**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.

- "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."
- "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."
- "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."
- "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?

- " If
*n*is a real number with*n*> 1, then*n*^{2}> 1. Suppose that*n*^{2}<= 1. Then*n*<= 1. - " If
*n*is a real number with*n*> 1, then*n*^{2}> 1. Suppose that*n*^{2}> 1. Then*n*> 1.