GATE CS 2011 Q.30 (CS Q.30): Propositional and First-Order Logic

Which one of the following options is CORRECT given three positive integers $x$ , $y$ and $z$ , and a predicate

P(x) = \neg(x = 1) \land \forall y \left( \exists z (x = y * z) \implies (y = x) \lor (y = 1) \right)

$P(x)$ being true means that $x$ is a prime number

$P(x)$ being true means that $x$ is a number other than 1

$P(x)$ is always true irrespective of the value of $x$

$P(x)$ being true means that $x$ has exactly two factors other than 1 and $x$

Official answer and solution

Solution

Translate the first-order logic predicate into natural mathematical language.

Analyze the first clause: $\neg(x = 1)$ asserts that $x$ is a positive integer strictly greater than 1 ( $x > 1$ ).

Analyze the quantified implication: $\forall y \left( \exists z (x = y * z) \implies (y = x) \lor (y = 1) \right)$ .

The condition $\exists z (x = y * z)$ means that $y$ is a divisor (factor) of $x$ .
The implication states that if $y$ divides $x$ , then $y$ must be either $x$ itself or $1$ .

Combining both clauses: $x$ is an integer strictly greater than 1 whose only positive divisors are 1 and $x$ . This is the definition of a prime number.

Final answer

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