Which one of the following options is CORRECT given three positive integers x, y and z, and a predicate P(x) = eg (x = 1) y (( z (x = y * z)) (y = x) (y = 1)) A) P(x) being true means that x is a prime number B) P(x) being true means that x is a number other than 1 C) P(x) is always true irrespective of the value of x D) P(x) being true means that x has exactly two factors other than 1 and x

Official answer: A) P(x) being true means that x is a prime number. Explanation: Let us analyze the components of the given predicate P(x) in the domain of positive integers: 1. $ eg(x = 1) asserts that x 1$. 2. z (x = y z) means that y is a factor (divisor) of x, since there exists a positive integer z such that y z = x. 3. The implication ( z (x = y z)) (y = x 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].

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

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

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

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

Engineering Mathematics · Propositional and First-Order Logic

Official IIT answer key · IIT Madras · Audited Aug 2026

Take topic test

GATE CSE 2011 Q30 · MCQ · 2 marks

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

Select an option to check

Key concept

Propositional and First-Order Logic

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Propositional and First-Order Logic

Across the provided GATE CS exam papers (2012–2026), Propositional and First-Order Logic is tested through two major avenues: formalizing natural language statements into first-order logic (FOL) expressions and analyzing algebraic/truth properties of propositional and predicate formulas (tautology checking, equivalence, and operator properties). Questions frequently test quantifier negation, conditional equivalences, and formal representations of existence, uniqueness, and contrapositives. The topic is evenly balanced between 1-mark conceptual items and 2-mark algebraic simplification or multi-quantifier questions.

Full Propositional and First-Order Logic guide →

More on Propositional and First-Order Logic

All 18 questions on Propositional and First-Order Logic →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

Engineering Mathematics · Propositional and First-Order Logic

Official IIT answer key · IIT Madras · Audited Aug 2026

Take topic test

GATE CSE 2011 Q30 · MCQ · 2 marks

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

Select an option to check

Key concept

Propositional and First-Order Logic

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Propositional and First-Order Logic

Full Propositional and First-Order Logic guide →

More on Propositional and First-Order Logic

All 18 questions on Propositional and First-Order Logic →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

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

Step-by-step verified solution

Check you can still solve Propositional and First-Order Logic under a clock

More from GATE CSE 2011

More on Propositional and First-Order Logic

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

Step-by-step verified solution

Check you can still solve Propositional and First-Order Logic under a clock

More from GATE CSE 2011

More on Propositional and First-Order Logic

Step-by-step verified solution

Check you can still solve Propositional and First-Order Logic under a clock

More from GATE CSE 2011

More on Propositional and First-Order Logic

Share this solution

Step-by-step verified solution

Check you can still solve Propositional and First-Order Logic under a clock

More from GATE CSE 2011

More on Propositional and First-Order Logic

Share this solution