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Let's analyze each statement to determine whether it is true or false.

### Given Statements:
1. **Statement (a):** \( \forall x \in \mathbb{N}, \exists y \in \mathbb{N} \, \text{such that} \, x - y > 0 \)
   - This statement asserts that for every natural number \( x \), there exists a natural number \( y \) such that \( x - y > 0 \).

2. **Statement (b):** \( \exists y \in \mathbb{N}, \forall x \in \mathbb{N} \, \text{such that} \, x - y > 0 \)
   - This statement asserts that there exists a natural number \( y \) such that for every natural number \( x \), \( x - y > 0 \).

### Analyzing Each Statement

#### Statement (a)
This statement is essentially saying that for every natural number \( x \), we can find another natural number \( y \) that is smaller than \( x \) (since \( x - y > 0 \) implies \( y < x \)).

- For example, if \( x = 2 \), we can choose \( y = 1 \), and \( x - y = 2 - 1 = 1 > 0 \).
- Similarly, if \( x = 3 \), we can choose \( y = 2 \), and \( x - y = 3 - 2 = 1 > 0 \).

Since we can always choose a \( y \) that is less than \( x \) (by picking \( y = x - 1 \) for any \( x > 1 \)), this statement is **true**.

#### Statement (b)
This statement suggests that there exists a specific natural number \( y \) such that for all natural numbers \( x \), \( x - y > 0 \). This would mean that \( y \) is smaller than every natural number \( x \).

- If we consider \( y = 1 \), then \( x - y > 0 \) for all \( x > 1 \).
- However, if we consider \( x = 1 \) and \( y = 1 \), then \( x - y = 1 - 1 = 0 \), which does not satisfy \( x - y > 0 \).

Since no single \( y \) can make \( x - y > 0 \) true for all \( x \in \mathbb{N} \) (particularly when \( x = 1 \)), this statement is **false**.

### Conclusion
- **Statement (a) is true.**
- **Statement (b) is false.**

Would you like further details on the reasoning? Here are some related questions for further exploration:

1. What would happen if we changed the condition \( x - y > 0 \) to \( x - y \geq 0 \)?
2. How would these statements change if the domain was the set of integers \( \mathbb{Z} \) instead of natural numbers?
3. What if we altered statement (a) to use "for some \( x \)" instead of "for all \( x \)"?
4. Could there be other values of \( y \) in statement (b) that make it work for more cases, if not all?
5. How does quantifier ordering affect the truth of mathematical statements in general?

**Tip:** In statements involving quantifiers, changing the order of "for all" (\( \forall \)) and "there exists" (\( \exists \)) can significantly alter the meaning and truth of the statement.

Perhatikan dua pernyataan berikut. a) ∀x∈N,∃y∈N ∃ x − y > 0 b) ∃y∈N, ∀x∈N ∃ x − y > 0 Tentukan apakah masing-masing pernyataan tersebut bernilai benar atau salah kemudian berikan penjelasan.

Examine the truth values of two statements involving quantifiers (∀, ∃) and inequalities with natural numbers. This problem covers advanced mathematical concepts like quantifier manipulation and inequality reasoning.

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