Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 3 - Logic - 3.3 Truth Tables for Negation, Conjunction and Disjunction - Exercise Set 3.3 - Page 150: 99


The statement is “10% named teaching or 4.8% named nursing, but 12% not named engineer.”

Work Step by Step

Assume that the compound statement consists of three simple statements: \[p\]: \[10%\] named teaching. \[q\]: \[4.8%\] named nursing. \[r\]: 12% not named engineer. The symbol ‘~’ is used for the word ‘not’, ‘\[\vee \]’ for the word ‘or’, and ‘\[\wedge \]’ for ‘and’. So, combine all the simple statements to write the compound statement in symbolic form by the use of the symbols ‘~’, ‘\[\vee \]’ and ‘\[\wedge \]’. The symbolic form of the statement, “\[10%\] named teaching or \[4.8%\] named nursing, but \[12%\] not named engineer” is \[\left( p\vee q \right)\wedge r\]. Now from given graph, the statement \[p\] is false, the statement \[q\]is true and the statement \[r\]is true. Consider the statement: \[\left( p\vee q \right)\wedge r\] Substitute the truth value true as T and false as F and use properties of conjunction and disjunction (conjunction gives truth value true only when all statements are true and disjunction gives truth value false only when all statements are false, and also use the negation property to make the statement negative). \[\begin{align} & \left( \text{F}\vee \text{T} \right)\wedge \text{T} \\ & \text{T}\wedge \text{T} \\ & \text{T} \\ \end{align}\] The truth value of the statement \[\left( p\vee q \right)\wedge \tilde{\ }r\] is true. Therefore, the assumed compound statement is true.
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