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)>0 for all x in (0,1), then ∫01f(x) dx>0' assuming f is integrable.

Example 2

hard
Find a counterexample to 'If f is differentiable on R, then f′ is continuous on R.'

Example 3

challenge
Find a counterexample to 'If a sequence {an} has an→0, then ∑an 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=1 a counterexample to 'For all real x, x2>x'?

Example 6

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

Example 7

medium
Find a counterexample to 'For all functions f, if f is increasing then f 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 4 are multiples of 8.'

Example 11

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

Example 12

easy
Does the single example 4+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∣bc then a∣b or a∣c.'

Example 15

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

Example 16

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

Example 17

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

Example 18

easy
Find a counterexample to 'If n2 is divisible by 4, then n is divisible by 4.'

Example 19

easy
Can a counterexample prove a universal statement TRUE?

Example 20

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