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.