To translate the given mathematical statement into propositional logic using the predicates:

• $\text{RelPrime}(x,y)$: $x$ and $y$ are relatively prime.
• $\text{Divisor}(d,x,y)$: $d$ is a divisor of both $x$ and $y$ (i.e., $d \mid x$ and $d \mid y$).

Given Statement:

"For any integers $x$ and $y$, if there is some integer $d > 1$ such that $d$ divides both $x$ and $y$, then $x$ and $y$ are not relatively prime."

Logical Translation:

$\forall x \forall y \left( \exists d \left( d > 1 \land \text{Divisor}(d, x, y) \right) \rightarrow \neg \text{RelPrime}(x, y) \right)$

Explanation:

• $\forall x \forall y$: The statement applies to all integers $x$ and $y$.
• $\exists d \left( d > 1 \land \text{Divisor}(d, x, y) \right)$: There exists an integer $d$ greater than 1 that divides both $x$ and $y$.
• $\rightarrow \neg \text{RelPrime}(x, y)$: If such a $d$ exists, then $x$ and $y$ are not relatively prime (i.e., they share a common divisor greater than 1).

Would you like me to clarify any part of this?

Related Questions:

1. How can this statement be expressed in set notation?
2. What is the contrapositive of this logical statement?
3. How does this statement relate to the definition of the greatest common divisor?
4. Can we express the concept of relative primeness using modular arithmetic?
5. How would we negate this logical statement?

Tip:

The contrapositive of a conditional statement $P \rightarrow Q$ is $\neg Q \rightarrow \neg P$, which is always logically equivalent.