Our professor gave us the following exercise:
"Statement P reads “x is a prime number”, statement Q reads “x +1 is a prime number"’. Variable x is an arbitrary positive integer. Is the statement P ⇒ ¬Q true of false for every value of x?"
The answer sheet states that the statement is indeed true. Me and my friend have spent quite some time wondering how can it be, since in case x = 1 or x = 2, both P and Q are true, therefore the statement should be false. What is wrong with our reasoning?
Your answer sheet is wrong, if you've quoted it correctly. Indeed $\forall x \in \mathbb N_+(P(x) \to \neg Q(x))$ is false: if $x = 2$ then $P(x)$ is true, because 2 is prime, yet $\neg Q(x)$ is false, because 3 is also prime.
However, note that you have also made a mistake: under the nowadays standard definition, 1 is not a prime number. (Historically 1 was sometimes considered prime, but this is not a standard definition anymore.)
The statement "for every $x$, $x$ prime implies $x+1$ is not prime" is false. $x=2$ is a counterexample.
Note: $1$ is not prime. Many people think it is since "its only divisors are itself and $1$" but the official definition excludes $1$. A positive integer is prime if it has exactly two positive divisors.
That's because if $1$ were a prime you could no longer say "every integer greater than $1$ can be written as a product of primes in just one way, if order doesn't matter."
