Question

Use existential and universal quantifiers to express the statement “Everyone has exactly two biological parents” using the propositional function\(P \left( {x, y} \right)\), which represents “x is the biological parent of y.”

Short answer

The required presentation is as,

\(\forall x\exists y\exists z\left( {y \ne z \wedge \forall w\left( {P\left( {w,x} \right) \leftrightarrow \left( {w = y \vee w = z} \right)} \right)} \right)\), where y is the biological parent of x and z is also another biological parent of x and\(y \ne z\).

Step-by-step solution

Introduction

Given statement:

Everyone has exactly two biological parents.

Propositional function\(P \left( {x, y} \right)\): x is the biological parent of y.

Using existential and universal quantifiers

Therefore, by the use of existential and universal quantifiers, we express this statement as:

\(\forall x\exists y\exists z\left( {y \ne z \wedge \forall w\left( {P\left( {w,x} \right) \leftrightarrow \left( {w = y \vee w = z} \right)} \right)} \right)\)

Here y is the biological parent of x and z is also another biological parent of x and\(y \ne z\).