A1PureFoundation

The Structure of a Mathematical Proof

Q: What is the difference between and?

P Q means P forces Q, but Q might be true for other reasons too. P Q means the two statements are equivalent — each is true exactly when the other is. Only use when both directions genuinely hold.

A proof is a chain of logic that takes you from things already known to be true to a new statement, with every link justified. Master the language of proof here and every technique that follows becomes a variation on one idea.

25 min Video by TLMaths Proof

The big picture

Mathematics is the only subject where you can be completely certain. A scientist gathers evidence and a verdict can be overturned tomorrow; a mathematician proves a statement and it is settled forever. That certainty is what proof buys you — and it is exactly why AQA puts proof first and threads it through the whole A-Level.

The catch is that certainty has a price: checking that a statement works for a few numbers is not proof, no matter how many you try. This lesson is about the difference between “it looks true” and “it is true”, and about the small, precise vocabulary you need to argue the second one.

A-Level Maths A1-01: Introduction to ProofWatch the walkthrough, then work through the notes below.

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What you'll be able to do

Explain what separates a proof from a collection of examples
Use the algebraic vocabulary of proof — even, odd, consecutive and rational numbers
Read and use the logic symbols $\Rightarrow$ , $\Leftarrow$ and $\Leftrightarrow$
Set out a proof so that every line follows from the last and earns the marks

What actually counts as a proof

A $conjecture$ is a statement we believe might be true. A $proof$ turns that belief into certainty: it is a sequence of steps, starting from definitions and results already established, where each step follows logically from the one before, ending at the statement we wanted. Once proved, a conjecture becomes a $theorem$ .

The single most important idea in the whole strand is this: $examples are not a proof$ . Suppose you claim “ $n^{2} - n + 41$ is prime for every positive integer $n$ ”. Test it: $n = 1$ gives $41$ , $n = 2$ gives $43$ , $n = 3$ gives $47$ — all prime. Test twenty more values; still prime. It looks certain. But at $n = 41$ the expression is $4 1^{2}$ , which is not prime. A pattern that holds for the first forty cases can still fail.

So to prove a statement about $all$ numbers, you cannot check them one at a time — there are infinitely many. You have to argue about a general number and show the claim is forced to hold whatever that number is.

Why this works

Checking cases can only ever do two things: it can $disprove$ a statement (one failure is fatal), or it can $suggest$ one worth proving. It can never confirm a “for all” claim, because you can never finish the list.

The raw material: how to write numbers algebraically

Almost every proof at A-Level is about integers, so you need to translate words like “even” into algebra you can manipulate. These are the workhorses:

Crucially, when two numbers must be different, give them different letters. “Two even numbers” is $2 m$ and $2 n$ , not $2 n$ and $2 n$ — writing $2 n$ twice secretly assumes they are equal and will cost you the proof.

even = 2 n, odd = 2 n + 1 (n \in Z)

Any even number is a multiple of 2; any odd number is one more than an even number.

n, n + 1, n + 2 are consecutive integers

Consecutive numbers differ by 1 — a single letter links them all.

rational = \frac{p}{q}, p, q \in Z, q \neq = 0

A rational number is an exact ratio of two integers.

1A multiple of 3 is

3 n

for some integer

n

2Three consecutive integers are

n

n + 1

and

n + 2

3Their sum is

n + (n + 1) + (n + 2) = 3 n + 3 = 3 (n + 1)

AnswerThe sum is

3 (n + 1)

— automatically a multiple of 3.

Tip — The letters $N, Z, Q, R$ name the natural numbers, integers, rationals and reals. Writing $n \in Z$ (“ $n$ is an integer”) tells the reader exactly what kind of number you are arguing about.

The logic symbols: implies and is equivalent to

Proof is about how statements connect, and three arrows capture almost all of it. $P \Rightarrow Q$ means “if $P$ is true then $Q$ must be true” — $P$ implies $Q$ . $P \Leftarrow Q$ is the same arrow reversed: “ $P$ is implied by $Q$ ”. And $P \Leftrightarrow Q$ means both at once — $P$ and $Q$ are $equivalent$ , each guaranteeing the other.

The direction matters enormously. “It is raining $\Rightarrow$ the ground is wet” is true, but the reverse is not: the ground could be wet from a hose. So this is $\Rightarrow$ , not $\Leftrightarrow$ . Confusing a one-way implication with a two-way one is the most common logical error at A-Level.

P \Rightarrow Q “ P implies Q ”

P

is enough (sufficient) to force

Q

P \Leftrightarrow Q “ P if and only if Q ”

Each implies the other — you can travel the logic both ways.

1(a)

x = 3 \Rightarrow x^{2} = 9

is true. But

x^{2} = 9 \Rightarrow x = 3

is false, because

x

could be

- 3

. So only

\Rightarrow

holds.

2(b)

2 x = 10 \Rightarrow x = 5

, and

x = 5 \Rightarrow 2 x = 10

. Both directions work.

Answer(a)

x = 3 \Rightarrow x^{2} = 9

(one-way). (b)

2 x = 10 \Leftrightarrow x = 5

(two-way).

