Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 4 - Elementary Number Theory and Methods of Proof - Exercise Set 4.6 - Page 205: 2

Answer

No, $\frac{1}{0}$ is not an irrational number.

Work Step by Step

Letting $R(x)$ denote "x is rational" and $I(x)$ "x is irrational, we have the following definition of the relationship between rational and irrational numbers: $\forall x\in\mathbb{R}(\neg R(x)\Rightarrow I(x))$, i.e., for all real numbers $x$, if $x$ is not rational, then it is irrational. But $\frac{1}{0}$ is not even in the domain of discourse, the real numbers, so we cannot use the definition to deduce that it is irrational.
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