## Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education

# Chapter 1 - Section 1.1 - Propositional Logic - Exercises: 4

#### Answer

a) Jennifer and Teja are not friends. b) There are not 13 items in a baker’s dozen. c) One day Abby sent less than -or exactly - 100 messages. d) 121 is not a perfect square

#### Work Step by Step

a) The word "friend" does not have a contrary (as for example "successful" and "unsuccessful"). So the only way to negate this sentence is by stating that "Jennifer and Teja are $not$ friends. Note that "Jennifer and Teja are enemies" is not a correct answer. What if they are just acquaintances? Or if they simply do not know each other? b) Once again, negating this sentence translate to state that "the number of items in a baker's dozen is not 13", or equivalently "There are not 13 items in a baker’s dozen". We cannot specify another quantity (ex 12 or 15) since that would be too strong (there could be 12 items, as 9, or 10, or 0 or 10000: just not 13). c) We want to express the fact that it is not true that a certain event happened everyday: that means that at least one day, this event did not occur: in our case that signifies that, at least one day, Abby hasn't sent more than 100 messages (so she has sent less than 100 of them, or maybe exactly 100) d) That is pretty self-explanatory. Since the sentence is equivalent to say that "the square root of 121 is an integer", its negation could also be stated as "the square root of 121 is not an integer".

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