Consider the sentences  ∀x∃yP(x,y)
  and  ∃y∀xP(x,y)
 . Are these sentences true or false under the interpretation where  x
  and  y
  range over the positive integers and  P(x,y)
  is " x
  divides  y
 "?

To determine the truth value of the sentences \(\forall x \exists y P(x,y)\) and \(\exists y \forall x P(x,y)\) under the interpretation where \(x\) and \(y\) range over the positive integers and \(P(x,y)\) means "x divides y," let's analyze each sentence separately.

### Sentence 1: \(\forall x \exists y P(x,y)\)

This sentence reads: "For every \(x\), there exists a \(y\) such that \(x\) divides \(y\)."

- **Analysis**: 
  - For any positive integer \(x\), we need to find at least one positive integer \(y\) such that \(x\) divides \(y\).
  - A simple choice for \(y\) is \(y = x\). Every positive integer divides itself.
  
Thus, for any \(x\), we can choose \(y = x\), making \(x\) divide \(y\) (since \(x\) divides itself).

- **Conclusion**: The sentence \(\forall x \exists y P(x,y)\) is **true**.

### Sentence 2: \(\exists y \forall x P(x,y)\)

This sentence reads: "There exists a \(y\) such that for every \(x\), \(x\) divides \(y\)."

- **Analysis**: 
  - We need to find a single positive integer \(y\) that is divisible by every positive integer \(x\).
  - The only number that could potentially be divisible by every positive integer would be infinitely large, which does not exist in the set of positive integers.
  
No such finite \(y\) exists that would be divisible by all positive integers simultaneously.

- **Conclusion**: The sentence \(\exists y \forall x P(x,y)\) is **false**.

### Summary

1. \(\forall x \exists y P(x,y)\) is true.
2. \(\exists y \forall x P(x,y)\) is false.

Would you like more details or have any questions?

Here are 8 related questions that expand on the information provided:

1. What is the general method for determining the truth value of quantified statements?
2. How do the domains of \(x\) and \(y\) affect the truth of quantified statements?
3. Can you provide another example with a different predicate \(P(x,y)\)?
4. How would the truth value change if \(x\) and \(y\) range over all integers (including negatives)?
5. What are some common pitfalls when interpreting quantified statements in mathematics?
6. How do nested quantifiers (\(\forall\) and \(\exists\)) affect the complexity of logical statements?
7. Can you explain the difference between "for all" (\(\forall\)) and "there exists" (\(\exists\)) in the context of logical statements?
8. How would the truth values change if the predicate \(P(x,y)\) were "x is greater than y"?

**Tip:** Always check the domains of the variables and the specific interpretation of the predicates when evaluating quantified statements.

Explore the truth values of quantified statements involving positive integers and divisibility in number theory. Analyze statements like ∀x∃yP(x,y) and ∃y∀xP(x,y) with explanations and examples. Suitable for advanced undergraduate students and researchers in mathematics.

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