Proof is how mathematics knows things. This mini-lesson covers proof by deduction, proof by exhaustion, disproof by counter-example and proof by contradiction, plus the logical language (⇒, ⇐, ⇔, ≡) that examiners expect you to use precisely.
Where this sits: At CCEA there is no separate 'Proof' content section. Proof is a cross-cutting skill that lives in the specification's overarching themes (mathematical argument, language and structure) and is assessed inside Pure topics in Units AS 1 and A2 1 — algebra, trigonometric identities, sequences and number.
Work through each screen, answer the questions as you go (some are reasoning, some are calculations) and collect ⭐ stars. Press Start when you are ready.
Structure of an argument
The language of proof
A proof is a complete, logical chain from what you are given to what you must show. Every line must follow from the last. Three symbols carry the logic:
P ⇒ Q · P ⇐ Q · P ⇔ Q⇒ "P implies Q" · ⇐ "P is implied by Q" · ⇔ "P if and only if Q" (both directions)
x = 3 ⇒ x² = 9 is true. The converse x² = 9 ⇒ x = 3 is false (x could be −3), so this is not an ⇔ statement.
≡ means identically equal — true for every value, e.g. (x + 1)² ≡ x² + 2x + 1. An equation (=) is only true for particular values.
Marks are lost for writing = where ⇒ is meant, and for a "proof" that starts by assuming what is to be proved. Start from the left-hand side or from the given, and work forwards.
Method 1
Proof by deduction
Deduction is direct proof: start from known facts and definitions and reason forwards.
Worked example — the product of two consecutive integers is even
Let the integers be n and n + 1, n ∈ ℤ.
One of n, n + 1 must be even. If n = 2k then n(n + 1) = 2k(n + 1), a multiple of 2.
If n = 2k + 1 then n + 1 = 2k + 2 = 2(k + 1), so n(n + 1) = 2(k + 1)n, a multiple of 2.
In both cases the product is even. ∎
Set up algebra properly: an even number is 2n; an odd number is 2n + 1; consecutive integers are n, n + 1; consecutive even numbers are 2n, 2n + 2. A generic example ("try n = 4") proves nothing.
Counter-example
Disprove it — the prime-generating polynomial
1Euler noticed that n² + n + 41 gives a prime for many values of n. Find the smallest positive integer n for which the statement "n² + n + 41 is prime for all positive integers n" fails — this single value is a counter-example.
n
Hint: test n = 1, 2, 3, … The expression stays prime for a long time. Look at what happens when n = 41 − 1, because then n² + n + 41 = n(n + 1) + 41 has a factor of 41: 40² + 40 + 41 = 1681 = 41².
Quick check
Is one example enough?
?A student writes: "n² − n + 2 is even, because when n = 3 you get 8, and when n = 5 you get 22." What is wrong with this?
Method 2
Proof by exhaustion
Exhaustion splits the problem into a finite, complete set of cases and checks every one. It is valid only when the cases really do cover everything.
Worked example — every square number is of the form 4k or 4k + 1
Case 1: n even. n = 2m ⇒ n² = 4m², which is 4k with k = m².
Case 2: n odd. n = 2m + 1 ⇒ n² = 4m² + 4m + 1 = 4(m² + m) + 1, which is 4k + 1.
Every integer is even or odd, so the cases are exhaustive. ∎
Consequence: no square number leaves remainder 2 or 3 on division by 4 — a fact worth remembering for number proofs.
Exhaustion
Exhaustion — last digits of squares
2By checking the ten cases n ≡ 0, 1, 2, …, 9 (mod 10), work out how many different digits can appear as the final digit of a square number.
digits
Hint: square 0–9: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 → final digits 0, 1, 4, 9, 6, 5, 6, 9, 4, 1. Count the DISTINCT digits in that list.
Method 3
Disproof by counter-example
To disprove a universal statement ("for all …") you need only one case where it fails. This is the cheapest tool in the box — always test small values first.
"All prime numbers are odd" — counter-example 2.
"If x² > 9 then x > 3" — counter-example x = −4: x² = 16 > 9 but x < 3.
"n² + n + 1 is always prime" — counter-example n = 4: 21 = 3 × 7.
Exam wording: you must state the counter-example and show it fails. Writing "x = −4" alone earns little; write "(−4)² = 16 > 9 but −4 is not greater than 3, so the statement is false."
Quick check
Pick the counter-example
?Which value disproves the statement "if n is an integer then n² > n"?
