GATE 2026 DA Q.14: solution and answer

GATE 2026 · DAQ.14 · 1 mark · MCQ

Which of the following statements is NOT true?

(The names of the predicates are intuitive.)

(A)
$\forall x \forall y C l a ss ma t e (x, y) \Rightarrow C l a ss ma t e (y, x)$
(B)
$\forall x L ik es (x, I cecr e am) \Rightarrow \neg\exists x \neg L ik es (x, I cecr e am)$
(C)
"Each king is a person" is equivalent to $\forall x I sK in g (x) \land I s P er so n (x)$
(D)
"All humans are mortal" is equivalent to $\forall x I sH u man (x) \Rightarrow I s M or t a l (x)$

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Answer and solution

ANSWER

The statement that is not true is (C): \"Each king is a person\" is not equivalent to $\forall x IsKing (x) \land IsPerson (x)$ .

Official answer: CSource: Official answer key, GATE 2026 DA (IIT Guwahati)

We need the statement that is NOT true, so check each option in turn.
(A) $\forall x \forall y Classmate (x, y) \Rightarrow Classmate (y, x)$ says that being classmates is symmetric. If $x$ is a classmate of $y$ , then $y$ is a classmate of $x$ . This is true.
(B) By quantifier duality, $\neg\exists x \neg P (x) \equiv \forall x P (x)$ . With $P (x) = Likes (x, Icecream)$ , the right side is equivalent to $\forall x Likes (x, Icecream)$ . The statement therefore has the form $Q \Rightarrow Q$ , which is always true.
(D) "All humans are mortal" is the standard universal-implication form $\forall x (IsHuman (x) \Rightarrow IsMortal (x))$ . This is true.
(C) "Each king is a person" translates to $\forall x (IsKing (x) \Rightarrow IsPerson (x))$ .
The option instead gives $\forall x (IsKing (x) \land IsPerson (x))$ . This says every object in the domain is both a king and a person. It fails in any domain that contains a non-king, for instance a person who is not a king. In such a domain the intended sentence is true and the option's sentence is false, so they are not equivalent. The statement is NOT true.
Hence the answer is (C).

KEY IDEA

A universal quantifier over a restricted class ("each king", "all humans") pairs with an implication $\Rightarrow$ , never a conjunction $\land$ . An existential statement ("some king is a person") pairs with $\land$ .

WATCH OUT FOR

Writing $\forall x (P (x) \land Q (x))$ for "every $P$ is a $Q$ ". That claims every object in the domain is a $P$ and a $Q$ , which is far stronger than the English sentence.

From the GATE 2026 DA paper. Every question of this paper · Download the full paper · All 7 Logic questions, with notes

GATE 2026 DA Q.14: solution and answer

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