Question

Describe a rule of inference that can be used to prove that there are exactly two elements \(x and y\)in a domain such that \(P \left( x \right)\) and \(P \left( y \right)\) are true. Express this rule of inference as a statement in English.

Short answer

There are exactly two elements that makeP true.

Step-by-step solution

Introduction

Let we consider on the fact that there are exactly two elements\(x\) and \(y\) in a domain such that \(P \left( x \right)\) and\(P \left( y \right)\) are true. Or we can say that two elements exist that make the statement true, and that every element that makes the statement true is one of these two elements.

Rule of inference

Also we can interpret as an element makes the statement true if and only if it is one of these two elements. In symbols this is \(\exists x\exists y\left( {x \ne y \wedge \forall z\left( {P\left( z \right) \leftrightarrow \left( {z = x \vee z = y} \right)} \right)} \right)\).

The hypotheses are that \(P \left( x \right)\) and \(P \left( y \right)\) are both true, when \(x \ne y\), and that every zthat satisfies \(P\left( z \right)\) must be either\(x\) or \(y\). The conclusion is that there are exactly two elements that make P true.