Quantifiers, Translating English to Symbolic Logic, Negating Statements

The following is the list of problems for Sections 2.7 and 2.10 of the Book of Proof (pages 53 and 60). 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.

Exercises for Section 2.7

Write the following as English sentences. Say whether they are true or false.

1. \( \forall x \in \mathbb{R}, x^2 > 0 \)
2. \( \forall x \in \mathbb{R}, \exists n \in \mathbb{N}, x^n \geq 0 \)
3. \( \exists \alpha \in \mathbb{R}, \forall x \in \mathbb{R}, \alpha x = x \)
4. \( \forall X \in \mathscr{P}(\mathbb{N}), X \subseteq \mathbb{R} \)
5. \( \forall n \in \mathbb{N}, \exists X \in \mathscr{P}(\mathbb{N}), \lvert X \rvert < n \)
6. \( \exists n \in \mathbb{N}, \forall X \in \mathscr{P}(N), \lvert X \rvert < n \)
7. \( \forall X \subseteq \mathbb{N}, \exists n \in \mathbb{Z}, \lvert X \rvert = n \)
8. \( \forall n \in \mathbb{Z}, \exists X \subseteq \mathbb{N}, \lvert X \rvert = n \)
9. \( \forall n \in \mathbb{Z}, \exists m \in \mathbb{Z}, m=n+5 \)
10. \( \exists m \in \mathbb{Z}, \forall n \in \mathbb{Z}, m=n+5 \)

Exercises for Section 2.10

Negate the following sentences.

1. The number \( x \) is positive but the number \( y \) is not positive.
2. If \( x \) is prime then \( \sqrt{x} \) is not a rational number.
3. For every prime number \( p \) there is another prime number \( q \) with \( q > p \).
4. For every positive number \( \varepsilon \), there is a positive number \( \delta \) for which \( \lvert x - \alpha \rvert < \delta \) implies \( \lvert f(x) - f(\alpha) \rvert < \varepsilon \).
5. For every positive number \( \varepsilon \) there is a positive number \( M \) for which \( \lvert f(x) - b \rvert < \varepsilon \), whenever \( x > M \).
6. There exists a real number \( a \) for which \( a + x = x \) for every real number \( x \).
7. I don’t eat anything that has a face.
8. If \( x \) is a rational number and \( x \neq 0 \) then \( \tan(x) \) is not a rational number.
9. If \( \sin(x) < 0 \), then it is not the case that \( 0 \leq x \leq \pi \).
10. If \( f \) is a polynomial and its degree is greater than 2, then \( f’ \) is not constant.
11. You can fool all of the people all of the time.
12. Whenever I have to choose between two evils, I choose the one I haven’t tried yet. (Mae West)