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

Explanation:

• ∃$x \in \mathbb{R}$ means "there exists an $x$ in the set of real numbers."
• ∃$y \in \mathbb{Z}$ 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 the integers ($\mathbb{Z}$), but they also include many other numbers, such as irrational numbers and non-integer rational numbers.

For example:

• Take $x = \sqrt{2} \in \mathbb{R}$ (which is not an integer).
• Take $y = 1 \in \mathbb{Z}$ (an integer).

Here, $x \neq y$ because $\sqrt{2} \neq 1$. Thus, there exist such $x$ and $y$ that satisfy the condition $x \neq y$.

So, the statement is true.

Would you like any further details or have any questions?

Here are five related questions you might find interesting:

1. How does the statement change if we reverse the sets, i.e., ∃$x \in \mathbb{Z}$ ∃$y \in \mathbb{R}$ : $x \neq y$?
2. What is the difference between a statement involving "for all" ($\forall$) and "there exists" ($\exists$)?
3. Can a similar statement be made about the set of rational numbers ($\mathbb{Q}$) instead of real numbers?
4. What are some examples of real numbers that are not integers?
5. How would the statement change if the inequality were $x = y$ instead of $x \neq y$?

Tip: When dealing with logical statements in mathematics, pay close attention to the quantifiers "∃" (there exists) and "∀" (for all), as they can significantly alter the meaning of a statement.