Let's solve these two questions systematically.

Question 3

• Given: $P(x)$ is the predicate $x^2 - 3x + 4 = 0$.
• Universe of discourse: The set of real numbers.
• Task: Find the truth value of $\exists x \, P(x)$ (there exists $x$ such that $P(x)$ is true).

Solution:
We need to solve the equation $x^2 - 3x + 4 = 0$. This is a standard quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 1$, $b = -3$, and $c = 4$. The roots are given by the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$

Substitute $a = 1$, $b = -3$, and $c = 4$:

$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(4)}}{2(1)}.$

Simplify:

$x = \frac{3 \pm \sqrt{9 - 16}}{2}.$

$x = \frac{3 \pm \sqrt{-7}}{2}.$

Here, $\sqrt{-7}$ is not a real number because it involves a negative square root. Therefore, the equation has no real solutions.

Conclusion: Since there is no real $x$ that satisfies $x^2 - 3x + 4 = 0$, the truth value of $\exists x \, P(x)$ is false.

Question 2

• Given: $P(x)$ is the predicate $x^2 + 1 \geq 0$.
• Universe of discourse: The set of all complex numbers.
• Task: Find the truth value of $\forall x \, P(x)$ (for all $x$, $P(x)$ is true).

Solution:
We analyze whether the inequality $x^2 + 1 \geq 0$ holds for all complex numbers $x$.

1. Recall that for real $x$, $x^2 \geq 0$, so $x^2 + 1 > 0$ for all real $x$.

2. For complex $x = a + bi$ (where $a, b \in \mathbb{R}$), we consider $x^2 + 1$. Expand:

   $x^2 = (a + bi)^2 = a^2 + 2abi - b^2.$ Adding 1:

   $x^2 + 1 = (a^2 - b^2 + 1) + 2abi.$

Here, $x^2 + 1$ is not necessarily real, so the inequality $x^2 + 1 \geq 0$ does not make sense directly for complex numbers because "greater than or equal to" is undefined in the complex field.

Conclusion: Since $\geq 0$ is undefined in the complex numbers, the predicate $P(x): x^2 + 1 \geq 0$ does not hold for all complex $x$. Thus, the truth value of $\forall x \, P(x)$ is false.

Final Answers

1. Question 3: The truth value of $\exists x \, P(x)$ is false.
2. Question 2: The truth value of $\forall x \, P(x)$ is false.

Do you need further explanations or clarifications?

Here are 5 related questions to expand your understanding:

1. What is the difference between existential ($\exists$) and universal ($\forall$) quantifiers?
2. How do you determine if a quadratic equation has real solutions?
3. How does the complex number system extend the real numbers?
4. What is the significance of the imaginary unit $i$ in solving quadratics?
5. Why are inequalities like $x^2 + 1 \geq 0$ undefined for complex numbers?

Tip: Always check the universe of discourse (real vs. complex numbers) when solving equations or verifying predicates.