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.
0
breaks inequalities and divisions; “always positive” claims often fail here
1
behaves oddly with powers (1n = 1) and fractions (½ behaves like a small number)
2
the only even prime — kills any “all primes are odd” type claim
−1
a small negative number — flips inequalities and signs
½
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 1Disprove 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 2Disprove 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 3Disprove 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 4Disprove 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 5Disprove 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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