Question

Show that these statements are inconsistent: “If Miranda does not take a course in discrete mathematics, then she will not graduate.” “If Miranda does not graduate, then she is not qualified for the job.” “If Miranda reads this book, then she is qualified for the job.” “Miranda does not take a course in discrete mathematics but she reads this book.”

Short answer

The given statements are inconsistent.

Step-by-step solution

Introduction

Prove this by using contradiction.

Proof using contradiction

First, translate the statements.

“If Miranda does not take a course in discrete mathematics, then she will not graduate” into symbols using the letters, we have\(\neg t \to \neg g\);

The statement “ If Miranda does not graduate, then she is not qualified for the job” into symbols using the letters, we have\(\neg g \to \neg q\);

The statement “If Miranda reads this book, then she is qualified for the job” into the symbols using the letters, we have\(r \to q\); and

The statement “Miranda does not take a course in discrete mathematics but she reads this book” into symbols using the letters, we have\(\neg t \wedge r\).

Assume that the statements are consistent.

The last statement tells us that\(r\)must be true, and so again modus ponens (third statement) makes\(q\)true.

This is contradiction to\(q \wedge \neg q\)

So the assumption the statements are consistent is false.

Hence the statements are consistent.