MATH 300 Fall 2018 Assignment 10

Quantifiers

The following is the list of problems for Section 2.7 of the Book of Proof (page 53). 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.

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. \( \forall X \in \mathscr{P}(\mathbb{N}), X \subset \mathbb{R} \)
  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. \( \exists m \in \mathbb{Z}, \forall n \in \mathbb{Z}, m=n+5 \)