Let's analyze the given logical proposition $a$ and its negation $\neg a$ to determine the correct answers.

Proposition $a$:

$a : \forall x \in \mathbb{Z}, \exists y \in \mathbb{N}_0 \text{ such that } x^2 = y$

This statement means that for every integer $x$, there exists a non-negative integer $y$ such that $x^2 = y$.

• For any integer $x$, $x^2$ is always a non-negative integer. Therefore, for each $x \in \mathbb{Z}$, we can always find a $y \in \mathbb{N}_0$ (specifically $y = x^2$) such that $x^2 = y$.
• This makes the proposition $a$ true ($a = T$).

Negation $\neg a$:

To find the negation $\neg a$, we apply logical negation:

$\neg a : \exists x \in \mathbb{Z} \text{ such that } \forall y \in \mathbb{N}_0, x^2 \neq y$

This means there exists some integer $x$ such that for all non-negative integers $y$, $x^2 \neq y$. However, this statement is false because $x^2$ will always equal some $y \in \mathbb{N}_0$.

Comparing with the Options:

Now let's compare the analysis with the given options:

• (a) $a = T$ and $\neg a = \forall x \in \mathbb{Z}, \exists y \in \mathbb{N}: x^2 \neq y$
  This option is incorrect because the negation is incorrectly formulated.

• (b) $a = T$ and $\neg a = \exists x \in \mathbb{Z}, \forall y \in \mathbb{N}_0: x^2 \neq y$
  This option is correct in stating $a = T$ and the correct formulation of the negation $\neg a$.

• (c) $a = F$ and $\neg a = \exists x \in \mathbb{Z}, \forall y \in \mathbb{N}_0: x^2 \neq y$
  This option is incorrect because $a$ is not false.

Correct Answer:

The correct answer is (b).

Would you like more details, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. What is the logical negation of the statement $\exists x \in \mathbb{Z}, \forall y \in \mathbb{N}_0: x^2 = y$?
2. How does the truth value of a proposition affect the truth value of its negation?
3. Can you provide an example of a proposition involving $\forall$ and $\exists$ that is false?
4. What is the difference between $\mathbb{N}$ and $\mathbb{N}_0$ in terms of set theory?
5. How would the proposition change if we were working with $\mathbb{R}$ instead of $\mathbb{Z}$?

Tip: When working with logical propositions, always be careful to apply negations correctly, especially when dealing with quantifiers like $\forall$ (for all) and $\exists$ (there exists).