Why this works

When you solve an equation by doing the same thing to both sides, you are chaining $\Leftrightarrow$ steps — which is why solving is really a proof that your answer is the only one. Squaring is the famous exception: $x = 2 \Rightarrow x^{2} = 2$ , but squaring can introduce extra solutions, so it is $\Rightarrow$ , not $\Leftrightarrow$ .

Setting out a proof that earns the marks

A full-mark proof reads like a short, self-contained argument. Three habits get you there. $Start from something true$ — a definition, a given fact, or a known result — never from the thing you are trying to prove. $Justify every line$ so a reader can see why each step follows. And $finish with a clear conclusion$ that names what you have shown, such as “which is a multiple of 3, as required”.

Examiners award marks for the logical journey, not just the destination. A correct final line with a broken chain above it scores badly; a clearly reasoned argument with one slip usually keeps most of the marks. Communicating the logic $is$ the skill being tested.

1Let the two odd numbers be

2 m + 1

and

2 n + 1

, where

m, n \in Z

. (Different letters — they need not be equal.)

2Their sum is

(2 m + 1) + (2 n + 1) = 2 m + 2 n + 2

3Factor out 2:

2 m + 2 n + 2 = 2 (m + n + 1)

4Since

m + n + 1

is an integer,

2 (m + n + 1)

is a multiple of 2, i.e. even.

AnswerThe sum of two odd numbers is

2 (m + n + 1)

, which is even.

■

Tip — End a proof with a full concluding sentence, not just a boxed answer. The symbol $■$ (or writing “as required”) signals “this is where the argument closes”.

Think like an examiner

The command word “ $Prove$ ” or “ $Show that$ ” means a general argument is required — a couple of numerical examples will score zero even if the statement is true.
Marks are split between $starting correctly$ (a general expression such as $2 n + 1$ ) and $reasoning to the conclusion$ . Setting up good algebra at the start earns method marks even if you stall.
A final “therefore” statement that refers back to the original claim (“…so the sum is always even”) is often worth a mark on its own. Never leave the proof hanging on a line of algebra.
Watch your quantifiers: a statement about “any” or “all” integers must be proved in general, whereas “there exists” only needs one example.

Common misconceptions

Proving a statement true by testing

n = 1, 2, 3

and a few more.

Use a general integer such as

2 n + 1

. Examples can disprove a claim but never prove a “for all” statement.

The one exception is proof by exhaustion, where the cases genuinely are finite — covered later in this chapter.

Using the same letter, e.g.

2 n

and

2 n

, for two numbers that should be independent.

Use different letters,

2 m

and

2 n

. Reusing a letter secretly assumes the two numbers are equal.

Writing

\Rightarrow

and

\Leftrightarrow

interchangeably.

Only use

\Leftrightarrow

when the implication genuinely runs both ways. If the converse can fail, it is

\Rightarrow

Starting the proof by writing down the thing you are trying to prove and rearranging it.

Begin from definitions or known facts and work towards the statement, so nothing is assumed.

The vocabulary of proof

2 n

Even number

(n \in Z)

2 n + 1

Odd number.

n, n + 1

Consecutive integers.

\frac{p}{q}, q \neq = 0

Rational number.

P \Rightarrow Q

P

implies

Q

(one-way).

P \Leftrightarrow Q

P

if and only if

Q

(both ways).

Stretch yourself

A student writes: “ $x^{2} > x$ , therefore $x > 1$ .” Using an implication arrow, explain precisely what is wrong, and state the correct relationship.

Hint — Try a negative value, or a value between 0 and 1, in $x^{2} > x$ .

Questions students ask

Why isn’t testing lots of examples enough to prove something?

Because a “for all” statement makes a claim about infinitely many numbers, and no finite list can cover them all. The expression

n^{2} - n + 41

is prime for the first 40 integers but fails at

n = 41

— proof that patterns can break exactly where you stop checking.

What is the difference between

\Rightarrow

and

\Leftrightarrow

P \Rightarrow Q

means

P

forces

Q

, but

Q

might be true for other reasons too.

P \Leftrightarrow Q

means the two statements are equivalent — each is true exactly when the other is. Only use

\Leftrightarrow

when both directions genuinely hold.

How much working do I need to show in a proof question?

Enough that a reader can follow every step without guessing. Marks are awarded for the reasoning, so justify each line and finish with a sentence that states what you have proved.

Do I have to write the little square

■

at the end?

No — it is optional. What matters is a clear concluding statement such as “as required” that signals the argument is complete and refers back to the original claim.

Key takeaways

A proof argues from known truths to a new statement, one justified step at a time.
Examples can disprove a claim but never prove a statement about all numbers.
Translate words into algebra: even $= 2 n$ , odd $= 2 n + 1$ , rational $= \frac{p}{q}$ ; use different letters for independent numbers.
$\Rightarrow$ is a one-way implication; $\Leftrightarrow$ works both ways — never swap them.
Start from definitions, justify each line, and close with a clear conclusion.

How this fits the course

Build on

Algebraic manipulation (GCSE)soon

Leads to

Proof by Deduction

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