Question

Let \(P \left( {x, y} \right)\) be a propositional function. Show that \(\exists x\forall y P \left( {x, y} \right) \to \forall y\exists x P \left( {x, y} \right)\) is a tautology.

Short answer

\(\exists x\forall y P \left( {x, y} \right) \to \forall y\exists x P \left( {x, y} \right)\) is a tautology.

Step-by-step solution

Introduction

A tautology is a logical argument that has the same conclusion as the premise.

Prove for tautology

Let us assume the hypothesis that \(\exists x\forall y P \left( {x, y} \right) \to \forall y\exists x P \left( {x, y} \right)\) is a tautology.

This means that there is some\({x_0}\) such that\(P\left( {{x_0},y} \right)\) holds for all y.

Then it is certainly true that for all y there exists an x such that \(P\left( {x,y} \right)\) is true, since each case we can take\(x = {x_0}\).

Hence, proved