IB Maths AA HL Topic 1 — Number & Algebra Paper 1 & 2 ~7 min read

Disproof by Counter Example

A claim that “for all numbers, something is true” is destroyed the moment you find a single number where it isn’t. That single number is called a counterexample, and producing one is a disproof — the opposite of a proof. The skill is two-fold: knowing where to look (which is much easier than it sounds — the suspects are usually the same kind of number every time), and writing the disproof properly so the IB awards full marks.

📘 What you need to know

To disprove a “for all” statement, find one counterexample — a single value of the variable for which the statement fails.
One is enough. You don’t need to find more than one. But you do need to show why your example fails — not just write down the number.
Look first at “unusual” values: 0, 1, 2 (the only even prime), negatives, fractions, and irrationals like √2 and π. The corner cases are usually where claims break.
The structure is always the same: state the claim → state the counterexample → demonstrate why it fails → conclude.
Disproof and proof are not symmetric. To prove a “for all” claim, you need a general argument; to disprove it, one specific failing case is enough.

What is disproof by counterexample?

Suppose someone makes a sweeping claim like “for all real x, x² > x“. To disprove this, you don’t need an argument that covers every case — you just need one bad apple. Pick x = ½: then x² = ¼, which is less than ½, not greater. The single example demolishes the claim.

Logic in a nutshell Claim: “for all x in S, P(x) is true”
Disproof: find one x₀ ∈ S where P(x₀) is false
Conclusion: claim is false ✗

🤔 Why is one counterexample enough?

Because the original claim says every case works. A single failing case shatters that. The mathematics doesn’t ask “how often does the claim work?” — only “does it work for absolutely all cases?”. One “no” turns the answer to “no”.

Where to look — the usual suspects

Most claims that fail break at the same kinds of “edge” numbers. Train your eye to test these first.

breaks inequalities and divisions; “always positive” claims often fail here

behaves oddly with powers (1ⁿ = 1) and fractions (½ behaves like a small number)

the only even prime — kills any “all primes are odd” type claim

−1

a small negative number — flips inequalities and signs

a fraction in (0, 1) — squaring makes it smaller, not larger

√2 or π

irrational — disproves any “rational” claim

When you see “for all real x“, quickly run through 0, 1, −1, ½, and 2 in your head. One of these will often falsify a careless claim. If those don’t work, try a slightly more exotic value (a small irrational, or a large negative).

How to write a disproof properly

The IB awards marks for the structure as much as the chosen number. A counterexample alone — without showing why it fails — usually scores zero.

🧭 Recipe — disproving a claim

Re-state the claim briefly so it’s clear what you’re disproving.
Choose a candidate number from the usual-suspect list. Try the simplest one that might fail.
Substitute and compute both sides of the claim explicitly. Show the working.
Point out the failure in plain words — “the LHS is X, the RHS is Y, so the claim does not hold for this value”.
Conclude with a line stating the claim has been disproved.

Bare minimum that earns marks: the chosen value, the substitution, the comparison, and the conclusion. Skipping any one of these typically loses a mark.

Worked examples

WE 1

Disprove a claim about multiples

Disprove the statement: “For every n ∈ ℤ⁺, if n is a multiple of 6, then n is also a multiple of 12.”

Step 1: Try a small multiple of 6 let n = 6 Step 2: Check both parts 6 is a multiple of 6 ✓ 6 ÷ 12 = 0.5 → not a whole number, so 6 is NOT a multiple of 12 Step 3: Conclude n = 6 disproves the claim ✗

WE 2

Disprove an inequality claim

Disprove the statement: “For all x ∈ ℝ, x² > x.”

Step 1: Try a value in (0, 1) let x = ½ Step 2: Compute both sides x² = (½)² = ¼ x = ½ Step 3: Compare ¼ < ½, so x² is NOT greater than x x = ½ disproves the claim ✗ x = 0 also works — both make the strict inequality fail. Either is fine.

WE 3

Disprove a claim about squares

Disprove the statement: “For all a, b ∈ ℝ, if a² = b², then a = b.”

Step 1: Use a positive and negative pair let a = 2 and b = −2 Step 2: Check the hypothesis a² = 4, b² = 4, so a² = b² ✓ Step 3: Check the conclusion but a = 2 ≠ −2 = b (a, b) = (2, −2) disproves the claim ✗ squaring loses sign information — this is the classic example showing why

WE 4

Disprove a claim about primes

Disprove the statement: “Every prime number is odd.”

Step 1: Try the smallest prime let n = 2 Step 2: Check both properties 2 is prime (its only factors are 1 and 2) ✓ but 2 is even, not odd n = 2 disproves the claim ✗ 2 is the only even prime — that’s why it always wins this type of question

WE 5

Disprove a false algebraic identity

Disprove the statement: “For all a, b ∈ ℝ, (a + b)² = a² + b².”

Step 1: Try a simple non-zero pair let a = 1, b = 1 Step 2: Compute both sides LHS: (1 + 1)² = 2² = 4 RHS: 1² + 1² = 1 + 1 = 2 Step 3: Compare 4 ≠ 2, so LHS ≠ RHS for this pair (a, b) = (1, 1) disproves the claim ✗ the missing 2ab term is what makes this claim false; any non-zero pair where both are non-zero will work

💡 Top tips

Try the suspects first. 0, 1, −1, ½, 2, √2, π. The simpler values work most of the time.
Show the working, not just the number. Substitute, compute, compare, conclude. Examiners want all four steps visible.
For claims with two variables, look for combinations that exploit the asymmetry — opposite signs, both equal, one zero.
Pay attention to the set. If the claim says “for all n ∈ ℤ⁺“, you can’t use n = 0 or n = −1 as a counterexample — they’re not in the set.
If a claim has the form “if A then B“, a counterexample needs the hypothesis A to hold while the conclusion B fails. Both parts matter.
One counterexample is enough. Don’t waste time finding more.
End with a clear concluding line: “this disproves the statement”, “the claim is false”, or “as a counterexample, …”.

⚠ Common mistakes

Just writing down a number. “x = 2″ by itself proves nothing. You must show how 2 makes the claim fail.
Picking a value outside the stated set. If the claim is about positive integers, your counterexample must be a positive integer.
For “if-then” claims, ignoring the hypothesis. A counterexample to “if A then B” needs to have A true and B false. If A is also false in your example, you haven’t disproved anything.
Trying to disprove with a general argument. Once you have one counterexample, the disproof is finished. No need for algebra.
Confusing disproof with proof of the negation. Disproving “for all x, P(x)” needs only one example. Proving “for some x, ¬P(x)” — the same thing — also needs only one. They’re equivalent.
Assuming the simplest number is always wrong. 0 and 1 break a lot of claims, but not all of them. If they don’t work, try a fraction, a negative, or 2.
Forgetting the conclusion. “Therefore the claim is false” — a single short line at the end. Don’t skip it.

Counterexamples are how mathematicians stress-test claims in everyday work — far before formal proofs. The skill of generating a quick mental list of “suspect values” carries beyond exams: it makes you a sharper thinker about any general claim, mathematical or otherwise.

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