Questions: Counterexamples in Reasoning

Question 1

What is a counterexample to the claim 'All multiples of 4 are also multiples of 8'?

• A. 16, because 16 is a multiple of both 4 and 8
• B. 12, because 12 is a multiple of 4 but not a multiple of 8
• C. 9, because 9 is odd
• D. 8, because 8 is a multiple of 8

Answer: (B)

A counterexample must satisfy the hypothesis (be a multiple of 4) but violate the conclusion (not be a multiple of 8). 12 = 4 x 3 is a multiple of 4, but 12/8 = 1.5, so 12 is not a multiple of 8. Options A and D confirm the claim rather than refuting it. Option C is irrelevant — 9 is not a multiple of 4, so it cannot serve as a counterexample.

Question 2

If you test 100 examples and few of them disprove a conjecture, the conjecture is proven true.

• T. True
• F. False

Answer: False

Testing examples can increase your confidence in a conjecture, but it never proves a universal claim true. Goldbach's conjecture (every even number greater than 2 is the sum of two primes) has been verified for trillions of numbers but remains unproven. The 101st case — or the trillionth-and-first — could be the counterexample. Only a proof can establish certainty; examples establish plausibility.

Question 3

A student claims: 'n² > n for all integers n.' Find a counterexample and explain why it works.

Model answer: n = 0 is a counterexample: 0² = 0, which is not greater than 0. Also n = 1: 1² = 1, which is not greater than 1. The claim requires n² to be strictly greater than n, but for n = 0 and n = 1, n² equals n.

The claim says 'for all integers,' so any single integer where n² is not strictly greater than n disproves it. Both 0 and 1 work. Negative integers do not work as counterexamples here because for negative n, n² is positive and thus greater than n. The correct statement would need to restrict the domain, for example: 'n² > n for all integers n > 1.'