Proof at A-level
A proof is a chain of logical steps from facts you know to the result you want. A-level uses four methods, and choosing the right one is usually the first mark.
Part 1 of 3: Learn it
In short
- Deduction: start from known facts and argue step by step.
- Exhaustion: split into cases that cover every possibility and prove each one.
- Contradiction: assume the opposite and show it leads to something impossible.
Where this is in your specification
Spec points: DfE A1 to A4 (AQA 7357 A1 to A4, Edexcel 9MA0 Pure topic 1, OCR A H240 1.01)
| Board | Topic: Algebra and functions |
|---|---|
| DfE content | A1-A4, B1-B6 |
| AQA 7357 | A1-A4, B1-B6 |
| Edexcel 9MA0 | Pure topics 1 and 2 |
| OCR H240 | 1.01, 1.02 |
| Higher C847 76 | Algebraic and trigonometric skills: quadratics, the discriminant |
The four methods
| Method | When to use it |
|---|---|
| Deduction | the statement follows from algebra or known results |
| Exhaustion | there is a small number of cases, such as odd and even, or n = 1 to 6 |
| Counterexample | to show a statement is false, one case is enough |
| Contradiction | statements with "not" or "no", such as irrationality or infinitely many primes |
Writing it up
- Define every variable: "let n be an integer".
- Show every step; do not skip the algebra.
- Finish with a conclusion that repeats the statement you have proved.
A classic contradiction proof
To prove √2 is irrational, assume it equals a/b with a and b whole numbers sharing no common factor. Squaring gives a² = 2b², so a² is even, so a is even: write a = 2k. Then 4k² = 2b², so b² = 2k² and b is even too. Both a and b are even, which contradicts having no common factor. So √2 is irrational.
Find a counterexample to: "if n is prime, 2ⁿ − 1 is prime".
Show the answer
n = 11: 2¹¹ − 1 = 2047 = 23 × 89.
Part 2 of 3: See it worked
Worked examples
Example 1
Prove that n³ − n is a multiple of 6 for every integer n.
- n³ − n = n(n² − 1) = (n − 1)n(n + 1)
- These are three consecutive integers, so one of them is a multiple of 3 and at least one is even
- So the product is a multiple of 2 × 3 = 6
Answer: n³ − n is the product of three consecutive integers, so it is always a multiple of 6.
Example 2
Prove by exhaustion that every square number is either a multiple of 4 or one more than a multiple of 4.
- Case 1, n even: n = 2k, so n² = 4k², a multiple of 4
- Case 2, n odd: n = 2k + 1, so n² = 4k² + 4k + 1 = 4(k² + k) + 1
- Every integer is even or odd, so all cases are covered
Answer: Proved: n² is 4k² or 4(k² + k) + 1.
Common mistakes
- Checking a few values and calling it a proof.
- In exhaustion, leaving out a case, such as negative numbers or zero.
- In contradiction, assuming the statement itself rather than its opposite.
- Missing the concluding sentence.
Which method would you use to prove there are infinitely many primes?
Show the answer
Proof by contradiction.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Find a counterexample to: "if n is prime, 2ⁿ − 1 is prime".
n = 11: 2¹¹ − 1 = 2047 = 23 × 89.
Which method would you use to prove there are infinitely many primes?
Proof by contradiction.
What must the cases in a proof by exhaustion do?
Cover every possible value, with none left out.
Jobs that use this
- Software developer (se abre en otra pestaña)
- IT security co-ordinator (se abre en otra pestaña)
- Actuary (se abre en otra pestaña)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Estos apuntes están en inglés porque siguen los programas de examen del Reino Unido.
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