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Proof
Proof
This lesson covers the topic of mathematical proof as required by the A-Level Mathematics specification. Proof lies at the heart of mathematics — it is the rigorous process by which we establish that a mathematical statement is always true, not just true for a few examples. Understanding proof is essential for developing logical reasoning and is examined directly in A-Level papers.
Why Do We Need Proof?
In mathematics, a statement may appear to be true based on a handful of examples, but this does not guarantee it holds in every case. A proof provides a logical argument that demonstrates a statement is true for all possible cases. Without proof, mathematics would be built on assumptions rather than certainty.
Example of a false conjecture:
"All prime numbers are odd."
This seems reasonable, but 2 is a prime number and it is even. A single counterexample is enough to disprove a statement.
Exam Tip: When asked to "prove" something, showing it works for a few values is not a proof. You must use a logical argument that covers all cases.
Types of Proof
1. Proof by Deduction
This is the most common type of proof at A-Level. You start from known facts, definitions, or previously proven results and use logical steps to arrive at the statement you want to prove.
Example: Prove that the sum of two consecutive integers is always odd.
Let the two consecutive integers be n and n + 1, where n is an integer.
Their sum is:
n + (n + 1) = 2n + 1
Since 2n is always even (it is a multiple of 2), 2n + 1 is always odd.
Therefore, the sum of two consecutive integers is always odd. ∎
2. Proof by Exhaustion
This involves checking every possible case. It is only feasible when the number of cases is finite and small.
Example: Prove that n² + n + 1 is odd for all integers n where 1 ≤ n ≤ 4.
| n | n² + n + 1 | Odd? |
|---|---|---|
| 1 | 1 + 1 + 1 = 3 | Yes |
| 2 | 4 + 2 + 1 = 7 | Yes |
| 3 | 9 + 3 + 1 = 13 | Yes |
| 4 | 16 + 4 + 1 = 21 | Yes |
All cases have been checked, so the statement is true for 1 ≤ n ≤ 4. ∎
Note: this does not prove the statement for all n — proof by exhaustion only covers the stated cases.
3. Proof by Contradiction
Assume the opposite of what you want to prove, and show that this leads to a logical contradiction. Since the assumption produces a contradiction, it must be false, and therefore the original statement is true.
Example: Prove that √2 is irrational.
Assume, for contradiction, that √2 is rational. Then we can write:
√2 = a/b
where a and b are integers with no common factors (i.e., the fraction is in its simplest form).
Squaring both sides: 2 = a²/b², so a² = 2b².
This means a² is even, so a must be even. Write a = 2k.
Then (2k)² = 2b², so 4k² = 2b², giving b² = 2k².
This means b² is even, so b must be even.
But if both a and b are even, they share a common factor of 2 — this contradicts our assumption that the fraction was in simplest form.
Therefore, √2 is irrational. ∎
4. Disproof by Counter-Example
To disprove a statement, you only need to find one example where it fails.
Example: Disprove the statement: "n² > 2n for all positive integers n."
Try n = 1: 1² = 1 and 2(1) = 2. Since 1 < 2, the statement is false.
Counter-example: n = 1. ∎
Algebraic Proof Techniques
When constructing algebraic proofs, the following representations are essential:
| Statement | Algebraic Representation |
|---|---|
| An even number | 2n (where n is an integer) |
| An odd number | 2n + 1 (where n is an integer) |
| Consecutive integers | n, n + 1, n + 2, ... |
| A multiple of 3 | 3n |
| The square of an integer | n² |
Example: Prove that the product of two odd numbers is always odd.
Let the two odd numbers be 2a + 1 and 2b + 1, where a and b are integers.
(2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1
Since 2(2ab + a + b) is even, adding 1 makes it odd.
Therefore, the product of two odd numbers is always odd. ∎
Example: Prove that (n + 1)² − (n − 1)² = 4n for all integers n.
LHS = (n + 1)² − (n − 1)²
= (n² + 2n + 1) − (n² − 2n + 1)
= n² + 2n + 1 − n² + 2n − 1
= 4n = RHS ∎
Summary
- Proof by deduction: Use logical steps from known facts to reach the conclusion.
- Proof by exhaustion: Check every possible case (only when finite).
- Proof by contradiction: Assume the opposite, derive a contradiction.
- Disproof by counter-example: Find one case where the statement fails.
- Always represent even numbers as 2n, odd numbers as 2n + 1, and consecutive integers as n, n + 1, ...
- Define your variables clearly (e.g. "Let n be an integer").
- Write a concluding sentence that directly addresses what was asked.
Exam Tip: In the exam, clearly state what you are assuming and what you are proving. Show every algebraic step and conclude with a clear statement such as "Therefore, the statement is true" or "This contradicts our assumption, so the original statement is proven." Look for the command word: "Prove" requires a full logical argument; "Show that" requires you to arrive at a given answer; "Disprove" requires a counter-example.