# Proof (Edexcel IAL Maths P2)

> Edexcel International A-Level Mathematics · IAL Maths P2
> Source: https://www.owlsprep.com/study/edexcel-ial-math-p2-proof/

This guide covers core proof skills required for Edexcel IAL Pure Maths 2: logical proof structure, proof by exhaustion for finite case sets, and disproof using counterexamples, with exam-aligned worked examples.

**Prerequisites:** [Basic integer properties (odd, even, prime, divisibility)](https://www.owlsprep.com/study/edexcel-ial-math-p1-number-basics/); [Algebraic substitution skills](https://www.owlsprep.com/study/edexcel-ial-math-p1-algebraic-expressions/)

## Learning objectives

- Understand the standard structure of a valid mathematical proof
- Construct valid proofs by exhaustion for problems with a finite set of cases
- Disprove false mathematical statements using counterexamples

## Structure of a Valid Mathematical Proof

All mathematical proofs follow a consistent structure: start with given assumptions (premises), use a sequence of logical, justified steps, and end with a clear conclusion that directly addresses the statement being proven. Every step must follow from the previous one without gaps in reasoning.

**Valid Proof** — A mathematical argument that starts from agreed or given assumptions, uses only logically sound deductions, and arrives unambiguously at the required conclusion with no unstated leaps of reasoning.

*Example:* To prove the sum of two even integers is even: assume integers a = 2k, b = 2m for integers k,m. Sum = 2k + 2m = 2(k+m), which is 2 times an integer, so even.

**Worked example:** Prove that for any integer n, the product n(n+1) is even.

1. Step 1: Start with the given: n is any integer. There are two mutually exclusive cases for n: n is even, or n is odd.
2. Case 1: n is even. Let n = 2k where k is an integer.
3. $$n(n+1) = 2k(2k + 1)$$
4. This is 2 multiplied by the integer k(2k+1), so it is even.
5. Case 2: n is odd. Let n = 2k + 1 where k is an integer.
6. $$n(n+1) = (2k+1)(2k + 2) = 2(2k+1)(k+1)$$
7. This is 2 multiplied by the integer (2k+1)(k+1), so it is even.
8. Conclusion: Both cases confirm n(n+1) is even for all integers n, so the statement is proven.

**Check your understanding**

Check your understanding of proof structure

1. Which of the following is a required part of a valid proof?

   - A: Testing at least 3 cases
   - B: A clear conclusion referencing the original statement
   - C: Using only algebraic steps
   - D: Including a counterexample

   *Why:* All valid proofs must end with a conclusion that directly addresses the statement being proven. Testing cases is only required for proof by exhaustion, and counterexamples are only used for disproof.

> **Exam tip:** Always explicitly state your conclusion at the end of a proof, referencing the original statement, to earn the final mark for proof questions.

*Calculator:* allowed

## Proof by Exhaustion

Proof by exhaustion (also called proof by cases) is used when the set of possible values for the variables in a statement is finite. You test every single possible case individually to confirm the statement holds for all of them.

**Proof by Exhaustion** — A proof method where you split the problem into a finite number of exhaustive (covering all possibilities) and mutually exclusive cases, then prove the statement holds for every case.

*Example:* Prove that for all odd integers x < 6, x² mod 4 = 1: test x = 1, 3, 5: 1²=1 mod4=1, 3²=9 mod4=1, 5²=25 mod4=1, so statement holds.

**Worked example:** Prove that for all positive integers x and y where x and y are both odd and less than 7, their sum is divisible by 2.

1. Step 1: List all valid values for x and y: odd integers less than 7 are 1, 3, 5.
2. Step 2: Calculate every possible sum of pairs of these values:
3. $$1+1=2, 1+3=4, 1+5=6, 3+1=4, 3+3=6, 3+5=8, 5+1=6,5+3=8,5+5=10$$
4. Step 3: Check divisibility by 2: all sums are even, so all are divisible by 2.
5. Conclusion: All pairs meet the condition, so the statement is proven.

> **tip**
>
> If cases can be grouped by shared properties (e.g. all single-digit even numbers), you can prove for the group instead of testing each individual value to save time, as long as you confirm all cases are covered.

