Question

Express the statement “There is exactly one student in this class who has taken exactly one mathematics class at this school” using the uniqueness quantifier. Then express this statement using quantifiers, without using the uniqueness quantifier.

Short answer

\(\exists !x\exists !yP\left( {x,y} \right)\) and \(\exists x\exists z\left( {\left( {\exists y\forall w\left( {T\left( {z,w} \right) \leftrightarrow w = y} \right)} \right) \leftrightarrow z = x} \right)\)

Step-by-step solution

Introduction

Consider the statement “There is exactly one student in this class who has taken exactly one mathematics class at this school”.

Presentation with uniqueness quantifier

The given statement above can be expressed with uniqueness quantifier as:

\(\exists !x\exists !yP\left( {x,y} \right)\), where \(P\left( {x,y} \right)\)means that student x has taken class y and the domain is all students in this class.

Presentation without uniqueness quantifier

The given statement above can be expressed without uniqueness quantifier as:

\(\exists x\exists z\left( {\left( {\exists y\forall w\left( {T\left( {z,w} \right) \leftrightarrow w = y} \right)} \right) \leftrightarrow z = x} \right)\), where \(T\left( {z,w} \right)\) means that student z has taken class w and the domain is all students in this class.