Question

a) What is the difference between the quantification\(\exists x\forall yP(x,y)\)and\(\forall y\exists xP(x,y)\), where\(P(x,y)\)is a predicate?

b) Give an example of a predicate P (x, y) such that\(\exists x\forall yP(x,y)\)and\(\forall y\exists xP(x,y)\)have different truth values.

Short answer

(a) The quantification\(\exists x\forall yP(x,y)\)denotes that “There exists\(x\)such that for all\(y\),\(P(x,y)\)”; and

The quantification\(\forall y\exists xP(x,y)\)denotes that “for every \(y\)such that there is a\(x\),\(P(x,y)\)”

(b) \(x > y\)is the required example.

Step-by-step solution

Introduction

A predicate is something that is affirmed or denied related to an argument of a proposition.

Part (a) Difference between the two quantifications

(a) Consider the following two quantifications\(\exists x\forall yP(x,y)\), where \(P(x,y)\)is a predicate.

The quantification\(\exists x\forall yP(x,y)\)denotes that “There exists\(x\)such that for all\(y\),\(P(x,y)\)”

The quantification\(\forall y\exists xP(x,y)\)denotes that “for every \(y\)such that there is a\(x\),\(P(x,y)\)”

Part (b) Example for different truth values

(b) There exists\(x\)such that for all\(y:x > y\)­­--(F)

For every\(y\)there is\(x\)such that:\(x > y\)--(T)

Thus, the truth values are different here in this example.