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Disproof by Counter-Example

Q: How do I disprove an “if P then Q” statement?

Find a case where P is true but Q is false. For “if a^2 > b^2 then a > b”, take a=-3, b=2: the hypothesis holds but the conclusion fails.

To destroy a “for all” statement you need just one example where it fails. This lesson is about that beautiful asymmetry — proving takes a general argument, but disproving takes a single well-chosen number.

20 min Video by TLMaths Proof

The big picture

There is a striking imbalance at the heart of proof. To show a statement is true for all numbers you must argue in general — no finite check will do. But to show it is $false$ , you only need one example that breaks it. A single failure is fatal to a universal claim. That is why the exam pairs “prove” questions (which demand generality) with “disprove” questions (which reward a sharp eye for the one case everyone forgets).

A-Level Maths A1-08: Introducing Disproof by Counter ExampleWatch the walkthrough, then work through the notes below.

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What you'll be able to do

Explain why one counter-example disproves a universal statement
Hunt efficiently for counter-examples using the usual suspects
Present a counter-example so it clearly settles the claim
Know the limits — a counter-example can disprove but never prove

Why one is enough

A statement like “for all integers $n$ , …” claims something about every single value. To knock it down, you only have to find $one$ value where the claim is false — because “for all” cannot survive a single exception. That value is called a $counter-example$ .

Compare the effort. Proving the statement true would mean arguing about every $n$ at once. Disproving it means producing one number and showing it breaks the rule. The asymmetry is the whole point: falsity is cheap to demonstrate, truth is expensive.

Why this works

This mirrors the very first idea of the strand — examples cannot prove a universal claim, but a single example $can$ disprove one. The counter-example is that idea turned into a weapon.

Where counter-examples hide

A good counter-example rarely comes from the “typical” numbers a statement was written with in mind. It comes from the edge cases people forget. Before believing any universal claim, deliberately test these usual suspects:

The number $0$ (breaks claims that assume positivity or division). The number $1$ (breaks claims about “getting bigger” when you multiply). $Negative$ numbers (break claims that quietly assume positivity, especially with squares and inequalities). $Fractions$ between 0 and 1 (break claims that squaring makes numbers larger). And the prime $2$ — the only even prime, which single-handedly falsifies “all primes are odd”.

1The claim quietly assumes

x \geq 1

. Test a fraction between 0 and 1.

2Let

x = \frac{1}{2}

. Then

x^{2} = \frac{1}{4}

3But

\frac{1}{4} < \frac{1}{2}

, so

x^{2} \geq x

fails here.

AnswerCounter-example

x = \frac{1}{2}

x^{2} = \frac{1}{4} < \frac{1}{2} = x

. The statement is false.

■

Tip — Read the statement and ask “what did the author probably forget?” It is almost always negatives, fractions, zero, or the number 2.

Presenting the disproof

A counter-example only earns marks if you show it fails. State the value clearly, substitute it in, and demonstrate that the claim is broken — then say so. You do not need to explain how you found it; you need to prove it works.

1Squares lose sign information, so try negatives.

2Let

a = - 3

and

b = 2

. Then

a^{2} = 9

and

b^{2} = 4

, so

a^{2} > b^{2}

holds.

3But

a = - 3

and

b = 2

, so

a > b

is false (

- 3 < 2

4The hypothesis is satisfied yet the conclusion fails.

AnswerCounter-example

a = - 3, b = 2

a^{2} > b^{2}

but

a < b

. The implication is false.

■

The limit of the method

A counter-example is a disproof and nothing more. Finding examples that $support$ a statement — even thousands — does not prove it, because the strand’s first principle still holds: you cannot check every case. So if you test the usual suspects and the statement survives, that is a hint it might be true, and a signal to switch to deduction, exhaustion or contradiction.

Think like an examiner

The command word “ $Disprove$ ” — or “ $Show that \dots is not always true$ ” — is an explicit invitation to find one counter-example. You do not need a general argument.
You must $show$ the example fails, not just state it: substitute the value and demonstrate the breakdown. A bare number scores nothing.
One counter-example is sufficient and complete — do not waste time finding several.
If asked to disprove an implication “if $P$ then $Q$ ”, your example must satisfy $P$ but break $Q$ .

Common misconceptions

Giving several examples that all support the statement and concluding it is true.

Supporting examples never prove a universal claim. To engage the statement you either find a counter-example or prove it generally.

Stating a counter-example without showing it fails.

Substitute the value and demonstrate the claim breaks — the working is the proof.

Only testing “nice” positive whole numbers.

Deliberately try 0, 1, negatives, fractions and the prime 2 — that is where counter-examples live.

For an implication, giving an example that breaks the hypothesis instead of the conclusion.

A valid counter-example must make the “if” part true while making the “then” part false.

The usual suspects

x = 0

Breaks positivity / division assumptions.

0 < x < 1

A fraction: squaring makes it smaller.

x < 0

Negatives: squares hide the sign.

2

The only even prime.

Stretch yourself

A student claims: “For all positive integers $n$ , the value $2^{n} + 1$ is prime.” Test it and either find a counter-example or explain why one is hard to find.

Hint — It works for $n = 1, 2, 4$ ( $3, 5, 17$ ). Try $n = 3$ , then $n = 6$ .

Questions students ask

How many counter-examples do I need to disprove a statement?

Exactly one. A universal (“for all”) statement is false the moment a single case breaks it, so one clearly-shown counter-example is a complete disproof.

Where should I look for a counter-example?

Start with the edge cases people forget: 0, 1, negative numbers, fractions between 0 and 1, and the prime 2. Statements usually fail on the values their author did not have in mind.

Can I use a counter-example to prove something is true?

No. Examples can only disprove a universal statement, never prove one. If your test cases all pass, that only hints the statement might be true — you then need deduction, exhaustion or contradiction.

How do I disprove an “if P then Q” statement?

Find a case where P is true but Q is false. For “if

a^{2} > b^{2}

then

a > b

”, take

a = - 3, b = 2

: the hypothesis holds but the conclusion fails.

Key takeaways

One counter-example is enough to disprove any “for all” statement.
Proving needs generality; disproving needs a single well-chosen case — the core asymmetry of proof.
Hunt among 0, 1, negatives, fractions and the prime 2.
Show the example fails by substituting it in; do not just state it.
Counter-examples disprove only — they can never prove a statement true.

How this fits the course

Build on

The Structure of a Proof

Proof by Exhaustion

Leads to

Proof by Contradiction

Test yourself

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