Method 4
Proof by contradiction
Contradiction: assume the negation of what you want to prove, reason correctly, and reach an impossibility. The assumption must therefore be false.
Worked example — √2 is irrational
Assume √2 is rational: √2 = a/b with a, b integers, b ≠ 0, and the fraction in lowest terms (no common factor).
Then 2b² = a², so a² is even, so a is even: write a = 2c.
Then 2b² = 4c² ⇒ b² = 2c², so b² is even, so b is even.
But then a and b share the factor 2 — contradicting "lowest terms". So √2 is irrational. ∎
Key sub-result used: if a² is even then a is even (if a were odd, a = 2k + 1, then a² = 4k² + 4k + 1 is odd). Quote it.
Contradiction
Euclid's contradiction — infinitely many primes
3Euclid assumes the primes are a finite list. Take the list 2, 3, 5, 7, 11, 13 and form N = (2 × 3 × 5 × 7 × 11 × 13) + 1. Calculate N.
Hint: 2 × 3 × 5 × 7 × 11 × 13 = 30030. Add 1. (N leaves remainder 1 when divided by each listed prime, so none of them divides N — yet N must have a prime factor. In fact N = 59 × 509, and 59 was not on the list: contradiction.)
Algebraic proof
Proving identities and divisibility
Two workhorse techniques appear all over CCEA Pure:
Identities (≡): work on one side only until it becomes the other. Never "do the same to both sides" — that assumes the result.
Divisibility: factorise to expose the factor. Example: (2n + 1)² − (2n − 1)² = 8n, so the difference of the squares of consecutive odd numbers is always a multiple of 8.
Worked example — a multiple of 4
Show n² + 2n is a multiple of 4 whenever n is even.
n even ⇒ n = 2k ⇒ n² + 2n = 4k² + 4k = 4(k² + k). ∎ It is 4 × an integer.
Algebraic proof
Divisibility proof
4Expand and simplify (2n + 1)² − (2n − 1)². The result is always a multiple of k. State the largest integer k for which this is guaranteed for all integers n.
Hint: (4n² + 4n + 1) − (4n² − 4n + 1) = 8n. Since n can be any integer (including 1), the largest guaranteed factor is the coefficient itself.
Sort it
Which method of proof?
Tap a task, then tap the method of proof it calls for.
🧠 Deduction
🔢 Exhaustion
❌ Counter-example
Counter-example
A famous false conjecture
5"If p is prime then 2ᵖ − 1 is prime." It works for p = 2, 3, 5 and 7. Find the smallest prime p for which it fails.
p
Hint: 2² − 1 = 3 ✓, 2³ − 1 = 7 ✓, 2⁵ − 1 = 31 ✓, 2⁷ − 1 = 127 ✓. Now test the next prime: 2¹¹ − 1 = 2047 = 23 × 89, which is not prime.
Quick check
Contradiction spotted
?In a proof by contradiction that there is no largest even number, what is the correct opening line?
Match it
Match the symbol to its meaning
Tap a statement on the left, then its meaning on the right.
Statement
Answer
Quick check
Which is a valid proof?
?Which of these is a valid proof that the sum of two odd numbers is even?
Examiner traps
Where proofs lose marks
Assuming the result. In an identity proof, never operate on both sides. Start with one side and transform it.
Using one letter twice. "Two odd numbers" are 2m + 1 and 2n + 1, not 2n + 1 and 2n + 1.
Examples instead of argument. Testing n = 1, 2, 3 proves nothing (see n² + n + 41, which survives to n = 39).
Counter-example without justification. State the value AND show the statement fails for it.
Contradiction with the wrong negation. The negation of "for all n, P(n)" is "there exists an n with not-P(n)" — not "for all n, not-P(n)".
Write the closing line. Finish with a sentence: "…so the statement is true for all integers n." Examiners look for the conclusion.
Recap
The big ideas to know
Deduction: reason forwards from definitions — even = 2n, odd = 2n + 1, consecutive = n, n + 1
Exhaustion: split into a complete finite set of cases and check every one
Counter-example: one failing case disproves a universal statement (state it AND show it fails)
Contradiction: assume the negation, derive an impossibility (√2 irrational; infinitely many primes)
Language: ⇒ one-way · ⇔ both ways · ≡ identity, true for all values
Proof is not a topic you meet once — at CCEA it is a skill examined inside algebra, trigonometry and sequences. Press Finish to see your score.
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