A counterexample is a single example that shows a general statement is false. It fits all the conditions of the statement but does not have the claimed result. To disprove ‘every prime number is odd,’ one counterexample is enough: 2 is prime and even. Examples that agree with a claim never prove it, but one counterexample disproves it.
In Spanish
contraejemplo
Also called
counter example, counter-example
Grade level
Geometry
Grade 8Algebra 1GeometrySATACTGEDPraxis
Counterexample in a picture
Agreeing examples, however many, cannot prove an all statement. One case that meets the condition and breaks the conclusion is enough to sink it.
Formal definition and notation
For a statement of the form ‘for every x with property P, property Q holds,’ a counterexample is a specific x for which P(x) is true and Q(x) is false. For a conditional ‘if P, then Q,’ it is a case with a true hypothesis and a false conclusion. Producing one counterexample proves that the general statement is false.
Symbol
Read it as
Meaning
∀ x
for all x
a universal claim; a counterexample targets this kind of statement
P ⇒ Q
if P, then Q
a counterexample makes P true and Q false
Key facts
A counterexample must satisfy the hypothesis. To disprove ‘if n is even, then n is divisible by 4,’ the number 6 works; the number 7 does not, because 7 is not even.
One counterexample is enough to disprove an all statement. No number of agreeing examples proves it; that takes a proof.
A statement can be true while its converse is false: ‘every square is a rectangle’ is true, but a 3-by-5 rectangle is a counterexample to ‘every rectangle is a square.’
Counterexamples also show that an algebra step is invalid: (a + b)2 = a2 + b2 fails for a = b = 1, since 4 ≠ 2.
Counterexample examples, step by step
Example 1: Number claim
Warm-up
Find a counterexample to: ‘The square of any number is greater than the number.’
Try a number between 0 and 1: x = 12.
(12)2 = 14.
14 < 12, so the square is smaller.
Answer:x = 12 is a counterexample: its square, 14, is smaller. (0 and 1 also work, since their squares are equal to them.)
Example 2: Algebra rule
Core
Is √a + b = √a + √b always true? Test a = 9, b = 16.
A proof shows a statement is true in every case. A counterexample shows a statement is false by giving one case where it fails.
Quick check: 3 questions
Question 1 of 3
Which number is a counterexample to ‘every odd number is prime’?
Show answer and explanation
Answer: B. 9 is odd but 9 = 3 × 3, so it is not prime. 7 and 11 are odd and prime, so they agree with the claim, and 2 is not odd, so it does not test it.
Question 2 of 3
Which value of x is a counterexample to ‘x2 > x for every positive x‘?
Show answer and explanation
Answer: D.0.42 = 0.16, which is less than 0.4. 2 and 10 agree with the claim, and −3 is not positive, so it does not satisfy the hypothesis.
Question 3 of 3
Which figure is a counterexample to ‘if a quadrilateral has four right angles, then it is a square’?
Show answer and explanation
Answer: A. A 2-by-6 rectangle has four right angles but is not a square. A square agrees with the claim, and the rhombus and trapezoid do not have four right angles, so they do not test it.
Counterexample: frequently asked questions
How many counterexamples do you need to disprove a statement?
Just one. A general statement claims something is true in every case, so a single case where the conditions hold and the result fails makes the statement false. For example, 2 + 3 = 5 alone disproves ‘the sum of any two primes is even.’ Adding more counterexamples is fine but proves nothing extra.
Can a counterexample prove a statement is true?
No. A counterexample can only show that a statement is false. To show a statement is true for every case, you need a proof, a general argument. If you search hard and find no counterexample, that is a hint the statement might be true, which is called a conjecture, not a proof.
How do you find a counterexample?
First list what the hypothesis requires, then try unusual cases that still meet it: 0, 1, 2, negative numbers, fractions between 0 and 1, very large numbers, or special shapes like a square or a long thin rectangle. Check each one against the conclusion. The first case that meets the hypothesis but fails the conclusion is your counterexample.
Updated November 2026·Reviewed by the Effortless Math team·Standards: n/a
Cite this entry
Effortless Math Team. “Counterexample.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/counterexample/
Flashcard
TermCounterexampleone case that breaks a rule
DefinitionAn example that meets a statement’s conditions but not its conclusion, proving it false. 2 is a counterexample to ‘all primes are odd.’
NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3RhombusA rhombus is a quadrilateral with four sides of equal length. It is a special parallelogram, so its opposite sides are parallel and its opposite angles are equal. Its diagonals cross at right angles and cut each other in half, and its area is half the product of the diagonals.Geometry · Grade 3, Grade 4, Grade 5Right TriangleA right triangle is a triangle with one right angle, an angle of exactly 90 degrees. The two sides that form the right angle are the legs, and the longest side, across from the right angle, is the hypotenuse. Its sides follow the Pythagorean theorem: a2 + b2 = c2.Geometry · Grade 4, Grade 6, Grade 8ProofA proof is a step-by-step logical argument that shows a math statement is true in every case it covers. It starts from given facts and uses definitions, postulates and statements already proved, with a reason for each step. A statement that has been proved is called a theorem.Geometry · Geometry, Algebra 1Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5TheoremA theorem is a math statement that has been proved true using definitions, postulates and statements already proved. Once proved, it is true in every case it covers, so you can use it to solve problems. The Pythagorean theorem and the Vertical Angles Theorem are famous examples.Geometry · Grade 8, GeometryTwo-Column ProofA two-column proof is a way of writing a proof as a table: numbered statements in the left column and the reason for each one in the right column. It begins with the given facts and ends with the statement to be proved. Every reason must be a given, a definition, a postulate, a property or a proved theorem.Geometry · GeometryPrime NumberA prime number is a whole number greater than 1 with exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13. The number 1 is not prime, and 2 is the only even prime number. Whole numbers greater than 1 that are not prime are composite.Number System · Grade 4, Grade 5, Grade 6Even NumberAn even number is an integer that can be divided by 2 with no remainder, so it can be split into two equal groups or into pairs. Even numbers end in 0, 2, 4, 6 or 8, such as 14, 236 and −10. Zero is even. Every even number can be written as 2n for some integer n.Number System · Grade 2, Grade 3, Grade 4