**Homework Help: Questions and Answers:** Express each statement or open sentence in a symbolic form such as P∧Q, P∨Q, P∨∼Q or ∼P, etc. Be sure to also state exactly what statements P and Q stand for.

- The number 8 is both even and a power of 2.
- The matrix A is not invertible.
- x = y
- x < y
- y ≥ x
- There is a quiz scheduled for Wednesday or Friday.
- The number x equals zero, but the number y does not.
- At least one of the numbers x and y equals 0.
- x ∈ A−B
- x ∈ A∪B
- A ∈ {X∈P(N) : ∣x̄∣ <∞}
- Happy families are all alike, but each unhappy family is unhappy in its own way. (Leo Tolstoy, Anna Karenina)
- Human beings want to be good, but not too good, and not all the time. (George Orwell)
- A man should look for what is, and not for what he thinks should be. (Albert Einstein)

**Answer:**

Let’s solve these problems step by step.

**1. The number 8 is both even and a power of 2.**

**Statement P**: 8 is even.**Statement Q**: 8 is a power of 2.**Symbolic Form**: P ∧ Q

**2. The matrix A is not invertible.**

**Statement P**: The matrix A is invertible.**Symbolic Form**: ¬P

**3. x / y**

- This is already in symbolic form. No further conversion is needed.

**4. x < y**

- This is also already in symbolic form.

**5. y ≥ x **

- This is already in symbolic form.

**6. There is a quiz scheduled for Wednesday or Friday.**

**Statement P**: The quiz is scheduled for Wednesday.**Statement Q**: The quiz is scheduled for Friday.**Symbolic Form**: P ∨ Q

**7. The number x equals zero, but the number y does not.**

**Statement P**: x = 0**Statement Q**: y ≠ 0**Symbolic Form**: P ∧ Q

**8. At least one of the numbers x and y equals 0.**

**Statement P**: x = 0**Statement Q**: y = 0**Symbolic Form**: P ∨ Q

**9. x ∈ A – B**

- This is already in symbolic form, where A – B represents the set difference.

**10. x ∈ A ∪ B**

- This is already in symbolic form, where A ∪ B represents the union of sets A and B.

**11. A ∈ {X ∈ P(N) : ∣x̄∣ < ∞}**

- This is already in symbolic form. It represents that set A belongs to a particular collection of sets.

**12. Happy families are all alike, but each unhappy family is unhappy in its own way.**

**Statement P**: Happy families are all alike.**Statement Q**: Each unhappy family is unhappy in its own way.**Symbolic Form**: P ∧ Q

**13. Human beings want to be good, but not too good, and not all the time.**

**Statement P**: Human beings want to be good.**Statement Q**: Human beings do not want to be too good.**Statement R**: Human beings do not want to be good all the time.**Symbolic Form**: P ∧ ¬Q ∧ ¬R

**14. A man should look for what is, and not for what he thinks should be.**

**Statement P**: A man should look for what is.**Statement Q**: A man should not look for what he thinks should be.**Symbolic Form**: P ∧ ¬Q

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