Counterexample Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Counterexample.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A counterexample is a specific instance that satisfies the hypothesis of a claim but contradicts its conclusion, thereby disproving the general statement.

One case where it fails is enough to kill a 'for all' claim.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A counterexample is a single instance fitting the hypothesis but breaking the conclusion, which disproves a universal claim outright.

Common stuck point: The procedure for counterexample is the easy part; the trap is giving a confirming example and thinking it proves the claim. Asking "Am I trying to kill a universal claim by exhibiting one case that fits the hypothesis but breaks the conclusion?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I trying to kill a universal claim by exhibiting one case that fits the hypothesis but breaks the conclusion?

Worked Examples

Example 1

easy

Disprove: 'All prime numbers are odd.'

Answer

\text{Counterexample: } 2 \text{ is prime and even.}

First step

To disprove a universal statement, find a single counterexample.

Full solution

2
Consider $2$ : it is prime (its only divisors are 1 and 2) and it is even.
3
Since $2$ is a prime number that is not odd, the statement is false.

A counterexample is a specific case that shows a universal statement is false. Only one counterexample is needed to disprove a 'for all' claim.

Example 2

medium

Disprove: 'For all real numbers

x

x^2 > x

Example 3

medium

Find a counterexample to '

\sin(x + y) = \sin x + \sin y

for all real

x, y

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find a counterexample to disprove: 'The sum of any two prime numbers is even.'

Example 2

easy

Find a counterexample to disprove: 'If an integer is divisible by 6, then it is divisible by 4.'

Example 3

easy

Find a counterexample to: 'Every prime number is odd.'

Example 4

easy

Find a counterexample to: 'If

n

is even, then

n

is divisible by 4.'

Example 5

easy

x=3

a counterexample to 'For all real

x

x^2 > x

Example 6

easy

Find a counterexample to: 'All multiples of 3 are odd.'

Example 7

easy

Find a counterexample to: 'If

a > b

then

a^2 > b^2

' using negatives.

Example 8

easy

Does the single example

4+4=8

(even+even=even) PROVE 'the sum of two evens is even'?

Example 9

easy

Find a counterexample to: 'Every quadrilateral with four equal sides is a square.'

Example 10

easy

n=1

a counterexample to 'Every natural number is greater than 0'?

Example 11

medium

Find a counterexample to: 'For all integers

n

n^2 - n + 11

is prime.'

Example 12

medium

Find a counterexample to: 'If

a \mid bc

then

a \mid b

a \mid c

Example 13

medium

Disprove: 'If

\frac{a}{b} = \frac{c}{d}

then

a=c

and

b=d

Example 14

medium

Find a counterexample to: 'For all real

x

\sqrt{x^2} = x

Example 15

medium

Disprove: 'Every function

f

with

f(0)=f(2)=0

has a root only at

0

and

2

Example 16

medium

Find a counterexample to: 'If

x^2 = y^2

then

x=y

Example 17

medium

Disprove: 'The product of two irrational numbers is always irrational.'

Example 18

medium

Find a counterexample to: 'If

a+b

is even and

a

is even, then

b

is odd.'

Example 19

medium

Disprove: 'For all sets

A,B

: if

A\cup B = A\cup C

then

B=C

Example 20

challenge

Disprove: 'For all positive integers

n

2^{2^n}+1

is prime' (Fermat's claim).

Example 21

challenge

Disprove with a counterexample: 'A set always has more elements than any of its proper subsets.'

Example 22

challenge

Find a counterexample to: 'If

f

and

g

are not differentiable at

0

, then

f+g

is not differentiable at

0

Example 23

easy

x = 1

a counterexample to 'For all real

x

x^2 > x

Example 24

easy

Find a counterexample to 'If

n^2

is divisible by

4

, then

n

is divisible by

4

Example 25

medium

Find a counterexample to 'For all real

x, y

: if

x < y

then

x^2 < y^2

Example 26

medium

Find a counterexample to 'For all integers

a, b

(a + b)^2 = a^2 + b^2

Example 27

medium

Find a counterexample to 'For all functions

f

, if

f

is increasing then

f

is one-to-one.'

Example 28

medium

Find a counterexample to 'For all matrices

A, B

AB = BA

Example 29

medium

Find a counterexample to 'For all positive integers

n

n^2 + n + 41

is prime.'

Example 30

hard

Find a counterexample to 'For all real

x

\frac{d}{dx}(f \cdot g) = f' \cdot g'

Example 31

hard

Find a counterexample to 'If

f(x) > 0

for all

x

(0, 1)

, then

\int_0^1 f(x)\,dx > 0

' assuming

f

is integrable.

Example 32

hard

Find a counterexample to 'For all real

a, b, c

: if

a > b

then

ac > bc

Example 33

hard

Find a counterexample to 'If

f

is differentiable on

\mathbb{R}

, then

f'

is continuous on

\mathbb{R}

Example 34

challenge

Find a counterexample to 'Every continuous function on

[0, 1]

is differentiable on

(0, 1)

Example 35

challenge

Find a counterexample to 'If a sequence

\{a_n\}

has

a_n \to 0

, then

\sum a_n

converges.'

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Related Concepts

Quantifiers Proof (Intuition)

Background Knowledge

These ideas may be useful before you work through the harder examples.

quantifiers