Exam (elaborations) TEST BANK FOR Discrete Mathematics and Its Applications 7th Edition By Kenneth H. Rosen and Jerrold W. Grossman (Student’s Solutions Guide)

Manipulating propositions and constructing truth tables are straightforward. A truth table is constructed by finding the truth values of compound propositions from the inside out; see the solution to Exercise 31, for instance. This exercise set also introduces fuzzy logic. 1. Propositions must have clearly defined truth values, so a proposition must be a declarative sentence with no free variables. a) This is a true proposition. b) This is a false proposition (Tallahassee is the capital). c) This is a true proposition. d) This is a false proposition. e) This is not a proposition (it contains a variable; the truth value depends on the value assigned to x). f) This is not a proposition, since it does not assert anything. 3. a) Mei does not have an MP3 player. b) There is pollution in New Jersey. c) 2+1#3 d) It is not the case that the summer in Maine is hot and sunny. In other words, the summer in Maine is not hot and sunny, which means that it is not hot or it is not sunny. It is not correct to negate this by saying "The summer in Maine is not hot and not sunny." [For this part (and in a similar vein for part (b)) we need to assume that there are well-defined notions of hot and sunny; otherwise this would not be a proposition because of not having a definite truth value.] 5. a) Steve does not have more than 100 GB free disk space on his laptop. (Alternatively: Steve has less than or equal to 100 GB free disk space on his laptop.) b) Zach does not block e-mails and texts from Jennifer. (Alternatively, and more precisely: Zach does not block e-mails from Jennifer, or he does not block texts from Jennifer. Note that negating an "and" statement produces an "or" statement. It would not be correct to say that Zach does not block e-mails from Jennifer, and he does not block texts from Jennifer. That is a stronger statement than just the negation of the given statement.) c) 7·11·13#999. d) Diane did not ride her bike 100 miles on Sunday. 7. a) This is false, because Acme's revenue was larger. b) Both parts of this conjunction are true, so the statement is true. c) The second part of this disjunction is true, so the statement is true. d) The hypothesis of this conditional statement is false and the conclusion is true, so by the truth-table definition this is a true statement. (Either of those conditions would have been enough to make the statement true.)