Exercise 1

Consider the following statements:

• (a) \( \forall x \in \mathbb{R}, \exists y \in \mathbb{R}, x + y > 0 \)
• (b) \( \forall x \in \mathbb{R}, \forall y \in \mathbb{R}, x + y > 0 \)
• (c) \( \exists x \in \mathbb{R}, \forall y^2 > x \)
• (d) \( \forall x \in \mathbb{R}, x^2 + x + 2 > 0 \)
• (e) \( \forall x \in \mathbb{R}^+, \forall y \in \mathbb{R}^+, \quad \frac{x}{x + 3} = \frac{y}{y + 3} \Rightarrow x = y \)
• (f) \( \forall x > 0, \quad x + \frac{1}{x} > 2 \)

1. Are the statements (a), (b), (c), (d), (e), (f) true or false?

2. Write their negation.

Exercise 2

Let \( a, b \in \mathbb{R}^+ \). Show that if \( a \leq b \), then \( a \leq \frac{a + b}{2} \leq b \) and \( a \leq \sqrt{ab} \leq b \).

Exercise 3

Show that if \( n \) is a strictly positive integer, then \( n^2 + 1 \) is not the square of a natural number.