Which one of the following options is CORRECT given three positive integers $x$, $y$ and $z$, and a predicate $$P(x) = eg (x = 1) \land \forall y ((\exists z (x = y * z)) \Rightarrow (y = x) \lor (y = 1))$$

Let us analyze the components of the given predicate $P(x)$ in the domain of positive integers:

1. $

eg(x = 1)$asserts that$x \neq 1$.

2. $\exists z (x = y \cdot z)$ means that $y$ is a factor (divisor) of $x$, since there exists a positive integer $z$ such that $y \cdot z = x$.

3. The implication $(\exists z (x = y \cdot z)) \implies (y = x \lor y = 1)$ asserts that for every positive integer $y$, if $y$ divides $x$, then $y$ must be either $x$ or $1$.

Combining these two parts:

• $x$ is a positive integer not equal to $1$.

• The only positive divisors of $x$ are $1$ and $x$ itself.

This is precisely the definition of a prime number.

Therefore, $P(x)$ being true means that $x$ is a prime number, which corresponds to option (A) [p1].