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Logic

Counterexamples

Lesson

A counterexampleis a single example that disproves a universal claim (“all,” “every”). One is enough.

The asymmetry of universal claims

To prove “all X are Y”, you must check every X. To disprove it, you only need one X that isn’t Y.

Worked example

Claim: “All primes are odd.” Counterexample: 2 is prime but not odd.

For each problem, find the smallest specific value that breaks the claim.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

“All primes are odd.” Counterexample prime?\text{``All primes are odd.'' Counterexample prime?}

Problem 2

“All natural numbers are positive.” Counterexample?\text{``All natural numbers are positive.'' Counterexample?}

Problem 3

“n2>n for all positive n.” Smallest counterexample?\text{``n}^2 > n \text{ for all positive n.'' Smallest counterexample?}

Problem 4

“All multiples of 4 are multiples of 8.” Smallest counterexample?\text{``All multiples of 4 are multiples of 8.'' Smallest counterexample?}

Practice

Standard problems matching the lesson.

Problem 5

“All squares are positive.” Counterexample?\text{``All squares are positive.'' Counterexample?}

Problem 6

“All odd numbers are prime.” Smallest odd composite?\text{``All odd numbers are prime.'' Smallest odd composite?}

Problem 7

“All naturals < 10 are < 5.” Counterexample (smallest)?\text{``All naturals < 10 are < 5.'' Counterexample (smallest)?}

Problem 8

“n2−n=0 for all n.” Smallest counterexample with n≥2?\text{``n}^2 - n = 0 \text{ for all n.'' Smallest counterexample with n≥2?}

Problem 9

“For all integers, |x| > 0.” Counterexample?\text{``For all integers, |x| > 0.'' Counterexample?}

Problem 10

“All multiples of 6 are multiples of 4.” Smallest counterexample?\text{``All multiples of 6 are multiples of 4.'' Smallest counterexample?}

Problem 11

“All even numbers are divisible by 4.” Smallest counterexample?\text{``All even numbers are divisible by 4.'' Smallest counterexample?}

Problem 12

“Sum of two primes is even.” Smaller prime in odd-sum pair?\text{``Sum of two primes is even.'' Smaller prime in odd-sum pair?}

Problem 13

“n3>1 for positive int n.” Counterexample?\text{``n}^3 > 1 \text{ for positive int n.'' Counterexample?}

Problem 14

“All multiples of 3 are odd.” Smallest counterexample?\text{``All multiples of 3 are odd.'' Smallest counterexample?}

Problem 15

“All powers of 2 are even.” Counterexample (smallest)?\text{``All powers of 2 are even.'' Counterexample (smallest)?}

Problem 16

“All squares are larger than 5.” Counterexample (smallest perfect square)?\text{``All squares are larger than 5.'' Counterexample (smallest perfect square)?}

Problem 17

“Squaring makes numbers bigger.” A fraction counterexample (1/n form, smallest n>1)\text{``Squaring makes numbers bigger.'' A fraction counterexample (1/n form, smallest n>1)}

Problem 18

“All natural numbers > 1 are prime.” Smallest counterexample?\text{``All natural numbers > 1 are prime.'' Smallest counterexample?}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

“n2≥2n for all naturals n.” Smallest positive counterexample?\text{``n}^2 \geq 2n \text{ for all naturals n.'' Smallest positive counterexample?}

Problem 20

“All natural n satisfy n2>n.” Smallest counterexample n≥0\text{``All natural n satisfy n}^2 > n. \text{'' Smallest counterexample n}\geq 0

Problem 21

“All odd n≥9 are composite.” Counterexample (smallest)?\text{``All odd n}\geq 9 \text{ are composite.'' Counterexample (smallest)?}

Problem 22

“Adding 1 to a prime gives a prime.” Counterexample prime?\text{``Adding 1 to a prime gives a prime.'' Counterexample prime?}

Problem 23

“All multiples of 5 are odd.” Smallest counterexample?\text{``All multiples of 5 are odd.'' Smallest counterexample?}

Problem 24

“All numbers ending in 1 are prime.” Smallest counterexample?\text{``All numbers ending in 1 are prime.'' Smallest counterexample?}

Problem 25

“n2+n+41 is prime for all n≥0.” Smallest counterexample n?\text{``n}^2 + n + 41 \text{ is prime for all n}\geq 0. \text{'' Smallest counterexample n}?

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