Proof / Proof By Cases
Least You Need to Know: Proof by Cases
When a statement splits naturally into a small number of possibilities, prove each case cleanly and make sure the cases cover everything.
The least you need to know
- A proof by cases starts by identifying cases that are exhaustive.
- Each case must end with the same target conclusion.
- Cases should be disjoint or at least clearly separated.
- You still need a reason that the listed cases cover all possibilities.
- Parity arguments are common proof-by-cases problems.
Key notation
n = 2k
n is even
n = 2k+1
n is odd
∨
or
Tiny worked example
- Claim: For every integer `n`, the product `n(n+1)` is even.\n- Case 1: `n` is even, so `n(n+1)` is even because one factor is even.\n- Case 2: `n` is odd, so `n+1` is even, and again the product is even.\n- The two cases cover every integer.
Common mistakes
- Students often forget to say why the cases cover all integers.
- Students sometimes prove different conclusions in different cases.
- A few examples are not a proof by cases.
How to recognize this kind of problem
- The domain naturally splits into even/odd, positive/negative/zero, or a few small congruence classes.
- The problem wording suggests 'consider separately'.
- One formula behaves differently across small categories.