> **Exam tip:** Never skip listing cases for proof by exhaustion: markers will check that you have explicitly accounted for every possible valid value, so list them clearly in your working.

*Calculator:* allowed

## Disproof by Counterexample

Many mathematical statements claim a property holds for *all* values of a variable. To disprove such a statement, you only need to find one single value (a counterexample) where the statement is false.

**Counterexample** — A specific example that satisfies the conditions of a mathematical statement but does not satisfy the conclusion, therefore proving the statement is false.

*Example:* Statement: All prime numbers are odd. Counterexample: 2 is a prime number and it is even, so the statement is false.

**Worked example:** Disprove the statement: For all positive integers n, n² - n + 1 is a prime number.

1. Step 1: Test small positive integer values of n to find one where n² - n +1 is not prime.
2. Test n=1: 1 - 1 +1 =1, which is not prime, but 1 is often excluded from prime testing, so try higher values.
3. Test n=2: 4 - 2 +1 =3, prime. n=3: 9 -3 +1 =7, prime. n=4: 16 -4 +1=13, prime. n=5: 25-5+1=21
4. $$21 = 3 × 7, so it is not a prime number.$$
5. Conclusion: n=5 is a counterexample that meets the conditions (positive integer) but makes the statement false, so the statement is disproven.

> **Exam tip:** You only need to provide one valid counterexample to earn full marks for a disproof question: you do not need to test multiple values, just clearly show your counterexample satisfies the conditions and breaks the conclusion.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Skipping some cases in a proof by exhaustion, e.g. only testing x=1 and 3 for odd x <7, missing x=5
  - Why it fails: Markers require confirmation you have checked every possible valid case, so partial testing invalidates the proof.
  - Correct: List all possible values or case groups explicitly before testing, and cross them off as you confirm each holds.
- **Wrong:** Using an invalid counterexample that does not meet the statement's conditions, e.g. using n=0 to disprove a statement about positive integers.
  - Why it fails: Counterexamples must satisfy all initial conditions of the statement to be valid.
  - Correct: Double-check that your counterexample fits all restrictions (e.g. positive integer, odd, less than 10) listed in the statement you are disproving.
- **Wrong:** Adding unstated assumptions to a proof, e.g. assuming n is positive when the statement says n is any integer.
  - Why it fails: Unjustified assumptions create gaps in reasoning that make your proof invalid.
  - Correct: Only use assumptions explicitly given in the question, or widely accepted mathematical facts (e.g. even numbers are multiples of 2).
- **Wrong:** Forgetting to state the final conclusion referencing the original statement at the end of a proof.
  - Why it fails: The final conclusion is a required marking point for all proof questions, so omitting it will lose you 1 mark even if your working is correct.
  - Correct: End every proof or disproof with a one-sentence conclusion that explicitly states you have proven/disproven the original statement.
- **Wrong:** Trying to use proof by contradiction or induction for P2 proof questions.
  - Why it fails: These methods are out of scope for P2 and will not be accepted as valid solutions for P2 proof questions, even if they are mathematically correct.
  - Correct: Only use the proof methods covered in P2: standard deductive proof, proof by exhaustion, and disproof by counterexample.

## Cheatsheet

| Method | Use Case | Key Steps |
| --- | --- | --- |
| Deductive Proof | General statements about all values | 1. State given assumptions 2. Use logical algebraic steps 3. State conclusion |
| Proof by Exhaustion | Statements about finite sets of values | 1. List all valid cases 2. Test each case 3. Confirm all hold, conclude |
| Disproof by Counterexample | Disproving 'for all' statements | 1. Find one valid value that breaks the statement 2. Show it meets conditions 3. Conclude the statement is false |

## What's next

Now that you have mastered core P2 proof skills, you can apply these reasoning techniques to all other areas of P2, including algebraic methods, trigonometric identities, and sequences and series. Proof questions are often embedded in questions on other topics, so practicing combining these proof skills with content from other P2 units will help you maximise your marks in the exam. Next, move on to algebraic manipulation skills, where you will frequently use proof to verify identity statements.

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