Question

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 ?

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

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

(c) Every rational number x ∈ S satisfies x ≤ 0.

(d) x ∈ S and x ≤ 0 ⇒ x is not rational.

Answer 1

(c) : The given statement is
P : at least one rational x ∈ S such that x > 0.
The negation would be : There is no rational number x ∈ S such that x > 0
which is equivalent to all rational numbers x ∈ S satisfy x ≤ 0.