Discrete Mathematics with Applications 4th Edition

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

Chapter 2 - The Logic of Compound Statements - Exercise Set 2.3 - Page 63: 44

Answer

a. p → q b. r ∨ s c. ∼s → ∼t d. ∼q ∨ s e. ∼s f. ∼p ∧ r → u g. w ∨ t h. ∴ u ∧ w d. ∼q ∨ s e. ∼s ∴ ∼q (1) Elimination a. p → q ∼q (1) ∴ ∼p (2) Modus Tollens c. ∼s → ∼t e. ∼s ∴ ∼t (3) Modus ponens g. w ∨ t ∼t (3) ∴ w (4) Elimination b. r ∨ s e. ∼s ∴ r (5) Elimination r (5) ∼p (2) ∴ ∼p ∧ r (6) Conjunction f. ∼p ∧ r → u ∼p ∧ r (6) ∴ u (7) Modus Ponens w (4) u (7) ∴ u ∧ w Conjunction = h

Work Step by Step

first step : by using (d)and (e ) as premises then the conclusion will be ∼q (1) by Elimination 2nd step: by using (a)and (1) as premises then the conclusion will be ∼p (2) by Modus Tollens 3rd step : by using (c)and (e) as premises then the conclusion will be ∼t (3) by Modus Tollens 4th step: by using (g)and (3) as premises then the conclusion will be w (4) by Elimination 5th step : by using (b)and (e) as premises then the conclusion will be r (5) by Elimination 6th step: by using (5)and (2) as premises then the conclusion will be ∼p ∧ r (6) by Conjunction 7th step: by using (f)and (6) as premises then the conclusion will be u (7) by Modus Ponens 8th step: by using (4)and (7) as premises then the conclusion will be u ∧ w by Conjunction
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