GATE CS 2025 Set 1 Q.48: Propositional and First-Order Logic

Which of the following predicate logic formulae/formula is/are CORRECT representation(s) of the statement: "Everyone has exactly one mother"?

The meanings of the predicates used are:

$\textit{mother}(y, x)$ : $y$ is the mother of $x$
$\textit{noteq}(x, y)$ : $x$ and $y$ are not equal

$\forall x \exists y \exists z (\textit{mother}(y, x) \wedge \neg \textit{mother}(z, x))$

$\forall x \exists y [\textit{mother}(y, x) \wedge \forall z (\textit{noteq}(z, y) \to \neg \textit{mother}(z, x))]$

$\forall x \forall y [\textit{mother}(y, x) \to \exists z (\textit{mother}(z, x) \wedge \neg \textit{noteq}(z, y))]$

$\forall x \exists y [\textit{mother}(y, x) \wedge \neg \exists z (\textit{noteq}(z, y) \wedge \textit{mother}(z, x))]$

Official answer and solution

Official answer: B and D
From: Official GATE 2025 answer key

Solution

Express both existence ( $\exists y\, \textit{mother}(y, x)$ ) and uniqueness (no $z \ne y$ satisfies $\textit{mother}(z, x)$ ) for every $x$ .

'Everyone ( $\forall x$ ) has at least one mother ( $\exists y\, \textit{mother}(y, x)$ ) and at most one mother (no $z \ne y$ is also a mother of $x$ )' is written as:

\forall x \exists y [\textit{mother}(y, x) \wedge \forall z (\textit{noteq}(z, y) \to \neg \textit{mother}(z, x))]

which is Option (B).

Since $\forall z (P(z) \to \neg Q(z)) \equiv \forall z (\neg P(z) \vee \neg Q(z)) \equiv \neg \exists z (P(z) \wedge Q(z))$ , Option (B) is logically equivalent to:

\forall x \exists y [\textit{mother}(y, x) \wedge \neg \exists z (\textit{noteq}(z, y) \wedge \textit{mother}(z, x))]

which is Option (D).

(Option (A) only says every $x$ has a mother $y$ and some non-mother $z$ , allowing multiple mothers; Option (C) is vacuously true if $x$ has no mother.)

Final answer

Options (B), (D)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 1 (CS1)
Question: 48 of 65
Section: Computer Science and Information Technology

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