Question

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

• A. \( \forall x \, \exists y \, \exists z \, [ {mother}(y, x) \land \neg {mother}(z, x)] \)
• B. \( \forall x \, \exists y \, [ {mother}(y, x) \land \forall z \, ({noteq}(z, y) \rightarrow \neg {mother}(z, x))] \)
• C. \( \forall x \, \forall y \, [ {mother}(y, x) \rightarrow \exists z \, ({mother}(z, x) \land \neg {noteq}(z, y))] \)
• D. \( \forall x \, \exists y \, [ {mother}(y, x) \land \exists z \, ({noteq}(z, y) \land {mother}(z, x))] \)

Hint:

To represent "exactly one mother" in predicate logic, ensure that for every person, there exists one mother, and for every other person, they cannot also be the mother of the same individual.

Answer 1

The Correct Option is B, D

Step 1: Understand the Meaning of the Statement
The statement "Everyone has exactly one mother" implies that for each individual \( x \), there exists one and only one person \( y \) such that \( y \) is the mother of \( x \). No one else can be the mother of \( x \).

Step 2: Analyzing Each Option
(A) \( \forall x \, \exists y \, \exists z \, [ \text{mother}(y, x) \land \neg \text{mother}(z, x)] \)
This formula suggests that there are two distinct mothers for \( x \), which violates the condition of exactly one mother. Therefore, this is incorrect.

(B) \( \forall x \, \exists y \, [ \text{mother}(y, x) \land \forall z \, (\text{noteq}(z, y) \rightarrow \neg \text{mother}(z, x))] \)
This formula correctly states that for each person \( x \), there exists one mother \( y \), and for every other \( z \neq y \), \( z \) cannot be the mother of \( x \). This captures the meaning of "exactly one mother". Thus, this is correct.

(C) \( \forall x \, \forall y \, [ \text{mother}(y, x) \rightarrow \exists z \, (\text{mother}(z, x) \land \neg \text{noteq}(z, y))] \)
This formula suggests that for every person \( x \) and mother \( y \), there exists another mother \( z \) who is the same as \( y \), which contradicts the idea of exactly one mother. Therefore, this is incorrect.

(D) \( \forall x \, \exists y \, [ \text{mother}(y, x) \land \exists z \, (\text{noteq}(z, y) \land \text{mother}(z, x))] \)
This formula states that for each person \( x \), there exists one mother \( y \), and for some \( z \neq y \), \( z \) is also a mother of \( x \). This captures the idea of "exactly one mother" by implicitly stating that no other \( z \) can be the mother of \( x \). Thus, this is correct.

Conclusion
The correct answers are (B) and (D).