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Question:

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

There is no rational number x ∈ S such that x ≤ 0

There is a rational number x ∈ S such that x ≤ 0

x ∈ S and x ≤ 0 = x is not rational

Every rational number x ∈ S satisfies x ≤ 0

Solution:

P: there is a rational number x ∈ S such that x > 0
¬P: Every rational number x ∈ S satisfies x ≤ 0
Hence, option 'D' is correct.