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 206: 19

Answer

a product of two positive real numbers is greater than 100, then at least one of the numbers is greater than 10. proving this statement by contrapositions.

Work Step by Step

If a product of two positive real numbers is greater than 100, then at least one of the numbers is greater than 10. Proof (by contraposition): -[To go by contraposition, we must prove that ∀ positive real numbers, r and s, if r ≤ 10 and s ≤ 10, then r s ≤ 100.] -Suppose r and s are positive real numbers and r ≤ 10 and s ≤ 10. -By the algebra of inequalities, rs ≤ 100. -[To derive this fact, multiply both sides of r ≤ 10 by s to obtatin r s ≤ 10s. - And multiply both sides of s ≤ 10 by 10 to obtain 10s ≤ 10·10 = 100. - By transitivity of ≤, then, r s ≤ 100.] But this is what was to be shown.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.