To solve this problem, we need to negate each of the propositions $P_1$, $P_2$, and $P_3$. Let’s go through each one:

Proposition $P_1$:

• Given Statement: $P_1: (\forall x \in \mathbb{R}) \, x^2 \geq x$.
• Negation: To negate a universal quantifier, we change $\forall$ to $\exists$, and negate the inequality.
  • Negated Statement: $\neg P_1: (\exists x \in \mathbb{R}) \, x^2 < x$.

Proposition $P_2$:

• Given Statement: $P_2: (\forall x \in \mathbb{R})(\exists y \in \mathbb{R}) \, x^2 = y - 3$.
• Negation: We negate the outer quantifier $\forall$ to $\exists$ and the inner quantifier $\exists$ to $\forall$, and then negate the equality.
  • Negated Statement: $\neg P_2: (\exists x \in \mathbb{R})(\forall y \in \mathbb{R}) \, x^2 \neq y - 3$.

Proposition $P_3$:

• Given Statement: $P_3: (\exists a \in \mathbb{R}) \, (a + 6 < 0 \text{ and } a(a + 2) - 6 = 17)$.
• Negation: We negate the existential quantifier $\exists$ to $\forall$ and then negate both parts of the conjunction (i.e., change $<$ to $\geq$ and $=$ to $\neq$).
  • Negated Statement: $\neg P_3: (\forall a \in \mathbb{R}) \, (a + 6 \geq 0 \text{ or } a(a + 2) - 6 \neq 17)$.

Let me know if you'd like further explanations or have any questions!

Related Questions

1. How do you apply De Morgan's laws in logical negations?
2. What are the general rules for negating quantified statements in mathematics?
3. How can you interpret the negation of statements in a real-world context?
4. Why is it important to correctly negate mathematical propositions?
5. How does the logic of "and" and "or" change when negated?

Tip

When negating statements with multiple quantifiers, always start from the outermost quantifier and proceed inward.