Question

Express this statement using quantifiers: “There is a building on the campus of some college in the United States in which every room is painted white.”

Short answer

The statement using quantifiers is:\(\exists z\exists y\forall xP\left( {x,y,z} \right)\).

Step-by-step solution

Introduction

Consider the statement “There is a building on the campus of some college in the United States in which every room is painted white”.

Let \(P\left( {x,y,z} \right)\) represents “Room \(x\)of building \(y\) of college \(z\) is painted white”.

Quantified presentation of the statement

From the statement, observe that there is a building on the campus represents the building that depends on the college.

So, use existential quantifier for building.

Some college in the United States means there exist a college\(z\)in the domain of all colleges in the United States.

So, use existential quantifier for college.

The room depends on the college and building, and every room is painted white represents a universal quantifier in the domain of all rooms.

Take college at the first, then building, and then rooms in the symbolic representation is \(\exists z\exists y\forall xP\left( {x,y,z} \right)\).

Therefore, the statement using quantifiers is \(\exists z\exists y\forall xP\left( {x,y,z} \right)\).