## Discrete Mathematics and Its Applications, Seventh Edition

a) May does $not$ have an MP3 player. b) There $is$ pollution in New Jersey. c) 2 + 1 $\ne$ 3 d) The summer in Maine is $not$ hot and sunny.
a) If it isn't true that May has an MP3 player, that means that "May does $not$ have an MP3 player" b) Similarly, if it isn't true that there is $no$ pollution in New Jersey, that means that there is (some) pollution in New Jersey, so this is the negation of b). c) Proposition c) states an equality: its negation is formed stating the equality doesn't hold, which is expressed by the sign $\ne$ ("it is not true that .... = ... " is $equivalent$ to ".... $\ne$ ...." ). d) The only valid negation of proposition d) is "the summer in Maine is not hot and sunny". For example, something on the lines of "the summer in Maine is cold and rainy" isn't a valid answer, as "hot" doesn't have a contrary (what would it be? "cold"? "lukewarm"?) and neither does "sunny" (would it be "cloudy"? "rainy"? "foggy"?). Note also that "the summer in Maine is $not$ hot $nor$ sunny" isn't the correct answer either. That is in fact much stronger than saying "the summer in Maine is not hot and sunny", which means that "the summer in Maine could be hot $but$ $not$ sunny" or "sunny $but$ $not$ hot" or, finally, "$neither$ hot $nor$ sunny".