∀𝑥∈ℝ∃𝑦∈ℤ:𝑥≠𝑦
true or false

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.

Explore the statement ∀𝑥∈ℝ ∃𝑦∈ℤ: 𝑥 ≠ 𝑦 in mathematical logic. Learn its implications with examples covering real numbers, integers, and logical quantifiers. Suitable for advanced high school or college-level mathematics.

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