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 integers and  P(x,y)
  is " 2x=3y
 "?

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 integers and \(P(x,y)\) means \(2x = 3y\), 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 \(2x = 3y\)."

- **Analysis**:
  - For each integer \(x\), we need to find an integer \(y\) such that \(2x = 3y\).
  - Rearrange the equation to solve for \(y\):
    \[
    y = \frac{2x}{3}
    \]
  - For \(y\) to be an integer, \(\frac{2x}{3}\) must be an integer. This means \(2x\) must be divisible by 3.
  - However, not all integers \(x\) will make \(2x\) divisible by 3. For example, if \(x = 1\), \(2 \cdot 1 = 2\), and 2 is not divisible by 3.
  
Thus, there exist values of \(x\) for which no integer \(y\) satisfies \(2x = 3y\).

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

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

This sentence reads: "There exists a \(y\) such that for every \(x\), \(2x = 3y\)."

- **Analysis**:
  - We need to find a single integer \(y\) such that \(2x = 3y\) holds for all integers \(x\).
  - Rearrange the equation to solve for \(x\):
    \[
    x = \frac{3y}{2}
    \]
  - For \(x\) to be an integer, \(\frac{3y}{2}\) must be

Explore the truth value of quantified statements ∀x∃yP(x,y) and ∃y∀xP(x,y) with the Diophantine equation 2x = 3y. Understand when these statements hold true or false under integer interpretations.

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