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 1.2 - Applications of Propositional Logic - Exercises - Page 23: 12

Answer

The system specifications are consistent

Work Step by Step

Let $p$ denote "The file system is locked" Let $q$ denote "New messages will be queued" Let $r$ denote "The system is functioning normally" Let $s$ denote "New messages will be sent to the message buffer" The system specification are as follows: 1) If the file system is not locked, then new messages will be queued. - $\neg p \to q$ 2) If the file system is not locked, then the system is functioning normally, and conversely. - $\neg p \leftrightarrow r $ 3) If new messages are not queued, then they will be sent to the message buffer. - $\neg q \to s $ 4) If the file system is not locked, then new messages will be sent to the message buffer.- $\neg p \to s$ 5) New messages will not be sent to the message buffer. - $\neg s$ For the last specification to be true, $s$ has to be false. For the fourth specification to be true, if $s$ is false, $\neg p$ has to be false, so $p$ has to be true. For the second specification to be true, if $\neg p $ is is false, $r$ has to be false as well. For the third specification to be true, if $s$ is false, $\neg q$ has to be false, so $q$ has to be true. The first specification is true for $p, q$ being true. Therefore, the system specifications are consistent for $p,q$ true and $r,s$ false.
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