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? 

Options

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

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

• Every rational number x ∈ S satisfies x ≤ 0

• x ∈ S and x ≤ 0 → x is not rational

Answer 1

Every rational number x ∈ S satisfies x ≤ 0

Explanation:

Given statement is

∃ x ∈ S, such that x > 0

∴ ∼(∃ x ∈ S, such that x > 0)

≡ ∀ x ∈ S, x ≤ 0

i.e., Every rational number x ∈ S satisfies x ≤ 0.