Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education
ISBN 10: 0073383090
ISBN 13: 978-0-07338-309-5

Chapter 1 - Section 1.4 - Predicates and Quantifiers - Exercises - Page 54: 19

Answer

a) There exists a value between 1 and 5 such that P(x) is true, this thus equivalent with $ P(1) \lor P(2) \lor P(3) \lor P(4) \lor P(5)$. b) For every value between 1 and 5 we need P(x) is true, this thus equivalent with $ P(1) \land P(2) \land P(3) \land P(4) \land P(5)$. c) This statement is negation of the statement in a, and is thus equivalent with $ \neg (P(1) \lor P(2) \lor P(3) \lor P(4) \lor P(5))$. d) This statement is negation of the statement in b, and is thus equivalent with $ \neg (P(1) \land P(2) \land P(3) \land P(4) \land P(5))$. e) This statement is equivalent with $( P(1) \land P(2) \land P(4) \land P(5)) \lor (\neg P(1) \lor \neg P(2) \lor \neg P(3) \lor \neg P(4) \lor \neg P(5))$, because the first statement means that P(x) is true when x is not equal to 3 and the second statement means that there has to be a value of x where P(x) is false and thus $\neg P(x) $ is true.

Work Step by Step

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