove: For every integer n, n^2 and n have the same parity.
- 2Prove: |x| >= 0 for all real x (cases x >= 0 and x < 0).
MCQ
Which case split is most natural to prove |x| >= 0 for all real x?