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: • $mother(y,x)$: $y$ is the mother of $x$ • $noteq(x,y)$: $x$ and $y$ are not equal

To state that "Everyone has exactly one mother", we need to ensure two conditions for every person $x$:

1. Existence: There exists a person $y$ who is the mother of $x$, i.e., $mother(y, x)$.

2. Uniqueness: Anyone $z$ different from $y$ ($noteq(z, y)$) is not the mother of $x$, i.e., $

eg mother(z, x)$. Alternatively, there is no person$z$different from$y$who is also the mother of$x$.

Let us analyze each option:

• Option (A): $\forall x \exists y \exists z (mother(y,x) \land

eg mother(z,x))$This states that for every $x$, there is some person $y$ who is a mother of $x$, and some person $z$ who is not a mother of $x$. This does not guarantee uniqueness, as $x$ could still have multiple mothers.

• Option (B): $\forall x \exists y [mother(y,x) \land \forall z(noteq(z,y) \rightarrow

eg mother(z,x))]$This states that every person $x$ has a mother $y$, and for any person $z$, if $z \neq y$, then $z$ is not a mother of $x$. This correctly guarantees both existence and uniqueness.

• Option (C): $\forall x \forall y [mother(y,x) \rightarrow \exists z(mother(z,x) \land

eg noteq(z,y))]$This does not enforce existence (it is vacuously true if nobody has a mother) and simplifies to saying if $y$ is a mother of $x$, then $y$ is a mother of $x$ and $y = y$, which is trivial.

• Option (D): $\forall x \exists y [mother(y,x) \land

eg \exists z(noteq(z,y) \land mother(z,x))]$Using the equivalence$\forall z (P(z) \rightarrow Q(z)) \equiv eg \exists z (P(z) \land eg Q(z))$, the uniqueness clause in Option (B),$\forall z (noteq(z,y) \rightarroweg mother(z,x))$, is logically equivalent to $eg \exists z (noteq(z,y) \land mother(z,x))$. Therefore, Option (D) correctly expresses existence and uniqueness.

Hence, both B and D are correct.