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 1.1 - Propositional Logic - Exercises - Page 12: 2


Only c) and e) are propositions. Their truth values are, respectively, False and False.

Work Step by Step

A sentence is called a proposition if it states a fact, and if we can assign it a truth value (i.e. if we can tell whether it is true of false). In other words, a sentence $p$ is a proposition if the question "is $p$ true?" makes sense (so that it accepts as an answer only a 'yes' or a 'no', and not other things like 'it depends', 'I don't agree' etc.). So, c) and e) are clearly propositions: it makes sense to ask ourselves whether they are true or false, and we can answer this question: they are both false. On the other hand, for a) and b) the question "is this sentence true" doesn't make any sense, so they are not propositions. Finally, for d) and f) the question $could$ make sense, but it is impossible to answer: the truth value of d) depends on the value of $x$ ( it would be true if $x = 1$, but false otherwise). Similarly the truth value of f) depends on the value of $n$ (for example it would be false for $n=2$, but true for $n=10$, since $2^{10} = 1024$).
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