# MATH 300 Fall 2018 Assignment 10

## 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.

- \( \forall x \in \mathbb{R}, x^2 > 0 \)
- \( \forall x \in \mathbb{R}, \exists n \in \mathbb{N}, x^n \geq 0 \)
- \( \exists \alpha \in \mathbb{R}, \forall x \in \mathbb{R}, \alpha x = x \)
- \( \forall X \in \mathscr{P}(\mathbb{N}), X \subseteq \mathbb{R} \)
- \( \forall n \in \mathbb{N}, \exists X \in \mathscr{P}(\mathbb{N}), \lvert X \rvert < n \)
- \( \exists n \in \mathbb{N}, \forall X \in \mathscr{P}(N), \lvert X \rvert < n \)
- \( \forall X \subseteq \mathbb{N}, \exists n \in \mathbb{Z}, \lvert X \rvert = n \)
- \( \forall n \in \mathbb{Z}, \exists X \subseteq \mathbb{N}, \lvert X \rvert = n \)
- \( \forall n \in \mathbb{Z}, \exists m \in \mathbb{Z}, m=n+5 \)
- \( \exists m \in \mathbb{Z}, \forall n \in \mathbb{Z}, m=n+5 \)

### Exercises for Section 2.10

Negate the following sentences.

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