GATE CS 2011 Q30 — Propositional and First Order Logic

Let us analyze the predicate $P(x)$ step-by-step:

1. First part: $\neg(x=1)$
   Since $x$ is a positive integer, this condition specifies that $x \neq 1$, meaning $x \ge 2$.
2. Second part: $\forall y \left(\exists z\left(x=y*z\right) \Rightarrow \left(y=x\right) \vee \left(y=1\right) \right)$
   • The term $\exists z (x = y * z)$ means that $y$ is a divisor (or factor) of $x$ in the domain of positive integers.
   • The implication states that if $y$ is a divisor of $x$, then $y$ must be either $x$ itself or $1$.

Combining both parts, $P(x)$ is true if and only if $x$ is a positive integer greater than $1$ whose only positive divisors are $1$ and $x$ itself. This is the definition of a prime number.

Thus, the correct option is A.