Practice Counterexample in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

hard
Find a counterexample to 'If f(x)>0f(x) > 0 for all xx in (0,1)(0, 1), then ∫01f(x) dx>0\int_0^1 f(x)\,dx > 0' assuming ff is integrable.

Example 2

hard
Find a counterexample to 'If ff is differentiable on R\mathbb{R}, then f′f' is continuous on R\mathbb{R}.'

Example 3

challenge
Find a counterexample to 'If a sequence {an}\{a_n\} has an→0a_n \to 0, then ∑an\sum a_n converges.'

Example 4

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

Example 5

easy
Is x=1x = 1 a counterexample to 'For all real xx, x2>xx^2 > x'?

Example 6

medium
Find a counterexample to 'sin⁡(x+y)=sin⁡x+sin⁡y\sin(x + y) = \sin x + \sin y for all real x,yx, y.'

Example 7

medium
Find a counterexample to 'For all functions ff, if ff is increasing then ff is one-to-one.'

Example 8

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

Example 9

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

Example 10

easy
Find a counterexample to 'All multiples of 44 are multiples of 88.'

Example 11

medium
Find a counterexample to: 'For all integers nn, n2−n+11n^2 - n + 11 is prime.'

Example 12

easy
Does the single example 4+4=84+4=8 (even+even=even) PROVE 'the sum of two evens is even'?

Example 13

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

Example 14

medium
Find a counterexample to: 'If a∣bca \mid bc then a∣ba \mid b or a∣ca \mid c.'

Example 15

challenge
Find a counterexample to 'Every continuous function on [0,1][0, 1] is differentiable on (0,1)(0, 1).'

Example 16

easy
Can a single counterexample disprove a universal claim '∀x P(x)\forall x\, P(x)'?

Example 17

easy
Find a counterexample to: 'If nn is even, then nn is divisible by 4.'

Example 18

easy
Find a counterexample to 'If n2n^2 is divisible by 44, then nn is divisible by 44.'

Example 19

easy
Can a counterexample prove a universal statement TRUE?

Example 20

medium
Find a counterexample to 'If ff and gg are continuous on [0,1][0,1], then f/gf/g is continuous on [0,1][0,1].'
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