## 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)