1. **Problem:** Negate the proposition $\forall x \in \mathbb{R}, \exists y \in \mathbb{R} : 2x + y > 3$. **Step 1:** Recall the negation rules for quantifiers: - Negation of $\forall x, P(x)$ is $\exists x$ such that $\neg P(x)$. - Negation of $\exists y, Q(y)$ is $\forall y$ such that $\neg Q(y)$. **Step 2:** Apply negation stepwise: $$\neg \left( \forall x \in \mathbb{R}, \exists y \in \mathbb{R} : 2x + y > 3 \right) = \exists x \in \mathbb{R}, \forall y \in \mathbb{R} : 2x + y \leq 3.$$ 2. **Problem:** Negate $\forall \varepsilon > 0, \exists \alpha > 0 : |x| < \alpha \Rightarrow |x^2| < \varepsilon$. **Step 1:** Use the same quantifier negation rules. **Step 2:** Negate the implication $P \Rightarrow Q$ which is $P \wedge \neg Q$. **Step 3:** Negation: $$\exists \varepsilon > 0, \forall \alpha > 0 : |x| < \alpha \wedge |x^2| \geq \varepsilon.$$ 3. **Problem:** Negate $\forall x \in \mathbb{R}, (x=0 \lor x \in ]2,4])$. **Step 1:** Negate the universal quantifier: $$\exists x \in \mathbb{R} : \neg (x=0 \lor x \in ]2,4]) = \exists x \in \mathbb{R} : (x \neq 0) \wedge (x \notin ]2,4]).$$ 4. **Problem:** Negate $\exists M \in \mathbb{R}^+, \forall n \in \mathbb{N} : |U_n| \leq M$. **Step 1:** Negate the existential quantifier: $$\forall M \in \mathbb{R}^+, \exists n \in \mathbb{N} : |U_n| > M.$$ **Summary:** 1. $\exists x \in \mathbb{R}, \forall y \in \mathbb{R} : 2x + y \leq 3$. 2. $\exists \varepsilon > 0, \forall \alpha > 0 : |x| < \alpha \wedge |x^2| \geq \varepsilon$. 3. $\exists x \in \mathbb{R} : x \neq 0 \wedge x \notin ]2,4]$. 4. $\forall M \in \mathbb{R}^+, \exists n \in \mathbb{N} : |U_n| > M$.