Let's examine the truth of each of the given propositions.

Proposition A: $\forall x \in \mathbb{R} : x^2 - 4x + 3 \geq 0$

1. Analyze the expression $x^2 - 4x + 3$: The expression $x^2 - 4x + 3$ is a quadratic function. We can rewrite it as: $x^2 - 4x + 3 = (x - 3)(x - 1)$

2. Determine when the expression is non-negative: The expression $(x - 3)(x - 1) \geq 0$ will be true if $x$ is outside the interval $(1, 3)$, because a product of two factors is positive if both factors are either positive or negative simultaneously.

   • For $x \leq 1$, both $x - 3$ and $x - 1$ are negative, so $(x - 3)(x - 1) \geq 0$.
   • For $x \geq 3$, both $x - 3$ and $x - 1$ are positive, so $(x - 3)(x - 1) \geq 0$.

   Thus, $x^2 - 4x + 3 \geq 0$ only for $x \leq 1$ or $x \geq 3$.

3. Conclusion for Proposition A: The statement $\forall x \in \mathbb{R} : x^2 - 4x + 3 \geq 0$ is false, because there exist values of $x$ (specifically, values in the interval $(1, 3)$) for which $x^2 - 4x + 3 < 0$.

Proposition B: $\exists y \in \mathbb{N} : y + 6 = 0$

1. Analyze the statement: Here, we are asked if there exists a natural number $y$ such that $y + 6 = 0$.

2. Evaluate the possibility: The natural numbers $\mathbb{N}$ are typically defined as $\{0, 1, 2, 3, \dots\}$. Since $y + 6 = 0 \Rightarrow y = -6$, this would imply $y = -6$, which is not a natural number.

3. Conclusion for Proposition B: The statement $\exists y \in \mathbb{N} : y + 6 = 0$ is false because there is no natural number $y$ that satisfies $y + 6 = 0$.

Summary

• Proposition A is false.
• Proposition B is false.

Would you like more details on any of the steps, or do you have any questions?

Follow-up Questions

1. How can we determine the sign of a quadratic expression like $x^2 - 4x + 3$ using its roots?
2. What does it mean for a proposition to be true for all $x \in \mathbb{R}$ or for some $y \in \mathbb{N}$?
3. How would the answer change if we altered Proposition B to involve integers $\mathbb{Z}$ instead of natural numbers?
4. What are the steps to factorize a quadratic expression?
5. Why is the interval $(1, 3)$ crucial in determining the truth of Proposition A?

Tip

For verifying propositions with inequalities, sketching the function’s graph can be helpful to visualize where the function is positive or negative.