Question 45

The predicate formula is:

Let $P(u) \equiv \forall z (z \mid u \implies (z = u \lor z = 1))$. This condition states that $u$ has no positive divisors other than $1$ and $u$ itself, which characterizes prime numbers (and 1).
Thus, the formula $\varphi$ states:
"For every prime $x$, there exists a prime $w$ such that $w > x$."

Now let us evaluate $\varphi$ for each domain:

• $S_1 = \{1, 2, 3, \dots, 100\}$: In this domain, $x = 97$ satisfies the antecedent $P(97)$ since $97$ is prime. However, the next prime is $101$, which is not in $S_1$. Thus, there is no $w \in S_1$ with $w > 97$ that is prime. Hence $\varphi$ is FALSE in $S_1$.

• $S_2$ (Positive integers): By Euclid's theorem, the set of primes is infinite. For every positive prime $x$, there exists a strictly greater prime $w > x$. Hence $\varphi$ is TRUE in $S_2$.

• $S_3$ (All integers): For negative integers and 0, the antecedent is false (for example, for $x = -2$, $z = -1$ divides $-2$, but $-1 \ne -2$ and $-1 \ne 1$), making the implication vacuously true. For positive primes $x$, larger positive primes $w > x$ always exist in $\mathbb{Z}$. Hence $\varphi$ is TRUE in $S_3$.

Therefore, $\varphi$ is satisfied by $S_2$ and $S_3$ (Option C).