The statement $\forall x \in \mathbb{R} \, \exists y \in \mathbb{Z} : x \neq y$ is true.

Explanation:

• $\forall x \in \mathbb{R}$ means "for every $x$ in the set of real numbers."
• $\exists y \in \mathbb{Z}$ means "there exists a $y$ in the set of integers."
• $x \neq y$ means that $x$ is not equal to $y$.

The real numbers ($\mathbb{R}$) include all possible numbers, including integers, rational numbers, and irrational numbers. For every real number $x$, there exists at least one integer $y$ that is not equal to $x$.

For example:

• If $x = 2$ (which is an integer), then we can choose $y = 3$ (another integer) so that $x \neq y$.
• If $x = \sqrt{2}$ (which is not an integer), then any integer $y$ will satisfy $x \neq y$.

Since for every real number $x$, you can always find an integer $y$ that is different from $x$, the statement is true.

Would you like more details or have any questions?

Here are five related questions that could deepen your understanding:

1. How does the statement change if we replace $\mathbb{Z}$ with $\mathbb{Q}$ (the set of rational numbers)?
2. What happens if the statement is reversed to $\exists y \in \mathbb{Z} \, \forall x \in \mathbb{R} : x \neq y$?
3. Is there a situation where $x = y$ for all $x$ in $\mathbb{R}$ and $y$ in $\mathbb{Z}$?
4. What would the statement mean if the inequality were $x = y$ instead of $x \neq y$?
5. Can you think of a real-world application where a similar logic is used?

Tip: When dealing with quantifiers in mathematical logic, it’s crucial to carefully consider the order of "for all" ($\forall$) and "there exists" ($\exists$), as changing their order can completely change the meaning of a statement.