1. Premises: (a) Every web server has an IP address; and (b) www.cia.gov is a web server. Conclusion: Therefore, www.cia.gov has an IP address.
2. Premises: (a) If a computer system is hacked, then there was user error or the system had a design flaw; and (b) A computer at NASA was hacked; and (c) That computer did not have a design flaw. Conclusion: Therefore, there was user error.
In the next several problems, you will be presented with a false claim and a bogus proof of that false claim. For each, you’ll be asked to (a) identify the precise error in the proof, and (b) give a counterexample to the claim. (Note that saying why the claim is false does not address (a) in the slightest—it would be possible to give a bogus proof a true claim!)
False Claim #1: Let n be a positive integer and let p, q ∈ Z
≥2
, where p and q are prime. If n is evenly divisible by both p and q, then n is also evenly divisible by pq.
Bogus proof of (FC-1). Because p | n, there exists a positive integer k such that n = pk. Thus, by assumption, we know that q | pk. Because p and q are both prime, we know that p does not evenly divide q, and thus the only way that q | pk can hold is if q | k. Hence k = qℓ for some positive integer ℓ, and thus n = pk = pqℓ. Therefore pq | n.