At GCSE, there are four approaches to proof and disproof. Deductive (algebraic) proof uses general algebraic expressions to show a result holds for all cases. Proof by exhaustion checks every possible case. Disproof by counter-example gives a single case where a statement fails. Knowing which method suits which statement is as important as carrying it out correctly.
What are the four GCSE proof methods?
| Method | When to use | What you produce |
|---|---|---|
| Deductive proof | Statement must hold for all cases (infinite set) | General algebraic argument |
| Proof by exhaustion | Finite set of cases to check | Verified list of all cases |
| Disproof by counter-example | Statement claimed to always be true, but you suspect it is false | One example that contradicts it |
| Conjecture testing | Exploring whether a pattern is always, sometimes or never true | Example-based reasoning leading to proof or disproof |
Method 1 — deductive algebraic proof
Deductive proof uses algebra to argue for all integers, all even numbers, all triangles, and so on. The key is to represent the general case with a variable.
Useful representations:
- Any integer: n
- Any even integer: 2n
- Any odd integer: 2n + 1 (or 2n − 1)
- Consecutive integers: n, n+1, n+2
- Consecutive even integers: 2n, 2n+2, 2n+4
Worked example 1: Prove that the sum of any two consecutive integers is always odd.
Let the integers be n and n + 1.
Sum = n + (n + 1) = 2n + 1.
2n is even (it is a multiple of 2), so 2n + 1 is odd. ✓
Worked example 2: Prove that the product of any two even numbers is divisible by 4.
Let the even numbers be 2m and 2n (where m and n are integers).
Product = 2m × 2n = 4mn.
4mn is divisible by 4 for all integer values of m and n. ✓
Method 2 — proof by exhaustion
This method lists and checks every possible case. It is only practical when there are finitely many cases — and usually a small number.
Worked example 3: Prove that all square numbers end in 0, 1, 4, 5, 6 or 9 (i.e., never in 2, 3, 7 or 8).
Any integer has one of ten possible last digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The last digit of its square depends only on the last digit of the integer:
| Last digit of n | Last digit of n² |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 6 |
| 5 | 5 |
| 6 | 6 |
| 7 | 9 |
| 8 | 4 |
| 9 | 1 |
All ten cases have been checked. n² ends in 0, 1, 4, 5, 6 or 9 — never in 2, 3, 7 or 8. ✓
Note: this only works by exhaustion because there are exactly 10 digit cases. If the claim were about all integers (no digit restriction), exhaustion would be impossible and deductive proof would be needed.
Method 3 — disproof by counter-example
To disprove a universal claim ("this is true for ALL…"), you only need to find ONE case where it fails. A single counter-example is sufficient — you do not need to show it fails for many cases.
Worked example 4: Disprove the claim: "The square of any integer is always greater than that integer."
Counter-example: let n = 0. Then n² = 0, which is NOT greater than 0 — it is equal. The statement is false. ✓
Alternative counter-example: n = −2. Then n² = 4, and 4 > −2, so this does not disprove it. Keep looking — n = 0 or any fraction between 0 and 1 works.
Worked example 5: Disprove: "n² + n + 41 is prime for all positive integers n."
Try n = 40: 40² + 40 + 41 = 1600 + 40 + 41 = 1681 = 41². This is 41², not prime. Counter-example found. ✓
How do you know which method to choose?
Ask yourself:
- Is the set of cases finite and small? → Exhaustion
- Does the statement claim to hold for all members of an infinite set? → Deductive proof
- Do you think the statement is false? → Look for a counter-example
- Are you unsure whether it is true or false? → Test several examples first, then prove or disprove depending on what you observe.
Important: testing examples only is NOT a proof. Checking that the statement works for n = 1, 2, 3, 4, 5 does not prove it works for all n. You need a deductive argument.
Frequently asked questions
Can I disprove a statement by showing it fails for most cases?
No. A disproof requires exactly one counter-example — no more is needed, and showing many failures (while valid) is not required. Conversely, showing many cases that work does not prove the statement; only a deductive argument or exhaustion does.
What makes an algebraic proof "rigorous" at GCSE?
State what your variable represents ("Let n be any integer"), perform valid algebraic steps (no jumps), and interpret your final expression clearly ("4mn is always divisible by 4"). The examiner is looking for a general argument, correct algebra and a clear conclusion — not just specific examples.
How do I prove something about consecutive even or odd numbers?
Use 2n and 2n + 2 for consecutive even numbers (not 2m and 2n, which are any two even numbers, not necessarily consecutive). Use 2n + 1 and 2n + 3 for consecutive odd numbers. The key is that the variables are linked — one consecutive pair differs by exactly 2.
Is there such a thing as "proof by example" at GCSE?
No — working out a specific example is never accepted as a proof at GCSE or beyond. However, examples can support a conjecture or help you find a counter-example. Once you have spotted the pattern from examples, you must then prove it algebraically or by exhaustion.
Explore proof questions with Professor Pi's Socratic guidance at aitutors.me.