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 4.1 - Page 161: 21

Answer

a. "If $x$ is a real number and $x\gt1$, then $x^{2}\gt{x}$." b. "Let $x$ be a real number such that $x\gt1$." "Therefore, for all real numbers $x$, if $x\gt1$, then $x^{2}\gt$$x$.

Work Step by Step

For part (a), we simply incorporate the fact that $x$ is a real number into the hypothesis (the "if" part of the statement). For parts (b) and (c), we need only to note that the proof in question is a proof by generalization from the generic particular (see page 151). Therefore, we begin the proof by asserting that a specific but arbitrary number that satisfies the hypothesis exists, and conclude the proof by restating what we set out to prove. For more on this, see example 4.1.8 under the heading "Getting Proofs Started" on page 158.
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