∃𝑥∈ℝ∃𝑦∈ℤ:𝑥≠𝑦 is is true or false

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.

Explore the statement ∃(x ∈ ℝ) ∃(y ∈ ℤ) : x ≠ y and understand its truth. Learn about real numbers, integers, and how this statement is verified. Suitable for advanced high school students and enthusiasts of set theory and mathematical logic.

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