Unit 9 Exercises

1. Show that the statements x y P(x, y) and x y P(x, y) have the same truth value.

2. For each of the following arguments, explain which rules of inference are used for each step. The universe is the set of people.

1. "John, a student in this class, is 16 years old.  Everyone who is 16 years old can get a driver's liscence.  Therefore, someone in this class can get a driver's liscence."
2. "Someboday in this class enjoys hiking. Every person who enjoys hiking also likes biking.   Therefore, there is a person in this class who likes biking."
3. "Every student in this class owns a personal computer.  Everyone who owns a personal computer can use the Internet.  Therefore, John, a student in this class, can use the Internet."
4. "Everyone in this class owns a personal computer.  Someone in this class has never used the Internet.  Therefore, someone who owns a personal computer has never used the Internet."

3. Determine whether each of the following arguments is valid.  If an argument is correct, what rule of inference is being used?  If it is not, what fallacy occurs?

1. " If n is a real number with n > 1, then n2 > 1.   Suppose that n2 <= 1. Then n <= 1.
2. " If n is a real number with n > 1, then n2 > 1.   Suppose that n2 > 1. Then n > 1.