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Which of the following predicate logic formulae/formulae 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

1. ∀x∃y∃z(mother(y,x) ∧ ¬mother(z,x))
2. ∀x∃y[mother(y,x) ∧ ∀z(noteq(z,y) → ¬mother(z,x))]
3. ∀x∀y[mother(y,x) → ∃z(mother(z,x) ∧ ¬noteq(z,y))]
4. ∀x∃y[mother(y,x) ∧ ¬∃z(noteq(z,y) ∧ mother(z,x))]

Correct answer: ∀x∃y[mother(y,x) ∧ ¬∃z(noteq(z,y) ∧ mother(z,x))]

Solution

The correct option accurately captures the idea that for every individual x, there exists a unique mother y such that no other individual z can also be a mother of x, ensuring that y is the only mother.

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