Let S be a non-empty subset of R. Consider the following statement:

P : There is a rational number x\in S such that x> 0.

Which of the following statements is the negation of the statement P ?

  • Option 1)

    There is a rational number x\in S such that x\leq 0.   

  • Option 2)

    There is no rational number x\in S such that x\leq 0.

  • Option 3)

    Every rational number x\in S satisfies  x\leq 0.

  • Option 4)

    x\in S and x\leq 0 \Rightarrow x is not rational.

 

Answers (1)

As we learnt in 

Negation of a Statement -

Symbol of negation is \sim

- wherein

We write it up as \simp

 

 P: There is rational numbers  x\epsilon S  such that x> 0.

negation is every rational number x\epsilon S  satisfies x\leq 0.

Since negation of x> 0 is x\leq 0 .


Option 1)

There is a rational number x\in S such that x\leq 0.   

Incorrect option

Option 2)

There is no rational number x\in S such that x\leq 0.

Incorrect option

Option 3)

Every rational number x\in S satisfies  x\leq 0.

Correct option

Option 4)

x\in S and x\leq 0 \Rightarrow x is not rational.

Incorrect option

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