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💡 Check Every Case!
Sometimes a claim is about a small, finite set of things. When there are only a few cases to check, you can prove the claim by checking every single one.
This is called proof by exhaustion (or proof by cases) — you "exhaust" all the possibilities!
⚠️ This only works when the number of cases is small enough to check.
Example: Every single-digit prime > 2 is odd
Single-digit primes: 2, 3, 5, 7 — only 4 cases to check!
| Prime | Is it > 2? | Even or Odd? | Claim holds? |
|---|
| 2 | No (skip) | Even | — |
| 3 | Yes | Odd | ✓ |
| 5 | Yes | Odd | ✓ |
| 7 | Yes | Odd | ✓ |
✅ All 3 cases check out — the claim is PROVED!
We didn't just try examples — we checked EVERY case. That's a proof!
Example: The square of any single-digit number ≤ 81
Single-digit numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9 — check all 9!
| n | n² | ≤ 81? |
|---|
| 1 | 1 | ✓ |
| 2 | 4 | ✓ |
| 3 | 9 | ✓ |
| 4 | 16 | ✓ |
| 5 | 25 | ✓ |
| 6 | 36 | ✓ |
| 7 | 49 | ✓ |
| 8 | 64 | ✓ |
| 9 | 81 | ✓ |
✅ All 9 cases work — proved by exhaustion!
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1 Prove It by Checking All Cases ✅
Complete the table to prove (or disprove!) each claim.
Claim: "Every even number from 2 to 12 can be written as the sum of two prime numbers."
Hint: The even numbers are 2, 4, 6, 8, 10, 12
| Even Number | Two Primes That Add to It | Works? |
|---|
| 2 | | |
| 4 | 2 + 2 | ✓ |
| 6 | | |
| 8 | | |
| 10 | | |
| 12 | | |
Claim: "For every whole number from 1 to 6, n² + n is always even."
| n | n² | n² + n | Even? |
|---|
| 1 | 1 | 2 | ✓ |
| 2 | 4 | | |
| 3 | | | |
| 4 | | | |
| 5 | | | |
| 6 | | | |
2 Can You Use Proof by Cases? 🤔
For each claim, decide: can it be proved by checking all cases? Why or why not?
a) "Every month of the year has at least 28 days."
Can you check all cases?
How many cases? Answer:
b) "Every even number is divisible by 2."
Can you check all cases?
Why or why not?
c) "Every two-digit perfect square is less than 100."
Can you check all cases?
The two-digit perfect squares are:
Answer Key:
1-Claim 1) 2 = not possible with two primes (trick! — 2 can't be written as sum of two primes if we require primes > 0, since 1 isn't prime; Goldbach's conjecture starts at 4); 4=2+2, 6=3+3, 8=3+5, 10=5+5 or 3+7, 12=5+7. —
1-Claim 2) All even: 2, 6, 12, 20, 30, 42 — PROVED —
2a) Yes, 12 months — PROVED — 2b) No! Infinite even numbers — need a different proof — 2c) Yes! 16,25,36,49,64,81 — all < 100 ✓