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Let S be a non-empty subset of R. Consider the following statement: P: There is a rational number x ∈ S such that x > 0. Which of the following statements is the negation of the statement P?

1. There is no rational number x ∈ S such that x ≤ 0.
2. Every rational number x ∈ S satisfies x ≤ 0.
3. x ∈ S and x ≤ 0 ⇒ x is not rational.
4. There is a rational number x ∈ S such that x ≤ 0.

Correct answer: Every rational number x ∈ S satisfies x ≤ 0.

Solution

The correct negation of statement P asserts that all rational numbers in the subset S must be less than or equal to zero, which directly contradicts the original claim that there exists at least one positive rational number in S.

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