And, Or, Not

The following is the list of problems for Section 2.2 of the Book of Proof (page 41). There is a forum open at the end, so you can ask questions. It is a great way to interact with the instructor and with other students in your class, should you need some assistance with any question. Please, do not post solutions.

Express each statement or open sentence in one of the forms \( P \lor Q, P \land Q \), or \( \lnot P \). Be sure to also state exactly what statements \( P \) and \( Q \) stand for.

1. The number 8 is both even and a power of 2.
2. The matrix \( A \) is not invertible.
3. \( x \neq y \)
4. \( x < y \)
5. \( x \geq y \)
6. There is a quiz scheduled for Wednesday or Friday.
7. The number \( x \) equals zero, but the number \( y \) does not.
8. At least one of the numbers \( x \) or \( y \) equals zero.
9. \( x \in A \setminus B \)
10. \( x \in A \cup B \)
11. \( A \in \{ X \in \mathscr{P}(\mathbb{N}) : \lvert X^\complement \rvert < \infty \} \)
12. Happy families are all alike, but each unhappy family is unhappy in its own way. (Leo Tolstoy, Anna Karenina)
13. Human beings want to be good, but not too good, and not all the time. (George Orwell)
14. A man should look for what it is, and not for what he thinks should be. (Albert Einstein)